ABSTRACT: A trading strategy represents a forecasting model and a forecasting model will generate profits in the future if it has demonstrated in the past that it can predict with a good probability of success the evolutions of the system on which it operates. Since the theory is so simple, why is it so difficult to evaluate a trading strategy? The reason is the following: in finance, we think we see the evolutions of the system but we are only observing the random component due to unpredictable events. Instead, the deterministic component that represents the true evolution of the system is not visible. Unfortunately, not seeing the deterministic component leads to a very important statistical problem: if I execute 10 winning predictions, I do not know if these operations concern 10 evolutions of the system or concern only one evolution of the system. Therefore, I do not know, if my forecasting algorithm acted intelligently by predicting 10 different evolutions or if it acted irrationally by dividing a bet into 10 smaller ones on the same evolution. Distinguishing between these two scenarios is essential to correctly evaluate the trading strategy and this discrimination can be done by understanding whether the trades are independent of each other. In fact, the condition of independence is the only statistical way to be sure that the trades concern forecasts of different evolutions. The Binomial evolution function solves this problem by transforming the succession of trades into a succession of independent results.

Introduction

The key to understanding whether an investment strategy can generate profits in the future is to shift the focus from the simple result (profit/loss) to the ability of the algorithm to predict the evolution of the market, under the assumption that markets are non-stationary [1][2]. Being “non-stationary” implies that operations are dependent on each other (market dynamics change over time), and consequently the data collected could lose statistical significance if treated as independent. When dependencies unknown to us are created, traditional statistical analysis risks leading to incorrect conclusions.

Consequently, the main goal of any function that wants to make a qualitative analysis about a trading algorithm is to transform the results into a series of independent data.

The Binomial evolution function solves this problem by transforming the succession of trades into a succession of independent outcomes. The data are then analyzed using the binomial distribution that allows us to calculate the probability of obtaining an equal or greater number of successes through a purely random process. The probability value obtained in this way turns out to be a statistically correct evaluation of the investment strategy.

Transformation of the Trade Series

The first step in obtaining a sequence of independent trades consists in applying the following transformation: if we have N forecasts on a system with K degrees of freedom, our predictions must be transformed into a series in which each forecast “bets” on a different evolution from the one immediately preceding it.

In the case of financial markets, there are two degrees of freedom: up (buy) or down (sell). 1 Consequently, in our new series, there should never be two consecutive operations of the same type (two buys or two sells). Therefore, the sequence must alternate buy and sell. From a practical point of view, this transformation is applied by inserting an opposite operation between two consecutive trades in the same direction. If, for example, we have two consecutive buys with a time interval ΔT between the closing of the first and the opening of the second, a sell trade of duration equal to ΔT is introduced in between.

Why this transformation? 

1. If after a bullish phase (buy) we think that the trend has ended, inserting a test opposite operation (sell), for the period ΔT, allows us to test our hypothesis of “end of trend”. 
2. If the market actually remains bullish, this intermediate operation will have a probability of loss greater than 50%. If, on the other hand, our assumption is correct and the bullish trend is over, the intermediate sell trade will have a probability of success that approaches 50%.

In conclusion, each intermediate operation inserted tests the hypothesis that the two consecutive operations (in the same direction) are truly independent.

Convert to a binary sequence

After creating the new sequence of operations (all alternating), we proceed to convert it into a binary sequence:

• We assign the value 1 to the trades closed in profit.
• We assign the value 0 to the trades closed in loss.

This additional step is necessary to not overestimate the impact of some individual results that may depend on random market events. By making each operation equal, we reduce the effect of large gains or isolated losses.

Selection of Independent Trades

In this phase, we identify the trades that we are sure are independent. In practice, this is done by selecting only the trades that were executed on different market evolutions. In this way, each trade represents an independent forecast with respect to the previous one.

To make this selection, we use the binary sequence of 0 and 1 obtained in the previous steps, where each symbol represents the result of a trade (for example, 1 for profit and 0 for loss). If a 0 is preceded by another 0, or a 1 is preceded by another 1, then we can be sure that the second trade refers to a different evolution of the market. This is because the system, by construction, generates alternating forecasts (buy/sell). Therefore, obtaining two consecutive equal results implies that the market must have changed.

Consequently, the principle on which this method is based is the following: two equal and opposite predictions cannot give the same outcome if the system has remained unchanged.

Conversely, if a 0 is preceded by a 1 or a 1 is preceded by a 0, the two results may be compatible with a stationary market situation. In this case, we cannot be sure that the two trades refer to distinct 2 evolutions, so we consider them potentially dependent and exclude them from further analysis.

At the end of this process, only independent trades remain, those that we are sure will be executed on different evolutions of the system.

Therefore, when we want to study a deterministic process, every time it determines a change in the system, we have the possibility of making a prediction on the evolution of the system whose probability of success randomly turns out to be less than 1. This leads to a decrease in the composite probability whose meaning, according to the axiom of disorder of Von Mises, is to indicate the presence of a deterministic process that acts on the system.

In practice, every time a deterministic process causes a change in the state of the system, it can be detected by making a prediction [3]. Consequently, a deterministic process to be detected with a low error must have produced a statistically significant number of variations in the system. Furthermore, our forecasting method must have made a sufficiently large number of independent predictions, each corresponding to a single evolution of the system.

Randomness test using the Binomial distribution

Finally, we calculate the probability of obtaining an equal or greater number of successes through a purely random process, as if we were tossing a fair coin (probability 50%). For this purpose, we use the binomial distribution, where:

• k is the number of successes (trades with value 1).
• n is the total number of trades.
• p is equal to 50% (assuming fair coin).

If the probability of observing similar or better results randomly is very low, it means that the algorithm has a good chance of being deterministic (able to predict the market). On the contrary, if the results can be obtained easily through a random process, we have no sufficient statistical reason to consider it capable of generating profits in the future.

Summary of the 4 fundamental steps of the Binomial evolution function

Step 1: Transformation of the Trade Series

Purpose: To test whether two consecutive forecasts of the same type (two buys or two sells) are truly independent.

Method: 

1. The trade sequence is transformed by always alternating buy and sell. 
2. If there are two consecutive buys or sells, an opposite “test” operation is inserted between the two, of the same duration as the waiting time between them (ΔT).

Example:

Original sequence:

Buy — ΔT1 — Buy — ΔT2 — Buy — ΔT3 — Buy

Transformed sequence:

Buy — Sell(ΔT1) — Buy — Sell(ΔT2) — Buy — Sell(ΔT3) — Buy

Step 2: Convert to binary sequence

Purpose: To prevent a single large gain (or loss) from influencing the statistical analysis.

Method:  Each trade is converted to: o

1. 1 if it closed in profit
2. 0 if it closed in loss

Example (results of each trade in the transformed sequence):

Buy (profit) → 1

Sell (loss) → 0

Buy (profit) → 1

Sell (loss) → 0

Buy (profit) → 1

Sell (profit) → 1

Buy (loss) → 0

Binary result:

[1, 0, 1, 0, 1, 1, 0]

Step 3: Selection of Independent Trades

Purpose: Select trades that have been executed on different market evolutions.

Method: 

1. In a binary sequence, only consecutive trade pairs having the same results are accepted (1 after 1 or 0 after 0). 
2. This is because, to have the same result with two opposite operations (buy/sell), the system must be changed ⇒ independence. 
3. Consecutive trade pairs where the results alternate (1 after 0 or 0 after 1) are discarded, because they are compatible with a stationary phase ⇒ possible dependence.

Example (using the sequence [1, 0, 1, 0, 1, 1, 0]):

Checks:

1 → 0 ⇒ rejected [1, 0, 1, 0, 1, 1, 0] 

0 → 1 ⇒ rejected [1, 0, 1, 0, 1, 1, 0] 

1 → 0 ⇒ rejected [1, 0, 1, 0, 1, 1, 0] 

0 → 1 ⇒ rejected [1, 0, 1, 0, 1, 1, 0] 

1 → 1 ⇒ included [1, 0, 1, 0, 1, 1, 0] 

1 → 0 ⇒ rejected [1, 0, 1, 0, 1, 1, 0]

Independent trades: only the pair 1 → 1 remain which corresponds to two consecutive profitable trades.

Step 4: Randomness test using the Binomial distribution

Purpose:

To assess whether the observed results can be attributed to a random process, for example a fair coin toss.

Method: 

1. k = number of successes (2)
2. n = number of independent trades selected (2)
3. p = 0.5 (fair coin) 
4. We calculate P(X ≥ k) with the binomial distribution.

Example: 

1. We obtained 2 successes on 2 independent trades. 
2. Using the binomial distribution: P(X ≥ 2) = 25%. 
3. The value is high ⇒ the result is compatible with a random process (there is no evidence of deterministic prediction).

If the observed number of successes is highly improbable under the assumption of randomness (probability <1%) this provides statistical evidence that the prediction algorithm may exhibit deterministic behavior. Conversely, high probability values indicate that the observed results may plausibly be the outcome of a random process.

How to distinguish a real “fortune teller” from simple luck using the Bionomial evolution function in the paradox of professional Traders

In the article, of the professional trader paradox [4], I develop a simple experiment based on the toss of a coin to explain the importance of considering financial markets as non-stationary. In this paragraph, I explain how the Binomial evolution function can correctly analyze the results of this experiment. In this way, we could understand if the strategy used has useful information that allows it to increase the probability of prediction.

The Experiment

Two people, A and B, sit opposite each other, separated by a black cloth.

1. A can toss a coin as many times as he wants during a fixed interval of time.
2. B does not see the coin but, at any time, can “bet” by declaring whether the coin at that moment shows Heads or Tails.
3. If the declaration coincides with the real state of the coin, B obtains a “success”.

The question: does B really know how to “read” the coin or does he win only by chance?

Why the Binomial evolution function is needed

In this experiment, A flips freely without being seen and B tries to guess Heads/Tails at any time. From the outside, we do not see the flips so If B hits two or more identical predictions in a row, we can interpret it in two opposite ways:

1. B is really skilled and anticipates every single flip.
2. B is just “breaking up” the same bet: the coin is not flipped again and he repeats the same answer on a single event already decided.

Confusing the two cases means mistaking a mere accounting trick for a “winning strategy.” Understanding when we are truly predicting new coin tosses and when we are simply spreading our bets over the same toss is crucial for evaluating any betting method.

Where the Binomial evolution function intervenes

The Binomial evolution function (BEF) was invented precisely to separate these two situations, here’s how:

1. Transformation: It imposes the alternation Heads-Tails between two successive predictions, inserting a “dummy toss” if necessary.
2. Binary conversion: each prediction becomes 1 (correct) or 0 (wrong); the individual amounts no longer matter, so each win and each loss have the same statistical value.
3. Selection of independent cases: compares the result of each prediction with the one immediately before: it only keeps 0–0 or 1–1. Indeed, two equal and opposite predictions give the same result, only if there has been a new toss.
4. Binomial test: calculates the probability that the independent successes of B are the result of a random process.

What we discover 

1. If the p-value remains large, the consecutive wins are nothing special: they are consistent with a simple “splitting” of a single bet over one toss.
2. If the p-value is low, then it means that the consecutive victories refer to effectively different throws and B has real information on the individual events.

When betting on non-stationary systems, one must analyze very carefully series of victories that derive from forecasts all in the same direction. In finance, the formation of groups of high returns, alternating with groups of low returns, creates a statistical phenomenon called clustering [5]. In other words, it means that the returns are not distributed homogeneously but tend to group together. The clustering phenomenon can have significative deleterious effects in finance, as it leads to the misperception that trades executed during winning streaks are statistically independent, when in fact they may be correlated. Consequently, it becomes essential to understand if the predictions were on truly distinct events by applying the Binomial evolution function to filter the self-delusion of “splitting the stakes” on the same roll.

Only in this way, do we know if we are acting as intelligent strategists or if we are simply multiplying bets on the same roll, convincing ourselves that we are winning when, in reality, we are acting irrationally.

Conclusion

The Binomial Evolution Function represents a new class of functions that address the problem of non independence of results in finance. Resolving this problem has become a central challenge in the modern approach to quantitative trading. As a result, there is growing interest in methods that can transform sequences of outcomes into statistically independent data. The method used by the Binomial evolution function to select independent data is conceptually simple: it is based on the idea that two identical but opposite predictions yield the same result only if the underlying system has evolved. This insight provides a practical criterion for identifying independent trades.

In conclusion, this function allows us to answer the following fundamental question:

Am I betting on different events or am I simply splitting a bet on the same event?

If we cannot answer this question, any analysis of the results will be useless.

References:

[1] Fama, E. F. (1965). “The Behavior of Stock-Market Prices.” The Journal of Business, 38(1), 34–105.

[2] O. Peters, M. Gell-Mann. “Evaluating gambles using dynamics”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016; 26 (2): 023103 DOI: 10.1063/1.4940236.

[3] Berdondini, Andrea, Application of the Von Mises’ Axiom of Randomness on the Forecasts Concerning the Dynamics of a Non-Stationary System Described by a Numerical Sequence (January 21, 2019). Available at SSRN: https://ssrn.com/abstract=3319864 or http://dx.doi.org/10.2139/ssrn.3319864.

[4] Andrea Berdondini, “The Professional Trader’s Paradox”, (November 20, 2018). Available at SSRN: https://ssrn.com/abstract=3287968.

[5] Marti, G., Nielsen, F., Bińkowski, M., & Donnat, P. (2017) “A review of two decades of 7 correlations, hierarchies, networks and clustering in financial markets”, arXiv preprint arXiv:1703.00485.

Binomial Evolution Function (The Only Function You Need to Evaluate a Trading Strategy) was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and responding to this story.

ABSTRACT: The fundamental problem of causal inference defines the impossibility of associating a causal link to a correlation, in other words: correlation does not prove causality. This problem can be understood from two points of view: experimental and statistical. The experimental approach tells us that this problem arises from the impossibility of simultaneously observing an event both in the presence and absence of a hypothesis. The statistical approach, on the other hand, suggests that this problem stems from the error of treating tested hypotheses as independent of each other. Modern statistics tends to place greater emphasis on the statistical approach because, compared to the experimental point of view, it also shows us a way to solve the problem. Indeed, when testing many hypotheses, a composite hypothesis is constructed that tends to cover the entire solution space. Consequently, the composite hypothesis can be fitted to any data set by generating a random correlation. Furthermore, the probability that the correlation is random is equal to the probability of obtaining the same result by generating an equivalent number of random hypotheses.

Introduction

The fundamental problem of causal inference defines the impossibility of associating causality with a correlation; in other words, correlation does not prove causality. This problem can be understood from two perspectives: experimental and statistical. The experimental approach suggests that this problem arises from the impossibility of observing an event both in the presence and absence of a hypothesis simultaneously. The statistical approach, on the other hand, suggests that this problem stems from the error of treating tested hypotheses as independent of each other.

Modern statistics tends to place greater emphasis on the statistical approach, as it, unlike the experimental approach, also provides a path to solving the problem. Indeed, when testing many hypotheses, a composite hypothesis is constructed that tends to cover the entire solution space. Consequently, the composite hypothesis can fit any data series, thereby generating a correlation that does not imply causality.

Furthermore, the probability that the correlation is random is equal to the probability of obtaining the same result by generating an equivalent number of random hypotheses. Regarding this topic, we will see that the key point, in calculating this probability value, is to consider hypotheses as dependent on all other previously tested hypotheses.

Considering the hypothesis as non-independent has fundamental implications in statistical analysis.

Indeed, every random action we take is not only useless but will increase the probability of a random correlation.

For this reason, in the following article [1], we highlight the importance of acting consciously in statistics.

Moreover, calculating the probability that the correlation is random is only possible if all prior attempts are known. In practice, calculating this probability is very difficult because not only do we need to consider all our attempts, but we also need to consider the attempts made by everyone else performing the same task. Indeed, a group of people belonging to a research network all having the same reputation and all working on the same problem can be considered with a single person who performs all the attempts made.

From a practical point of view, we are almost always in the situation where this parameter is underestimated because it is very difficult to know all the hypotheses tested.

Consequently, the calculation of the probability that a correlation is casual becomes something relative that depends on the information we have.

The Fundamental Problem of Causal Inference

The fundamental problem of causal inference [2] defines the impossibility of associating causality with a correlation, in other words: correlation does not prove causality. From a statistical point of view, this indeterminacy arises from the error of considering the tested hypotheses as independent of each other. When a series of hypotheses is generated, a composite hypothesis is formed that tends to fit any data series, leading to purely random correlations.

For example, you can find amusing correlations between very different events on the internet; these correlations are obviously random. These examples are often used to demonstrate the fundamental problem of causal inference. In presenting this data, the following information is always omitted: how many hypotheses did I consider before finding a related hypothesis.

This is essential information because if I have a database comprising a very high number of events, for any data series, there will always be a hypothesis that correlates well with my data. Thus, if I generate a large number of random hypotheses, I will almost certainly find a hypothesis that correlates with the data I am studying. Therefore, having a probability of about 100% of being able to obtain the same result randomly, I have a probability of about 100% that the correlation does not also imply causation.

On the other hand, if we generate a single hypothesis that correlates well with the data, in this situation, almost certainly, the correlation also implies causation. This is because the probability of obtaining a good correlation by generating a single random hypothesis is almost zero.

This result is also intuitive, because it is possible to achieve a good correlation with a single attempt only if one has knowledge of the process that generated the data to be analyzed. And it is precisely this knowledge that also determines a causal constraint.

The following figure summarizes the basic concepts showing how the correct way to proceed is to consider the hypotheses as non-independent.

Calculating the probability that the correlation is random

Correctly calculating the probability of getting an equal or better result randomly involves changing our approach to statistics. The approach commonly used in statistics is to consider the data produced by one method independent of the data produced by different methods. This way of proceeding seems the only possible one but, as we will show in the following paradox, it leads to an illogical result, which is instead solved by considering the data as non-independent.

We think to have a computer with enormous computational capacity that is used to develop hypotheses about a phenomenon that we want to study. The computer works as follows: it creates a random hypothesis and then performs a statistical test. At this point, we ask ourselves the following question: can there be a useful statistical test to evaluate the results of the hypothesis generated?

If we answer yes, we get an illogical result because our computer would always be able, by generating a large number of random hypotheses, to find a hypothesis that passes the statistical test. In this way, we arrive at the absurd conclusion that it is possible to create knowledge randomly, because it is enough to have a very powerful computer and a statistical test to understand every phenomenon.

If we answer no, we get another illogical result because we are saying that no hypothesis can be evaluated. In practice, the results of different hypotheses are all equivalent and indistinguishable.

How can we solve this logical paradox? The only way to answer the question, without obtaining an illogical situation, is to consider the results obtained from different methods depending on each other. A function that meets this condition is the probability of getting an equal or better result at random. Indeed, the calculation of this probability implies the random simulation of all the actions performed. Hence, random attempts increase the number of actions performed and consequently increase the probability of obtaining an equal or better result randomly.

For this reason, generating random hypotheses is useless, and therefore if you use this parameter, it is possible to evaluate the data and at the same time it is impossible to create knowledge by generating random hypotheses.

Considering the hypothesis as non-independent is a fundamental condition for correctly calculating of the probability that the correlation is random.

The probability of getting an equal or better result at random meets this condition.

The dependence of hypothesis on each other has profound implications in statistics, which will be discussed in the next section.

Consequences of the non-independence of the hypothesis

Consider the tested hypotheses to be dependent on each other when calculating the probability that the correlation is causal leads to three fundamental consequences in statistics.

First fundamental consequence of the non-independence of the hypothesis: our every random action always involves an increase in the probability of a random correlation.

Example: We need to analyze a statistical datum represented by 10 predictions about an event that can only have two results. The 10 predictions are divided into 8 successes and 2 failures. To calculate the probability of obtaining an equal or better result randomly we use the binomial distribution and we get the following value 5.5%. If before making these 10 predictions, we tested a different hypothesis with which we made 10 other predictions divided into 5 successes and 5 failures, the uncertainty of our result changes. Indeed, in this case, we must calculate the probability of obtaining a result with a number of successes greater than or equal to 8 by performing two random attempts consisting of 10 predictions each. In this case, the probability becomes 10.6%, so the fact of having first tested a random hypothesis almost doubled the probability of a random correlation of our second hypothesis. Consequently, increasing the random hypotheses increases the number of predictions that we will have to make, with the true hypothesis, to have a low probability that the correlation is coincidental.

Second fundamental consequence of the non-independence of the hypothesis: every random action of ours and of every other person equivalent to us, always involves an increase of the probability that the correlation is random.

By the equivalent term, we mean a person with the same reputation as us, therefore the data produced by equivalent people are judged with the same weight.

Example: 10 people participate in a project whose goal is the development of an algorithm capable of predicting the outcome of an event that can have only two results. An external person who does not participate in the project but is aware of every attempt made by the participants evaluates the statistical data obtained. All participants make 100 predictions, 9 get a 50% chance of success, one gets a 65% chance of success. The probability that a 65% success is due to a random correlation is obtained by calculating the probability of obtaining a result with a number of successes greater than or equal to 65 by performing ten random attempts consisting of 100 predictions each. The probability obtained, in this way, is 16% instead if he was the only participant in the project the probability would have been 0.18%, therefore about 100 times lower.

Third fundamental consequence of the non-independence of the hypothesis: the calculation the probability that the correlation is random varies according to the information possessed.

Example: 10 people participate in a project whose goal is the development of an algorithm capable of predicting the outcome of an event that can have only two results. In this case, people do not know the other participants and think they are the only ones participating in the project. All participants make 100 predictions, 9 get a 50% chance of success and one gets a 65% chance of success. The participant who obtains a probability of success of 65% independently calculate the probability that the correlation is coincidental. Not knowing that other people are participating in the project, calculate the probability of obtaining a result with a number of successes greater than or equal to 65 by performing a single random attempt consisting of 100 predictions; the probability obtained is 0.18%. An external person who is aware of every attempt made by the participants calculate the probability that the 65% success rate of one of the participants was due to a random correlation. knowing the number of participants in the project calculates the probability of obtaining a result with a number of successes greater than or equal to 65 by making ten random attempts consisting of 100 predictions each. The probability obtained, in this way, is 16%, a much higher value than the probability calculated by the participant. The probability calculated by the external person using more information is most accurate than the probability calculated by the individual participant. Consequently, the probability obtained by exploiting the greatest number of information must always be considered, in the case of the example, the probability that the 65% success is due to a random correlation is 16%. Therefore, the participant having less information underestimates this probability.

The first and second fundamental highlighting consequence of the non-independence of the hypothesis can be redefined by highlighting the non-randomness of the action.

First fundamental consequence of the non-independence of the hypothesis: our every non-random action always involves a reduction in the probability that the correlation is random.

Second fundamental consequence of the non-independence of the hypothesis: every non-random action of ours and of every other person equivalent to us, always involves a reduction in the probability that the correlation is random.

How to perform correctly the statistical hypothesis test

About to perform correctly the statistical hypothesis test, It is interesting to note how the nonindependence of the hypothesis can be seen as something extremely obvious or as something extremely innovative. Indeed, it may seem absolutely banal to consider all the hypotheses that have been tested, for the obvious reason that by running a large number of random hypotheses sooner or later there will be some hypothesis that will fit the data quite well. On the other hand, also considering the previous hypotheses represents a revolution in the evaluation of a hypothesis. In fact, from this point of view, the mere knowledge of the hypothesis that makes the prediction does not allow us to define its real complexity. Therefore, if in the statistical hypothesis test the p-value [3], [4], used as a threshold to reject the null hypothesis, is calculated considering only the hypothesis that actively participates in the prediction, it means, that we are underestimating the complexity of the hypothesis. Consequently, the p-value, thus calculated, is wrong and therefore determines a false evaluation of the hypothesis. It is therefore believed that this systematic error, in the execution of the hypothesis test, is responsible for the high number of non-reproducible results [5], [6].

Taking advantage of these considerations it is understood that evaluating a statistical result can be very difficult because some information can be hidden. For example, we are obliged to report the mathematical formula that makes the prediction but, instead, we may not report all previous failed attempts. Unfortunately, this information is essential for evaluating the hypothesis, because they are an integral part of the hypothesis. Indeed, if we test 10 hypotheses, we simply interpolate the data with those ten hypotheses and choose the hypothesis that passes the chosen evaluation test.

This problem also depends on the increasing use of statistical software capable of quickly executing a huge number of mathematical models.

Consequently, there is the risk of “playing” with this software by performing a multitude of analyzes and this sooner or later leads to a random correlation. For these reasons, the evaluation of statistical results represents one of the most important challenges for scientific research.

Unfortunately, it is a difficult problem to solve because, as mentioned, some information can always be hidden when writing an article. The simplest solution adopted is to use more selective evaluation parameters, which in practice means making it unlikely to pass the evaluation test by developing random hypotheses. However, this solution has a big problem: by acting in this way there is the risk of discarding a correct hypotheses and cannot be applied to all fields of research. For example, in finance where the possible inefficiencies of the markets [7], which can be observed, are minimal, adopting very restrictive valuation methods means having to discard almost any hypothesis.

Conclusion

In this article, we analyzed the fundamental problem of causal inference from a statistical perspective. From this point of view, the problem arises from treating all tested hypotheses as independent of each other. This way of acting is wrong because when we generate a series of hypotheses we are building a composite hypothesis that will tend to adapt and therefore give a random correlation to each of our series of data.

It is believed that this incorrect approach is the cause of the problem of non reproducibility of scientific results.

Moreover, the increase in computational capacity speeds up hypothesis development, inadvertently creating composite hypotheses that can lead to random correlations.

The probability that a correlation is random is obtained by calculating the probability of obtaining an equal or better result randomly. This calculation can be done, correctly, only by knowing all the hypotheses tested, unfortunately this information is very difficult to have.

For this reason, in modern statistics, it is considered fundamental to develop the awareness that each of our compulsive and irrational actions, which leads us to develop and test a large quantity of hypotheses, has as a consequence the generation of random correlations that are difficult to detect.

Bibliography:

[1] Berdondini, Andrea, “Statistics the Science of Awareness” (August 30, 2021). Available at SSRN: https://ssrn.com/abstract=3914134.

[2] Holland, P. W. (1986) — Statistics and Causal Inference. Journal of the American Statistical Association, 81(396), 945–960.

[3] Hung, H.M.J., O’Neill, R.T., Bauer, P., & Kohne, K. (1997). “The behavior of the p-value when the alternative hypothesis is true.” Biometrics, 53(1), 11–22.

[4] Harlow, L.L., Mulaik, S.A., & Steiger, J.H. (1997). “What if there were no significance tests?” Psychological Methods, 2(4), 315–328.

[5] Munafò, M., Nosek, B., Bishop, D. et al. “A manifesto for reproducible science”. Nat Hum Behav 1, 0021 (2017). https://doi.org/10.1038/s41562-016-0021.

[6] Ioannidis, J. P. A. “Why most published research findings are false”. PLoS Med. 2, e124 (2005).

[7] Black, F. (1971) “Random Walk and Portfolio Management,” Financial Analyst Journal, 27, 16–22

A Modern Approach To The Fundamental Problem of Causal Inference was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and responding to this story.

AI Solves The P Versus NP Problem

I want to start this article by asking the following question: what is the most important open problem? The problem that would most likely win would be P Vs NP problem[1][2]. This problem is considered the most important unsolved problem in the field of theoretical computer science.

PVsNP problem is a question regarding the resolution times of a problem. The term NP (nondeterministic polynomial time) indicates problems in which the correctness of the solution found is easy to verify, therefore requiring polynomial time. Instead, the term P refers to problems that have polynomial resolution times (easy to solve) and represent a subclass of the set NP.

Within the class of NP problems there is a very important subclass NP-C (NP-complete) made up of problems that have non-polynomial resolution times (difficult to solve). The PVsNP problem concerns precisely this class of problems and is stated as follows:

if the solution of a problem can be verified quickly (polynomial calculation time) then it is also possible to solve the problem quickly (P=NP).

In simpler terms, if I can verify the solution to a problem quickly, then I can also solve it quickly.

We are living in a time where artificial intelligence is applied in many different situations [3][4]. To understand the new method that exploits artificial intelligence, we need to reformulate the problem from the point of view of information theory.

Let’s start by asking the following question: what does non-polynomial computation time mean?

It means the computation time is exponential, but what does an exponential computation time mean?

It means that we must consider every solution and therefore, it is not possible a priori to discriminate some solutions. This means that we do not have any useful information that can be used to discriminate some solutions and, therefore, reduce the set of solutions.

Using this simple concept, we can reformulate the definition ofNP-C problem as follows: a problem is NP-C if there is not enough information to significantly reduce the set of solutions.

So, in other words, an NP-C problem is a problem that we don’t understand and therefore we are forced to iteratively work out all the solutions.

With these simple concepts let’s rephrase the PVsNP problem:

If there is a set of information that allows us to verify the correctness of a solution to a problem quickly then there is a set of information that allows us to solve it quickly.

I love this definition because, in just a few lines, it makes us fully understand the problem.

In fact, from this definition, we immediately understand that the main element in order to solve this problem is the search for information that can allow us to understand and therefore solve the problem quickly.

Now, let’s ask ourselves the following question: what type of characteristics must the information we are looking for have?

This question is important because not all information is “useful” for our purpose. Indeed, the calculation of information such as a statistical indicator (mean, variance, etc.) requires a number of calculations.

Therefore, for our purpose, we are only interested in “efficient” information that reduces the problem resolution times by a greater amount than the calculation times used to obtain the information itself.

Searching for this particular type of “efficient” information is almost impossible to do without the use of an automated method that can also “evaluate” the information.

And, it is precisely in this fundamental step (search for efficient information) that artificial intelligence comes into play through the development of artificial neural networks. The fundamental aspect of artificial intelligence is to be able to evaluate the information given; this aspect is fundamental. Indeed, the characteristic of neural networks that exploit reinforcement learning is that they reward the parameters (information) that are most efficient (fewer number of steps) in solving the problem.

Before, the arrival of these methods, the computer always and only had a supporting role, it never entered actively into the development of logical processes, which were instead always developed by man.

If this type of information found by artificial intelligence is implemented in an algorithm, we can say that the logical process, which led to the development of the algorithm, is of a hybrid type; partly generated by AI and partly generated by man.

With this method, it is possible to create algorithms that represent real collections of this efficient information. Therefore, they are algorithms with a level of complexity and abstraction much greater than those normally developed by humans.

This method has proven capable of finding this “efficient” information; however, this is not enough. Indeed, to go from an exponential to a polynomial calculation time, the efficient information must reduce the set of solutions to a fraction of the initial set.

In other words, to solve a problem quickly, there must be a high understanding of the problem; having only some efficient information can help us optimize the problem, which, however, can always remain with an exponential calculation time.

The step that still needs to be done is to understand, if we can generate such a large amount of efficient information capable of leading us to a full understanding of the problem.

Until now, we have always talked about solving a problem and we have never talked about mathematical proof, so it may come naturally to ask ourselves: how can this method give a definitive answer regarding the P Vs Np problem?

On this aspect, there is good news because the NP-C problems have the following characteristics: we just need to solve one of them, and we are sure that there is a fast way to solve all the other Np problems [5][6]. In fact, it has been shown that any NP problem can be reformulated as any NP-C problem.

Therefore, we don’t need a mathematical proof, we just to concentrate our forces on a single NP-C problem to be able to definitively solve P Vs NP problem.

The most famous example of an NP-C problem is the partial sum problem in which, given a set of numbers, we must search for subsets of numbers whose sum is equal to a constant value.

Now, let’s return to the P Vs NP problem and see some consequences of its possible resolution.

Case 1) P=NP it is possible to quickly solve an NP-C problem. This result implies that the condition of fast verification of the result is a sufficient condition to generate a non-random structure between the parameters that constitute the problem. In simpler words, parameters can be seen as variables dependent on each other. Therefore, it is possible to exploit this dependence, through the use of the efficient information described previously, to reduce the solution space significantly, making the problem solvable quickly.

Case 2) P≠NP it is not possible to quickly solve an NP-C problem. This result implies that the condition of fast verification of the result is a condition that is not sufficient to generate a non-random structure between the parameters that constitute the problem. Thus, the parameters can be viewed as independent random variables. Consequently, we cannot do anything other than take into account all possible combinations of these parameters (exponential calculation time).

In conclusion, solving this problem means knowing whether intrinsically random problems exist. Looking at this problem from this point of view, we notice some similarities with Godel’s incompleteness theorem [7], in which it is shown that there exist unprovable theorems. Therefore, just as there are unprovable theorems, there may be intrinsically random problems, in which it is not possible to define a logical scheme useful in helping us solve the problem quickly.

So, solving the PVsNP problem allows us to understand the nature of problems, and since problems are the elements that underlie human evolution, we can immediately understand why this problem is defined as the most important unsolved problem.

Like any unsolved problem, there are people who think that the case P=NP is true and others who think the case P≠NP is true. Since an algorithm capable of quickly solving an NP-C problem has never been found, supporters of the P≠NP case have always been in the majority. However, in recent times, supporters of the P=NP case have increased. Because the awareness is growing that NP-C problems are difficult to solve not because of their intrinsic randomness but because they require a level of complexity and abstraction that is difficult to understand for the human mind. Difficulty that can be overcome with the development of hybrid logical models, partly developed by man and partly developed by AI. In this dispute, I also, as a good gambler, want to make a bet on this problem and I bet on the P=NP case.

Bibliography:

1. Cook, Stephen (April 2000). “The P versus NP Problem” (PDF). Clay Mathematics Institute. Archived (PDF) from the original on 16 December 2013. Retrieved 18 October 2006.
2. Fortnow, Lance (2009). “The status of the P versus NP problem” (PDF). Communications of the ACM. 52 (9): 78–86. CiteSeerX 10.1.1.156.767. doi:10.1145/1562164.1562186. S2CID 5969255. Archived from the original (PDF) on 24 February 2011. Retrieved 26 January 2010.
3. Wang, H., Fu, T., Du, Y., Gao, W., Huang, K., Liu, Z., Chandak, P., Liu, S., Van Katwyk, P., Deac, A., Anandkumar, A., Bergen, K., Gomes, C. P., Ho, S., Kohli, P., Lasenby, J., Leskovec, J., Liu, T.-Y., Manrai, A., Marks, D., Ramsundar, B., Song, L., Sun, J., Tang, J., Veliˇckovi´c, P., Welling, M., Zhang, L., Coley, C. W., Bengio, Y., and Zitnik, M. (2023). Scientific discovery in the age of artificial intelligence. Nature, 620(7972):47–60
4. Yao, S., Yu, D., Zhao, J., Shafran, I., Griffiths, T. L., Cao, Y., and Narasimhan, K. (2023). Tree of thoughts: Deliberate problem solving with large language models. arXiv preprint arXiv:2305.10601.
5. Fortune, S. (1979). “A Note on Sparse Complete Sets” (PDF). SIAM Journal on Computing. 8 (3): 431–433. doi:10.1137/0208034. hdl:1813/7473.
6. Arora, Sanjeev; Barak, Boaz (2009). Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978–0–521–42426–4.
7. Adrian William Moore, 2022. Gödel´s Theorem: A Very Short Introduction. Oxford University Press, Oxford, 2022.

AI solves P Versus NP problem was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and responding to this story.

This book consists of a selection of articles divided into three main themes: Statistics, Quantitative Trading, Psychology. These three arguments are indispensable for the development of a quantitative trading system. The order of the articles was chosen so as to constitute a single logical reasoning that develops progressively.

The Theory of Quantitative Trading

“The difference between an amateur trader and a professional trader is that the first is obsessed by the result while the second is obsessed by the knowledge”

Introduction

The first step that you must take when you want to learn how to develop a quantitative trading system is to answer the following question: what characteristics must a system have to represent the theoretically most difficult situation possible where to make predictions?

The first characteristic that this system must possess is that of having a very low ratio between the deterministic component and the random component. Therefore, the random component must overcome the deterministic component.

The second characteristic that we need to determine concerns the number of degrees of freedom. This parameter determines how much difficult it is to test hypotheses. This consideration derives from the fundamental problem of statistics defined as follows: “A statistical data does not represent useful information, but becomes useful information only when it is shown that it was not obtained randomly”. Therefore, given a result, the probability of obtaining it randomly decreases as the degrees of freedom of the system increase. Consequently, systems with a low number of degrees of freedom are particularly dangerous, because it is particularly easy to get good results randomly and therefore you risk overestimating an investment strategy. For this reason, a system that wants to be as difficult as possible where making predictions must have a low number of degrees of freedom.

The third characteristic we have to choose is if to consider the system as ergodic (stationary) or non-ergodic (non-stationary). This choice is easy because it is much more difficult to make forecasts on a non-ergodic system. Indeed, in this case, past results may not be significant with respect to future results.

In conclusion, the system that represents, from the theoretical point of view, the most difficult situation in which to make predictions is a system in which there is a predominant random component, with a low number of degrees of freedom and not ergodic. Is there a system that has all these 3 characteristics? The answer is yes indeed, the financial markets represent a system that respects all these conditions.

I started this book with this consideration because I believe that the most important thing to understand, for any person who wants to develop a quantitative trading system, is to know that you are facing the most difficult situation theoretically possible where to make predictions.

It is this awareness that must guide us in the study of the financial markets. Without this awareness, we will inevitably be led to underestimate the difficulty of the problem and this will lead us to make mistakes.

The theory of quantitative trading was originally published in ILLUMINATION on Medium, where people are continuing the conversation by highlighting and responding to this story.

ABSTRACT: Any result can be generated randomly and any random result is useless. Traditional methods define uncertainty as a measure of the dispersion around the true value and are based on the hypothesis that any divergence from uniformity is the result of a deterministic event. The problem with this approach is that even non-uniform distributions can be generated randomly and the probability of this event rises as the number of hypotheses tested increases. Consequently, there is a risk of considering a random and therefore non-repeatable hypothesis as deterministic. Indeed, it is believed that this way of acting is the cause of the high number of non-reproducible results. Therefore, we believe that the probability of obtaining an equal or better result randomly is the true uncertainty of the statistical data. Because it represents the probability that the data is useful and therefore the validity of any other analysis depends on this parameter.

Introduction

Any result can be generated randomly and any random result is useless. Traditional methods [1] and [2] define uncertainty as a measure of the dispersion around the true value and are based on the hypothesis that any divergence from uniformity is the result of a deterministic event. The problem with this approach is that even non-uniform distributions can be generated randomly and the probability of this event rises as the number of hypotheses tested increases. Consequently, there is a risk of considering a random and therefore non-repeatable hypothesis as deterministic. Indeed, it is believed that this way of acting is the cause of the high number of non-reproducible results [3] and [4]. Therefore, we believe that the probability of obtaining an equal or better result randomly is the true uncertainty of the statistical data, because it represents the probability that the data is useful and therefore the validity of any other analysis depends on this parameter.

In addition, we will also address the problem of determining the correct method of calculating the probability of obtaining an equal or better result randomly. Regarding this topic, we will see that the fundamental point, in calculating this probability value, is to consider the statistical data dependent on all the other data generated by all the tested hypotheses.

In this way, we obtain a ‘paradoxical’ situation, because we can have a series of results where the outcome does not depend on the previous events, but the uncertainty associated with the single result turns out to be dependent on the previous events. So, how can we consider these results as dependent or independent? Two events are independent if there is no kind of dependence between them. In this case, since a dependency has been generated in the calculation of their uncertainty, the events can no longer be considered independent of each other.

Furthermore, as we will see later, the problem of causal inference defined as the inability to understand whether a correlation also implies causation depends precisely on considering individual hypotheses as independent

Considering the statistical data as non-independent has fundamental implications in statistical analysis. Indeed, all our random actions are not only useless but will increase the uncertainty of the statistical data. For this reason, in the following article [5], we highlight the importance of acting consciously in statistics.

Furthermore, the evaluation of the uncertainty of the statistical data will be possible only by knowing all the attempts made. In practice, the calculation of uncertainty is very difficult because not only we must consider all our attempts, but we must also consider the attempts made by every other person who is performing the same task as us. In this way, the uncertainty of our statistical data also depends on the actions performed by the people who are working on our own analysis. Indeed, a group of people who belong to a research network all having the same reputation who all work on the same problem can be considered with one person who carries out all the attempts made. Consequently, the calculation of uncertainty becomes something relative that depends on the information we have.

Finally, we will see how this new definition of uncertainty allows us to solve the fundamental problem of causal inference (correlation does not prove causality). Indeed, the indeterminacy between correlation and causality is not something absolute but derives from the error of considering the hypotheses as independent of each other. For example, if I test ten hypotheses and the tenth hypothesis tested has a good correlation, this does not mean that the tenth hypothesis is correlated but it means that among ten hypotheses there is one that is well correlated. If instead of testing 10 hypotheses I test 10 thousand whatever my data will be, there will always be a hypothesis, among these 10 thousand, which will be well correlated with my data. In this case, the probability of obtaining an equal or better result randomly, generating 10 thousand random hypotheses, will be very high. Therefore, I will be sure that, in this case, correlation does not imply causation. As a result, statistical data uncertainty defined as the probability of obtaining the same or better result randomly also represents the probability that correlation does not imply causation.

Definition of uncertainty

The aim of the definition of uncertainty of the statistical data that we are going to give is to determine a parameter that is linked to the repeatability of the result and that is universal and therefore, independent of the system in which we perform the statistical analysis.

We define the uncertainty of the statistical data as the probability of obtaining an equal or better result randomly.

This definition considers the statistical data as a forecast, so a forecast is repeatable only if the process that generated it is non-random. Consequently, the calculation of uncertainty involves determining the type of process that generated the result. We can distinguish cognitive processes from random processes by their statistical property of generating non-reproducible results in a random way. Indeed, by using the information on the system, on which we are performing a measurement, we can increase our probability of forecasting and this leads to a consequent decrease in the probability of obtaining the same result randomly.

It is interesting to note that the repeatability of the statistical data and non-randomness of the process that produced it are two equivalent concepts. Indeed, the information leads to the repeatability of the result and at the same time generates results that cannot be reproduced randomly.

To understand the definition given, we report the following example: We have to analyze a statistical datum represented by 1000 predictions on an event that can have only two results. The 1000 predictions are divided into 600 successes and 400 failures. To calculate the probability of obtaining an equal or better result in a random way, we use the binomial distribution and we obtain the following value 1.4∙10−8%.

Now, instead, let us consider a statistical datum represented by 10 predictions divided into 8 successes and 2 failures. In this case, the probability of getting an equal or better result randomly is 5.5%.

Comparing the two results, we note that in the first case, although the number of successes is only 60%, the uncertainty is almost zero, while in the second case, with a probability of success of 80%, the uncertainty is much higher. This difference is due to the fact that the definition given, as mentioned, concerns only the repeatability of the result and not its accuracy. Therefore, it is a value that decreases as the repetition of the result increases. The approach presented is very different from the classic approach, where uncertainty is seen as a measure of the dispersion of the data with respect to the true value. The definition we have given of uncertainty represents a qualitative and not a quantitative measure of the statistical data. In practice, it tells us if there is a deterministic component in the data collected.

The fundamental point to understand is that the probability that statistical data is completely random and the estimate of its random component (dispersion around the true value) are two parameters that are only partially dependent on each other. The first decreases as the number of repetitions of the measurement increases, the second does not and this is one of the reasons, why the traditional definition of uncertainty, in many cases, is not significant with regard to the repeatability of the result.

The problem, as we have seen in the examples, is that there is always a greater or lesser probability that a purely random process generates the result. In this case, any analysis turns out to be wrong, for this reason, this value is considered the true uncertainty of the statistical result.

Calculation of the uncertainty of the statistical data

Correctly calculating the probability of getting an equal or better result randomly involves changing our approach to statistics. The approach commonly used in statistics is to consider the data produced by one method independent of the data produced by different methods. This way of proceeding seems the only possible one but, as we will show in the following paradox, it leads to an illogical result, which is instead solved by considering the data as non-independent.

We think to have a computer with enormous computational capacity that is used to develop hypotheses about a phenomenon that we want to study. The computer works as follows: it creates a random hypothesis and then performs a statistical test. At this point, we ask ourselves the following question: can there be a useful statistical test to evaluate the results of the hypothesis generated?

If we answer yes, we get an illogical result because our computer would always be able, by generating a large number of random hypotheses, to find a hypothesis that passes the statistical test. In this way, we arrive at the absurd conclusion that it is possible to create knowledge randomly because it is enough to have a very powerful computer and a statistical test to understand every phenomenon.

If we answer no, we get another illogical result because we are saying that no hypothesis can be evaluated. In practice, the results of different hypotheses are all equivalent and indistinguishable.

How can we solve this logical paradox? The only way to answer the question, without obtaining an illogical situation, is to consider the results obtained from different methods depending on each other. A function that meets this condition is the probability of getting an equal or better result at random. Indeed, the calculation of this probability implies the random simulation of all the actions performed. Hence, random attempts increase the number of actions performed and consequently increase the probability of obtaining an equal or better result randomly. For this reason, generating random hypotheses is useless, and therefore if you use this parameter, as a measure of uncertainty, it is possible to evaluate the data and at the same time, it is impossible to create knowledge by generating random hypotheses.

Considering the statistical data as non-independent is a fundamental condition for correctly calculating the uncertainty. The probability of getting an equal or better result at random meets this condition.

The dependence of statistical data on each other has profound implications in statistics, which will be discussed in the next section.

Consequences of the non-independence of the statistical data

Considering the statistical data dependent on each other in the calculation of uncertainty leads to three fundamental consequences in statistics.

First fundamental consequence of the non-independence of the statistical data: our every random action always involves an increase in the uncertainty of the statistical data.

Example: We need to analyze a statistical datum represented by 10 predictions about an event that can only have two results. The 10 predictions are divided into 8 successes and 2 failures. To calculate the probability of obtaining an equal or better result randomly we use the binomial distribution and we get the following value 5.5%. If before making these 10 predictions, we tested a different hypothesis with which we made 10 other predictions divided into 5 successes and 5 failures, the uncertainty of our result changes. Indeed, in this case, we must calculate the probability of obtaining a result with a number of successes greater than or equal to 8 by performing two random attempts consisting of 10 predictions each. In this case, the probability becomes 10.6%, so the fact of having first tested a random hypothesis almost doubled the uncertainty of our second hypothesis. Consequently, increasing the random hypotheses increases the number of predictions that we will have to make, with the true hypothesis, to have an acceptable uncertainty.

Second fundamental consequence of the non-independence of the statistical data: every random action of ours and of every other person equivalent to us, always involves an increase in the uncertainty of the statistical data.

By the equivalent term, we mean a person with the same reputation as us, therefore the data produced by equivalent people are judged with the same weight.

Example: 10 people participate in a project whose goal is the development of an algorithm capable of predicting the outcome of an event that can have only two results. An external person who does not participate in the project but is aware of every attempt made by the participants evaluates the statistical data obtained. All participants make 100 predictions, 9 get a 50% chance of success, one gets a 65% chance of success. The uncertainty of the static data of the participant who obtains a probability of success of 65% is obtained by calculating the probability of obtaining a result with a number of successes greater than or equal to 65 by performing ten random attempts consisting of 100 predictions each. The probability obtained, in this way, is 16% instead if he was the only participant in the project the probability would have been 0.18%, therefore about 100 times lower.

Third fundamental consequence of the non-independence of the statistical data: the calculation of the uncertainty varies according to the information possessed.

Example: 10 people participate in a project whose goal is the development of an algorithm capable of predicting the outcome of an event that can have only two results. In this case, people do not know the other participants and think they are the only ones participating in the project. All participants make 100 predictions, 9 get a 50% chance of success and one gets a 65% chance of success. The participant who obtains a probability of success of 65% independently calculates the uncertainty of the result obtained. Not knowing that other people are participating in the project, calculate the probability of obtaining a result with a number of successes greater than or equal to 65 by performing a single random attempt consisting of 100 predictions; the probability obtained is 0.18%. An external person who is aware of every attempt made by the participants calculates the uncertainty of the participant’s statistical data, which obtains a probability of success of 65%. It then calculates the probability of obtaining a result with a number of successes greater than or equal to 65 by making ten random attempts consisting of 100 predictions each. The probability obtained, in this way, is 16%, a much higher value than the uncertainty calculated by the participant. The uncertainty value calculated by the external person using more information is most accurate than the uncertainty value calculated by the individual participant. Consequently, the uncertainty value obtained by exploiting the greatest number of information must always be considered, in the case of the example, the most accurate uncertainty is that of 16%.

The first and second fundamental highlighting consequence of the non-independence of the statistical data can be redefined by highlighting the non-randomness of the action.

First fundamental consequence of the non-independence of the statistical data: our every non-random action always involves a decrease in the uncertainty of the statistical data.

Second fundamental consequence of the non-independence of the statistical data: every non-random action of ours and of every other person equivalent to us, always involves a decrease in the uncertainty of the statistical data.

Solving the fundamental problem of causal inference

The fundamental problem of causal inference defines the impossibility of associating a link between correlation and causality, in other words: correlation does not prove causality. In this paragraph, we will see how this problem arises from an error and how the uncertainty of the statistical data (the probability of obtaining the same result randomly) represents the probability that correlation does not imply causality.

On the internet, you can find hilarious correlations between very different events, these correlations are obviously random. These examples are often used to demonstrate the fundamental problem of causal inference. In presenting this data, the following information is always omitted: how many hypotheses did I consider before finding a related hypothesis.

This is a fundamental piece of information, because if I have a database of a million events, whatever my data is, there will always be a event that will be well correlated with my data.

So, if I generate a million random hypotheses, I will succeed with a near 100% probability of finding a hypothesis that is correlated with my data. So, having about a 100% chance of being able to get the same correlation randomly, I have about a 100% chance that the correlation doesn’t also imply causation.

Instead, if we generate a single hypothesis and it is well correlated with the data, in this situation, almost certainly the correlation also implies causality. This is because the probability of obtaining a good correlation by generating a single random hypothesis is almost zero.

This result is also intuitive, because it is possible to obtain a good correlation with a single attempt, only if we have knowledge of the process that generates the data. And it is precisely this knowledge that also determines a bond of causality.

Conclusion

The traditional definition of uncertainty implies considering true, for non-homogeneous data dispersions, the hypothesis that the result is not completely random. We consider this assumption the main problem of the definition of uncertainty and the primary cause of the high number of nonreproducible results. Indeed, whatever the statistical data obtained, there is always a possibility that they are completely random and therefore useless.

This error stems from the fact that the definition of uncertainty was developed in an environment where each method had a strong deterministic component. Therefore, calculating the probability of obtaining an equal or better result at random might seem useless. However, when we apply statistics in fields such as finance, where the random component is predominant the traditional approach to uncertainty turns out to be unsuccessful. It fails for the simple reason that the hypothesis on which it is based may not be true. For this reason, we have defined the uncertainty of the statistical data as the probability of obtaining an equal or better result randomly. Since this definition of uncertainty is not linked to any hypothesis, it turns out to be universal. The correct calculation of this probability value implies considering the statistical data dependent on each other. This assumption, as we have shown through a paradox, makes the definition of uncertainty given consistent with the logical principle that it is not possible to create knowledge randomly.

The non-independence of the statistical data implies that each action performed has an effect on the calculation of uncertainty. The interesting aspect is that a dependence is also created between actions performed by different people. Consequently, the calculation of uncertainty depends on the information in our possession, so it becomes something relative that can be determined absolutely only with complete knowledge of the information.

Finally, we have shown how the fundamental problem of statistical inference (correlation does not mean causality) arises from the error of considering the hypotheses tested as independent of each other. This problem is solved by using the probability of obtaining an equal or better result randomly as the uncertainty of the statistical data. Indeed, in this case, uncertainty also defines the probability that the correlation does not imply causality.

Bibliography:

[1] Bich, W., Cox, M. G., and Harris, P. M. Evolution of the “Guide to the Expression of Uncertainty in Measurement”. Metrologia, 43(4):S161–S166, 2006.

[2] Grabe, M .,”Measurement Uncertainties in Science and Technology”, Springer 2005.

[3] Munafò, M., Nosek, B., Bishop, D. et al. “A manifesto for reproducible science”. Nat Hum Behav 1, 0021 (2017). https://doi.org/10.1038/s41562-016-0021.

[4] Ioannidis, J. P. A. “Why most published research findings are false”. PLoS Med. 2, e124 (2005). [5] Berdondini, Andrea, “Statistics the Science of Awareness” (August 30, 2021). Available at SSRN: https://ssrn.com/abstract=3914134.

[5] Berdondini, Andrea, “Statistics the Science of Awareness” (August 30, 2021). Available at SSRN: https://ssrn.com/abstract=3914134.

The uncertainty of the statistical data was originally published in CodeX on Medium, where people are continuing the conversation by highlighting and responding to this story.

Statistics the science of awareness

ABSTRACT: The uncertainty of the statistical data is determined by the value of the probability of obtaining an equal or better result randomly. Since this probability depends on all the actions performed, two fundamental results can be deduced. Each of our random and therefore unnecessary actions always involves an increase in the uncertainty of the phenomenon to which the statistical data refers. Each of our non-random actions always involves a decrease in the uncertainty of the phenomenon to which the statistical data refers.

Introduction

This article proves the following sentence:

“The only thing that cannot be created randomly is knowledge”

A true story of a true coincidence

Ann is a researcher, is a clever and beautiful researcher, one day she decides to do the following experiment: she wants to understand if she has some special abilities that allow her to extract the number 1 from a bag containing one hundred different numbers mixed in a random way.

Day 1, Ann takes the bag and randomly pulls out a number. The drawn number is not 1, so she failed, the drawn number is put back into the bag.

Day 2, Ann takes the bag and randomly pulls out a number. The drawn number is not 1, so she failed, the drawn number is put back into the bag.

……………

……………

The days pass and Ann fails every attempt but continues his experiment.

……………

……………

Day 100, Ann takes the bag and randomly pulls out a number. The number drawn is 1, so she is successful. The probability of finding number 1 by performing a random extraction is 1/100; this value represents an acceptable error that makes the result significant to support the hypothesis that the extraction is non-random. But Ann knows that this probability does not represent the uncertainty of her success because this value does not take into account previous attempts. Therefore, she calculates the probability of randomly extracting the number 1, at least once, in a hundred attempts. The probability value thus calculated is 63%, since this value is very high, she cannot consider the success obtained as significant statistical data to support the hypothesis that the extraction is non-random.

Ann concludes the experiment and deduces, from the results obtained, that she has no special ability and the extractions are all random.

John is a data scientist, one day he is entrusted with the following task: he has to develop an algorithm capable of predicting the result of an experiment whose result is determined by a value from 1 to 100.

Day 1, an incredible coincidence begins, at the same time that Ann pulls a number John tests his own algorithm. The number generated by the algorithm does not coincide with the result of the experiment, so he failed.

Day 2, the incredible coincidence continues, at the same time that Ann pulls a number John tests a new algorithm. The number generated by the algorithm does not coincide with the result of the experiment, so he failed.

……………

……………

The days pass, the coincidence continues and John fails every attempt.

……………

……………

Day 100, the incredible coincidence continues, at the same time that Ann pulls a number John tests his new algorithm. The number generated by the algorithm coincides with the result of the experiment, so he is successful. The probability of predicting the result of the experiment by running a random algorithm is 1/100; this value represents an acceptable error that makes the result significant to support the hypothesis that the algorithm used is non-random.

For this reason, John writes an article in which presents the result obtained. The article is accepted, John is thirty years old and this is his hundredth article.

Awareness breeds awareness

We call “researcher” a person who knows only his own attempts regarding the study of a certain phenomenon.

We call “reviewer” a person who does not actively participate in the study of a particular phenomenon but knows every single attempt made by each researcher.

Researcher 1: develops an algorithm that obtains the result R1 with respect to a phenomenon F. The probability of getting a result equal to or better than R1 in a random way is 1%.

Researcher 2: develops an algorithm that obtains the result R2 with respect to a phenomenon F. The probability of getting a result equal to or better than R2 in a random way is 1%.

Reviewer: defines a new result RT= R1∩R2. The probability of getting a result equal to or better than RT in a random way is 0.01%. Consequently, the uncertainty of the result RT is 0.01%.

The absence of awareness reduces awareness

We call “researcher” a person who knows only his own attempts regarding the study of a certain phenomenon.

We call “reviewer” a person who does not actively participate in the study of a particular phenomenon but knows every single attempt made by each researcher.

Researcher 1: develops an algorithm that obtains the result R1 with respect to a phenomenon F. The probability of getting a result equal to or better than R1 in a random way is 1%.

Researcher 2: develops an algorithm that obtains the result R2 with respect to a phenomenon F. The probability of getting a result equal to or better than R2 in a random way is 100%.

Reviewer: defines a new result RT= R1∩R2. The probability of getting a result equal to or better than RT in a random way is 2%. Consequently, the uncertainty of the result RT is 2%.

Statistics — the science of awareness was originally published in CodeX on Medium, where people are continuing the conversation by highlighting and responding to this story.

Use of the Fundamental Problem of Statistics to Define the Validity Limit of Occam’s Razor Principle

An innovative method to define the validity limit of Occam’s razor principle

ABSTRACT: In statistics, to evaluate the significance of a result, one of the most used methods is the statistical hypothesis test. Using this theory, the fundamental problem of statistics can be expressed as follows: “A statistical data does not represent useful information, but becomes useful information only when it is shown that it was not obtained randomly”. Consequently, according to this point of view, among the hypotheses that perform the same prediction, we must choose the result that has a lower probability of being produced randomly. Therefore, the fundamental aspect of this approach is to calculate correctly this probability value. This problem is addressed by redefining what is meant by hypothesis. The traditional approach considers the hypothesis as the set of rules that actively participate in the forecast. Instead, we consider as hypotheses the sum of all the hypotheses made, also considering the hypotheses preceding the one used. Therefore, each time a prediction is made, our hypothesis increases in complexity and consequently increases its ability to adapt to a random data set. In this way, the complexity of a hypothesis can be precisely determined only if all previous attempts are known. Consequently, Occam’s razor principle no longer has a general value, but its application depends on the information we have on the tested hypotheses.

Introduction

The logical principle of Occam’s razor [1], [2], suggests choosing the simplest hypothesis among those available. In this article, we will analyze this principle using the theory of statistical hypothesis test [3], [4]. By exploiting this theory, we will reformulate the fundamental problem of statistics in such a way as to bring attention to the link between the statistical data and the probability that it was produced randomly. Consequently, according to this point of view, among the hypotheses that perform the same prediction, we must choose the result that has a lower probability of being produced randomly. Therefore, it becomes essential to calculate this probability value correctly.

This problem is addressed by redefining what is meant by hypothesis. The traditional approach considers the hypothesis as the set of rules that actively participate in the forecast. Instead, we consider as hypotheses the sum of all the hypotheses made, also considering the hypotheses preceding the one used. Therefore, each time a prediction is made, our hypothesis increases in complexity and consequently increases its ability to adapt to a random data set. In this way, the complexity of a hypothesis can be precisely determined only if all previous attempts are known. Consequently, Occam’s razor principle no longer has a general value, but its application depends on the information we have on the tested hypotheses.

Finally, we use this new definition of hypothesis to understand the reason for the high percentage of non-reproducible results, in which the hypothesis test was used.

The fundamental problem of statistics

In statistics, to evaluate the significance of a result, one of the most used methods is the statistical hypothesis test. Using this theory, the fundamental problem of statistics can be expressed as follows: “A statistical data does not represent useful information, but becomes useful information only when it is shown that it was not obtained randomly”.

This definition is particularly significant, because it highlights the two fundamental aspects of statistics, which are its uncertainty and the reason for its uncertainty.

Indeed, the purpose of statistics is the study of phenomena in conditions of uncertainty or non-determinism by exploiting the sampling of events related to the phenomenon to be studied. Knowing that the observed events can be randomly reproduced with a probability that will never be zero, we understand the reason for the indeterminism that characterizes the statistics. This probability value is called universal probability [5].

Through this definition of the fundamental problem of statistics, it is also possible to formulate the following paradox [6], which highlights how the evaluation of statistical results is dependent on each action performed on the analyzed data.

The validity limit of Occam’s razor principle

In this paragraph, we will see how the information regarding the development of a hypothesis is fundamental to define the validity limit of Occam’s razor principle.

Let us start by giving some definitions useful to formalize our theory.

Given an experiment that measures N values of a discrete variable X with cardinality C, we call D the set of dimension 𝐶^𝑁, which includes all possible sequences 𝑋S of length N that can be observed.

Now, we redefine the concept of hypothesis in order to define a chronological succession among the tested hypothesis.

We call H(t) the hypothesis developed at time t.

We call PH(t) the set of sequences 𝑋S∈𝐷 that hypothesis H(t) is able to predict.

We call NPH(t) the cardinality of the set PH(t).

We call TH(t) the set that includes all the hypotheses up to time t.

𝑇𝐻(𝑡)={𝐻(𝑖1),𝐻(𝑖2),……,𝐻(𝑖𝑡)}

We call TPH(t) the union of all the sets PH(t) relating to all the hypotheses H(t)∈TH(t).

We call NTPH(t) the cardinality of the set TPH(t). Consequently, NTPH(T) defines the number of sequences, belonging to D, that the hypothesis TH(t) is able to predict. It may happen that different hypotheses forecast the same sequence of values of X, having made the union of the sets PH(t) these sequences are calculated only once.

If we have only made a hypothesis H(t)=TH(t) and NPH(t)=NTPH(t). If, on the other hand, more than one hypothesis has been tested H(t)≠TH(t) and NPH(t)≤NTPH(t).

We define the ability of the hypothesis TH(t) to predict a sequence of N casual observations of the i.i.d. random variable X with discrete uniform distribution and cardinality C, the ratio:

This ratio also defines the probability that the hypothesis TH(t) can predict the results of an experiment, in which the cardinality of D is equal to 𝐶^𝑁, in a completely random way.

Knowing that a hypothesis TH(t) can predict the results of an experiment only in the following two conditions:

1) TH(t) is true.

2) TH(t) is false and the prediction occurs randomly; the probability of this event is given by equation (1).

Under these conditions, the probability that the hypothesis TH(t) is true turns out to be:

Consequently, this equation defines the parameter that must be used in the evaluation of H(t).

So, if we want to compare two hypotheses H1(t) and H2(t), we have 4 possible results:

1) NPH1(t)>NPH2(t) and NTPH1(t)>NTPH2(t)

2) NPH1(t)>NPH2(t) and NTPH1(t)<NTPH2(t)

3) NPH1(t)<NPH2(t) and NTPH1(t)<NTPH2(t)

4) NPH1(t)<NPH2(t) and NTPH1(t)>NTPH2(t)

NPH(t) and NTPH(t) define the number of sequences that hypothesis H(t) and the hypothesis TH(t) are able to predict. Consequently, they can be used as a measure of their complexity, in fact, the more complex a hypothesis is, the greater the number of results it can predict.

Analyzing the four possible results, we note that even if a hypothesis H1(t) is less complex than a hypothesis H2(t) (NPH1(t)<NPH2(t)), it is possible to have a hypothesis TH1(t) more complex than a hypothesis TH2(t) (NTPH1(t)>NTPH2(t)). Consequently, using equation (2) as an evaluation method, hypothesis H1(t) should be discarded in favor of hypothesis H2(t). This situation can happen, for example, if H1(t) is the last hypothesis of a long series of other hypotheses tested previously.

In the event that there is no information on the hypotheses to be evaluated, it must be assumed that the hypotheses have been developed under the same conditions. Therefore, in this case, not being able to calculate TH(t), it is recommended to choose the simpler hypothesis H(t).

Finally, from equation (2), we can deduce the following result: given a hypothesis H(t) the probability that is true can be calculated only if all the previously tested hypotheses are known.

Consequently, the complexity of a hypothesis not only depends on the mathematical formula that makes the prediction, but also depends on all the attempts made previously.

Therefore, Occam’s razor principle does not have an absolute value but its application depends on the information about the hypotheses.

How to perform correctly the statistical hypothesis test

It is interesting to note how the definition of hypothesis, which was given in the previous paragraph, can be seen as something extremely obvious or as something extremely innovative. Indeed, it may seem absolutely banal to consider all the hypotheses that have been tested, for the obvious reason that by running a large number of random hypotheses sooner or later there will be some hypothesis that will fit the data quite well. On the other hand, also considering the previous hypotheses represents a revolution in the evaluation of a hypothesis. In fact, from this point of view, the mere knowledge of the hypothesis that makes the prediction does not allow us to define its real complexity.

Therefore, if in the statistical hypothesis test the p-value [7], [8], used as a threshold to reject the null hypothesis, is calculated considering only the hypothesis that actively participates in the prediction, it means, that we are underestimating the complexity of the hypothesis. Consequently, the p-value, thus calculated, is wrong and therefore determines a false evaluation of the hypothesis. It is therefore believed that this systematic error, in the execution of the hypothesis test, is responsible for the high number of non-reproducible results [9], [10].

Taking advantage of these considerations it is understood that evaluating a statistical result can be very difficult because some information can be hidden. For example, we are obliged to report the mathematical formula that makes the prediction but, instead, we may not report all previous failed attempts. Unfortunately, this information is essential for evaluating the hypothesis, because they are an integral part of the hypothesis. Indeed, if we test 10 hypotheses, we simply interpolate the data with those ten hypotheses and choose the hypothesis that passes the chosen evaluation test.

This problem also depends on the increasing use of statistical software capable of quickly executing a huge number of mathematical models. Consequently, there is the risk of “playing” with this software by performing a multitude of analyzes and this sooner or later leads to a random correlation.

For these reasons, the evaluation of statistical results represents one of the most important challenges for scientific research. Unfortunately, it is a difficult problem to solve because, as mentioned, some information can always be hidden when writing an article. The simplest solution adopted is to use more selective evaluation parameters, which in practice means making it unlikely to pass the evaluation test by developing random hypotheses. However, this solution has different problems in fact, in this way, we can discard correct hypotheses and cannot be applied to all fields of research. For example, in finance where the possible inefficiencies of the markets [11], which can be observed, are minimal, adopting very restrictive valuation methods means having to discard almost any hypothesis.

Conclusion

In this article, we have discussed the logical principle of Occam’s razor using the hypothesis test theory. This allowed us to reformulate the fundamental problem of statistics, in such a way as to make us understand the importance of correctly calculating the probability of obtaining the same results randomly. Solving this problem involved redefining the concept of hypothesis. According to this point of view, by hypothesis we mean the sum of all tested hypotheses. Consequently, the complexity of a hypothesis not only depends on the mathematical formula that makes the prediction but also depends on the previous hypotheses tested.

Therefore, according to this approach, the logical principle of Occam’s razor no longer has a general value if one considers as a hypothesis only the set of rules that actively participate in the prediction. If, on the other hand, the hypothesis is considered as the sum of all the tested hypotheses, in this case, Occam’s razor principle returns to have a general value.

Finally, it is noted that not considering all the tested hypotheses causes a systematic error in the application of the statistical hypothesis test. Therefore, it is hypothesized that this error, which leads to underestimate the complexity of a hypothesis, is the cause of the high percentage of non-reproducible scientific results.

References

[1] Roger Ariew, “Ockham’s Razor: A Historical and Philosophical Analysis of Ockham’s Principle of Parsimony”, 1976.

[2] Sober, Elliott (2004). “What is the Problem of Simplicity?”. In Zellner, Arnold; Keuzenkamp, Hugo A.; McAleer, Michael (eds.). Simplicity, Inference and Modeling: Keeping it Sophisticatedly Simple. Cambridge, U.K.: Cambridge University Press. pp. 13–31. ISBN 978–0–521–80361–8. Retrieved 4 August 2012ISBN 0–511–00748–5 (eBook [Adobe Reader]) paper as pdf.

[3] Fisher, R (1955). “Statistical Methods and Scientific Induction” (PDF). Journal of the Royal Statistical Society, Series B. 17 (1): 69–78.

[4] Borror, Connie M. (2009). “Statistical decision making”. The Certified Quality Engineer Handbook (3rd ed.). Milwaukee, WI: ASQ Quality Press. pp. 418–472. ISBN 978–0–873–89745–7.

[5] Cristian S. Calude (2002). “Information and Randomness: An Algorithmic Perspective”, second edition. Springer. ISBN 3–540–43466–6.

[6] Berdondini Andrea, “The Information Paradox”, (July 8, 2019). Available at SSRN: https://ssrn.com/abstract=3416559.

[7] Wasserstein, Ronald L.; Lazar, Nicole A. (7 March 2016). “The ASA’s Statement on p-Values: Context, Process, and Purpose”. The American Statistician. 70 (2): 129–133. doi:10.1080/00031305.2016.1154108.

[8] Hung, H.M.J.; O’Neill, R.T.; Bauer, P.; Kohne, K. (1997). “The behavior of the p-value when the alternative hypothesis is true”. Biometrics (Submitted manuscript). 53 (1): 11–22.

[9] Munafò, M., Nosek, B., Bishop, D. et al. “A manifesto for reproducible science”. Nat Hum Behav 1, 0021 (2017). https://doi.org/10.1038/s41562-016-0021.

[10] Ioannidis, J. P. A. “Why most published research findings are false”. PLoS Med. 2, e124 (2005).

[11] Black, F. (1971) “Random Walk and Portfolio Management,” Financial Analyst Journal, 27, 16–22.

Use of the fundamental problem of statistics to define the validity limit of Occam’s razor… was originally published in CodeX on Medium, where people are continuing the conversation by highlighting and responding to this story.

Application of the Von Mises’ axiom of randomness on the forecasts concerning the dynamics of a non-stationary system described by a numerical sequence

Use of the Von Mises’ axiom of randomness as an analysis method concerning the forecasts on the evolutions of a non-stationary system.

ABSTRACT: In this article, we will describe the dynamics of a non-stationary system using a numerical sequence, in which the value of terms varies within the number of degrees of freedom of the system. This numerical sequence allows us to use the Von Mises’ axiom of randomness as an analysis method concerning the results obtained from the forecasts on the evolutions of a non-stationary system. The meaning of this axiom is as follows: when we understand a pattern about a numerical sequence, we obtain results, intended as forecast on the next sequence number, which cannot be reproduced randomly. In practice, this axiom defines a statistical method capable of understanding, if the results have been obtained by a random algorithm or by a cognitive algorithm that implements a pattern present in the system. This approach is particularly useful for analysing non-stationary systems, whose characteristic is to generate non-independent results and therefore not statistically significant. The most important example of a non-stationary system are financial markets, and for this reason, the primary application of this method is the analysis of trading strategies.

Introduction

The first problem that must be faced, in order to apply the axiom of randomness of Von Mises as an analysis method, is to be able to describe the evolution of a system using a numerical sequence. For this purpose, we use a sequence in which the value of terms can change within the number of degrees of freedom of the system. Then we also introduce a temporal progression represented by a series of increasing integers. With these two sequences, we can characterize a dynamic that describes the evolution of the system that we are studying. In this way, we can apply the Von Mises’ axiom of randomness, which defines the statistical characteristic of a random sequence. The axiom is as follows: “the essential requirement for a sequence to be defined as random consists in the complete absence of any rules that may be successfully applied to improve predictions about the next number”.

This axiom tells us that when we understand a pattern about a numerical sequence, we can obtain results, intended as forecasts on the next sequence number, which cannot be reproduced randomly. Knowing that the values of our numerical sequence represent the dynamics of a system, we have found a method to evaluate the results. Therefore, if the probability of obtaining equal or better results randomly is very small (i.e. tends to zero as the number of times the strategy is used increases), it means that the forecasting method uses rules present in the system. Consequently, we can use the developed method in order to predict the evolution of the system; this approach is fundamental in the study of non-stationary systems. In fact, if we are in a condition of non-stationarity, the results can be non-independent, the consequence of which is a reduction in their statistical value. In order to explain such a complex topic in a simple way, I have created a paradox [1], in which two players challenge each other in a game, where is possible to move from a stationary to a non-stationary condition by changing the rules of the game. The consequence is that the data at the beginning are independent, and then once the changes are made, they become dependent on each other and therefore useless for statistical purposes.

For that reason, a large number of data obtained under these conditions can be not statistically useful in order to understand if the method that produced the data is not random. In this case, the performance of the forecasting algorithm is no longer a reliable data; therefore we need to find a new statistical indicator in order to evaluate the results. In fact, it happens very often in finance, that a strategy that despite having obtained excellent results by performing a numerous number of operations, suddenly stops working generating large losses, proving in this way to be a completely random strategy. There are many researches on the non-stationarity of the financial markets; some articles about this topic are [2], [3], [4], [5].

Description of the dynamics of a non-stationary system by means of a numerical sequence

We start by considering a non-stationary system characterized by two degrees of freedom. We made this choice for two reasons: first in order to simplify the treatment, second among the systems with two degrees of freedom there are the financial markets, that represent the primary application of the method that we will go to expose.

Then we associate to this system a succession of values that can vary between 1 and 2. In practice when on this system there is a change in the direction in which a deterministic process is acting, we have the result that the value of the succession change respect to the previous term. When we speak of “deterministic process”, we indicate a deterministic and non-random force that acts on the system. We also introduce the simplification that the deterministic process acts as a constant force, without variations in intensity in the direction of the two degrees of freedom; this allows us to characterize the dynamics of the system with a single numeric succession.

Finally, we add a temporal metric to the system; we can do this by using a series of increasing integer numbers, in this way every value of my succession unambiguously corresponds to a value in my temporal progression.

The result obtained is the following: we have two sequences, one that describes the direction of the force acting on the system F and the other one that describe the progression of the time T:

Analysing these two sequences we can note that the force (deterministic process) that acts on the system changes direction three times, the first at time T = 8, the second at time T = 13, and the third at time T = 20. Figure 1 shows the dynamics described by the two sequences; the temporal progression has also been included in the graph.

Application of the Von Mises’ axiom of randomness as an analysis method concerning the results obtained by a forecasting algorithm

At this point, we are going to apply the Von Mises’ axiom of randomness in order to evaluate the results obtained by a forecasting algorithm. According to the axiom, reported and analyzed in the introduction of this article, if we have developed a non-random method that implements a pattern present in the system, we will be able to predict when the term of the succession varies respect the previous term.

This statement is crucial because we are saying that only the forecasts concerning system changes are useful in order to understand whether the method used is random or not. In fact, if we make two forecasts about two terms of the succession at time 8 and 12, betting that the value of the succession is two, we obtain two successes. Are these two results independent of each other? This question is fundamental because only if they are independents are useful for statistical purposes in order to evaluate the forecasting method.

In order to answer this question we must use the formula of the compound probability that we report:

P (E1∩E2) = P (E1 | E2) P (E2) = P (E2 | E1) P (E1) (1)

The probability that both events (E1, E2) occurs, is equal to the probability that E1 occurs P(E1), multiplied by the conditional probability of E2 given E1 occurs P(E2 | E1).

Seeing formula (1) we understand that the correct calculation of conditional probability P (E2 | E1) depends on whether the events are independent or not. In fact, in the case of independent events P(E2 | E1) = P(E2), instead in the case of events completely dependent on each other P (E2 | E1) = 1 or 0.
In our example the probability P(E1) = 1/8. In fact, the probability of winning in a random way betting that the value of the sequence within the first 8 time positions is equal to 2 is 1/8, since in the first seven positions the sequence has value 1 and only at the eighth position has value 2. In practice, we can see this value like the probability of a random draw of a marble with the value 2 from a bag that contains eight marbles, seven of them with the value 1 and one with the value 2. Therefore, P (E1) represents the probability of obtaining the same result using a random strategy. In this way, we are able to derive a probability from a succession of data concerning the dynamics of the system.

Now we try to calculate the value of P (E2); studying the time interval from the ninth position to the twelfth position, we see that the system remains constant. Therefore, the probability of betting randomly on the value 2 and win, within this time interval, is equal to 1. This is because all positions have value two, so a random strategy obtains the same result with a probability of 100%. In this case, we can treat the second bet as completely dependent from the first bet. This means that we can consider the two bets, from the statistical point of view, like a single bet.

Now let’s try to shift the second bet from the twelfth position to the twentieth position, in this case we will have P(E2) = 5/7. In fact, in the time interval from the ninth position to the twentieth position, there are seven values equal to 1 and five values equal to 2. With this variation in the second forecast, the compound probability will be equal to:

P (E1∩E2) = P (E1) P (E2 | E1) = P (E1) P (E2) = 1/8 ∙ 5/7 = 0.089

The two events in this case are independent of each other, so they are useful for statistical purposes in order to understand if the method used can predict the evolution of the system.

From these two examples, we can deduce the following conclusion: when we want to study a deterministic process, every time that it determines a change in the system, we have the possibility to execute a prediction on the evolution of the system, whose probability of success with a random strategy turns out to be minor 1. This involves a decrease in the compound probability whose meaning, according to the Von Mises’ axiom of randomness, is to indicate the presence of a deterministic process that acts on the system.

In practice, whenever that a deterministic process changes the system status, we can detect it by making a prediction. Consequently, in order to detect a deterministic process with a low error, it is necessary that it has produced a statistically significant number of variations in the system. On which our forecasting method has carried out a large number of independent predictions. Therefore, every forecast must concern a single variation of the system.

This approach is crucial for non-stationary systems where the forecasts may be not independent. For that reason, it is fundamental to calculate the compound probability, which is the probability of obtaining the same results in a random way. We also remember that within the compound probability (1), there is the calculation of the conditional probability, in which it is taken into account if the events are dependent or independent from each other. In fact, the goal of this method is to discard all non-independent forecasts, whose contribution is to make me overestimate the forecast skill of the algorithm. The consequence of this erroneous evaluation can lead us, for example in the financial field, to bear excessive risks.

Conclusion

The analysis of results, concerning the evolution of a non-stationary system, represents one of the most important problems of applied mathematics still unresolved. In this article, I propose the use of the Von Mises’ axiom of randomness as a method of evaluating the data obtained under these conditions. This axiom, as explained previously, defines a statistical characteristic that assumes a well-defined behaviour when we operate consciously. In practice, the results generated by a non-random forecasting algorithm, that implements knowledge about the considered system, cannot be reproduced randomly. Consequently, the probability of obtaining equal or better results randomly tends to zero as the number of times the strategy is used increases. In this way, we shift our attention from the value of the result, which under the non-stationary condition may have been produced with non-independent forecasts, to its statistical characteristic correlated to its non-random behaviour.

In order to apply this method, we have defined a simple mathematical model whose task is to describe the dynamics of a system by means of a numerical sequence. Using this mathematical model, we have shown how to compute compound probability from a series of results obtained with a forecasting algorithm. Then, analysing some examples, we have deduced some important considerations.

In particular, we have seen that when there is a change in the system, we have the possibility to make a prediction whose probability of success is less than 1. The consequence of this is a decrease of the compound probability whose meaning, according to the Von Mises’ axiom of randomness, is to indicate the presence of a pattern on the system.

Therefore, if we want to detect it with a low error we must have made a statistically significant number of independent predictions, concerning every single variation of the system. In this way, the forecast algorithm proves to be able to predict future system evolutions. This characteristic, therefore, becomes fundamental in estimating the risk of a strategy that operates on a non-stationary system, such as the financial markets. In fact, a correct risk assessment must always take into consideration the forecast skill of the algorithm about the future evolutions of the system.

In the next article, we will introduce a purely random component within the dynamics of a non-stationary system. In this way, the deterministic process will be replaced by a stochastic process described by a probability density function. This step is essential in order to apply this method in the financial field.

References

[1] Berdondini Andrea, “The Professional Trader’s Paradox”, (November 20, 2018). Available at SSRN: https://ssrn.com/abstract=3287968.

[2] J. Barkley Rosser Jr. ‘Reconsidering ergodicity and fundamental uncertainty’. In: Journal of Post Keynesian Economics 38 (3 2015), 331–354. doi: 10.1080/01603477.2015.1070271 (1).

[3] M. Sewell. “History of the efficient market hypothesis”. Technical report, University College London, Department of Computer Science, January 2011.

[4] O. Peters, M. Gell-Mann. “Evaluating gambles using dynamics”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016; 26 (2): 023103 DOI: 10.1063/1.4940236

[5] Berdondini, Andrea, “Description of a Methodology from Econophysics as a Verification Technique for a Financial Strategy “,(May 1, 2017). Available at SSRN: https://ssrn.com/abstract=3184781.

Application of the Von Mises’ axiom of randomness on the forecasts concerning the dynamics of a… was originally published in TDS Archive on Medium, where people are continuing the conversation by highlighting and responding to this story.

In this article, we will analyze a mental attitude that distinguishes professional gamblers. This characteristic can be summarized with the following concept: “dissociation from the result”. This type of mental predisposition is very important because the way we relate to the result affects our ability to behave rationally.

The basic feature of the human mind

To understand how the mind behaves in relation to the result of the actions, we must understand the environment in which it evolved. Man has evolved in a context in which the link between action and result has always been very strong.

The non-randomness of the result has an important consequence: if the action leads to a benefit, the action that generated it turns out to be correct.

Therefore, a very strong connection is created between the action and the result. Consequently, to a useful result for our survival, we are led to consider the action performed as rational and correct. The brain also strengthens this bond in a physiological way by producing a physical sensation of happiness. In practice, the brain rewards us for the action done and wants it to be repeated over time because considered useful for our survival.

How the mind is deceived

Now we analyze how a mind, evolved under the circumstances described in the previous paragraph, behaves when it has to face situations in which the outcome of the result is subject to a significant random component.

In these situations, an extremely important thing happens: the result of our action will never be constant but will have a certain degree of randomness. In this situation, the link between action and result is broken, and it is no longer true that a useful result corresponds to rational action.

Situations in which there is no link between the rationality of the action and the result are situations almost exclusively created by man as in the case of gambling. For example, if we play heads or tails, we have 50% of probability to win and 50% to lose, and obviously, there is no rational link between action and result. However, for the brain it makes no difference, if you win a bet it will reward you with a feeling of happiness and this is because for our mind the link between result and action still exists. For this reason, the psychological pathology called ludopathy, which afflicts gamblers, is the direct consequence of an evolved mind in an environment where there is a very strong link between action and result.

In this situation, the mind is deceived and considers actions that are irrational and destructive as rational and useful. This behavior is also called cognitive distortion, and the people who are affected are really convinced that their irrational actions can lead them to victory.

At this point, you will surely have understood that the main problem is due to the direct link that the mind associates between action and result. Therefore, it is precisely on this key element that arises the difference in approach between an ordinary person and a professional gambler.

People who understand this problem understand the importance of training their mind about the dissociation from the result. In this way, they develop a new awareness about the result, which allows them to maintain a rational and winning response even in situations where the result is subject to a random component.

The mind of a professional gambler

In the previous chapter, the cognitive characteristic that distinguishes an ordinary person from a professional gambler was identified. This characteristic can be summarized with the following concept: “dissociation from the result”. In other words, it is about breaking the link that our brain creates between action and result.

This mental approach is present among professionals from very different sectors; for example, it is normal for a professional poker player to get angry when he makes a stupid play and to be happy when he makes an intelligent play, regardless of the outcome of the individual plays. In fact, these players have perfectly understood that there is no direct link between the result of a single operation and the action performed. What makes them successful is always being able to perform a series of rational and correct actions without getting involved in emotions. About this argument, I also report an interesting statement by the professional trader Linda Raschke whose interview was featured in the 06–2017 issue of Trader Magazine. When asked, “Is Trading a game for you?” Linda Raschke replies “Yes. I barely check the account balance. Because this unnecessarily affects me both positively and negatively and does not change the fact that I have to make the right decision again. And if you always make the right decisions, performance becomes simple in the long run”. As you can see, there is such a strong dissociation from the single result that she does not even care about its balance (an amateur trader checks his balance every 5 seconds).

Viewed from this point of view, gambling is no longer random, but is a purely cognitive process, where you only operate if you understand the game by always performing rational and never emotional actions. At this point, becomes easy to understand the difference between an amateur gambler and a professional gambler: the first is obsessed by the result, while the second is obsessed by the knowledge.

The psychology of the professional gambler was originally published in ILLUMINATION on Medium, where people are continuing the conversation by highlighting and responding to this story.

How meditation can improve our problem-solving skills

The goal of this article is to try to explain easily, how the practice of meditation affects our problem-solving skills. To do this, I use a mathematical logical method to characterize problems based on the number of their possible solutions. In this way, it easy to explain the two primary approaches that the mind uses in solving problems. The first approach is iterative, optimal in solving simple problems (problems with a low number of solutions). The second approach is the logical one, optimal in solving complex problems (problems with a high number of solutions). The interesting aspect of these two methods is that their mode of action is the opposite. The iterative method is based on action, while the logical method is a reflective approach, in which any unnecessary action takes us away from solving the problem. Consequently, we will see how through the practice of meditation, we can shift our mental predisposition towards the logical approach by inhibiting our propensity for the iterative approach. This is a very important result because it can allow us to improve our ability to solve complex problems. This type of attitude is fundamental in a society where technological progress is making all simple and repetitive jobs less indispensable.

The two fundamental methods used by the mind in solving problems

From a mathematical logical point of view, the problems can be divided according to the number of possible solutions they can have. Therefore, using this approach, we can define two classes of problems:

1) “Simple” problems: the problems in which the space of possible solutions is constituted by a small number of elements. Where the time required to try all possible solutions iteratively, is limited and acceptable. Example: a padlock that has 100 possible combinations, if I can try a different combination every 5 seconds, I will try all the combinations in an acceptable time.

2) “Complex” problems: the problems in which the space of possible solutions is constituted by a large number of elements. Where the time to try all possible solutions iteratively, tends to infinity or to an unacceptable time. Example: proving a mathematical theorem, doing it iteratively trying random solutions, takes a time that tends to infinity.

Thanks to this classification, we can study, in a simple way, the two fundamental approaches that the mind uses in solving problems.

The first approach is iterative, the mind does not try to solve the problem but tries all possible combinations. This approach is the optimal one for solving “simple” problems and was fundamental in the initial part of human evolution.

The second approach is the logical one; the mind creates a model of the problem and tries to solve it. This approach is the optimal one for solving “complex” problems, and its importance has increased in the course of human evolution. Consequently, this category of problems is also the one that most characterizes our problem-solving skills.

Comparing these two methods, the interesting thing we notice is that they act oppositely. The iterative method is based on action, faster I act, faster I solve the problem. Instead, the logic-based method is thoughtful; any incorrect action takes us away from the solution.

To explain the importance of not acting irrationally, when trying to solve a complex problem, I like to give the following example: imagine that you are a hiker who got lost in the jungle, what are you going to do? If we try to ask this question to a survival expert, he will answer that the best thing to do is to do nothing, and wait for help, because any of your actions will only tire you and put you in danger. The same thing happens in solving complex problems, in which every irrational action is not only useless but it makes us lose energy and time. This example makes us understand how different the two mental approaches are, and how fundamental our mental predisposition is to be successful in situations where a type of problem predominates.

Another useful point of view, to understand the importance of these two mental approaches, is to comprehend why there is so much interest in algorithms based on artificial intelligence. The reason for such interest stems from the fact that through artificial intelligence the algorithms are moving from an iterative approach to a logical approach. In fact, for example, the software developed to play chess, until recently used iterative approaches. In practice, the software simulates all possible combinations and chose the best move. This method had two important limitations: it needed a very powerful computer and could not be applied to games like the “go” in which the possible move combinations are very high. With the advent of artificial intelligence, these virtual players have gone from an iterative approach to a logical approach with incredible results. Google’s Deepmind research team has developed the first software capable of beating the human champion of “go”, on this topic I recommend reading the article published on nature “Mastering the Game of Go without Human Knowledge”.

Now you can understand why the knowledge of these two different mental approaches is fundamental for studying the dynamics that involve our problem-solving skills.

Meditation as mental training to improve the problem-solving skills

In this section, we will try to explain the implications of meditation on problem-solving skills. The term meditation refers to a large number of techniques, even very different from each other, whose task is to bring complete awareness of the present moment. One of the oldest and best-known techniques, and consequently among the most practiced, is called vipassana. The practice of this meditation is performed by sitting cross-legged while remaining completely still in a mental state in which we observe everything that happens. Mainly the observation is directed towards thoughts that tend to manifest themselves and towards one’s breathing.

If we now analyze the two mental approaches, described in the previous paragraph, it is easy to understand how the practice of this type of meditation tends to be in contrast with the iterative method used in solving problems. As described in the previous paragraph, this approach is based on action, in practice, I act as quickly as possible without ever stopping. Hence, sitting still for no purpose represents the opposite of this method.

Consequently, the constant and repeated practice of this type of meditation leads over time to inhibit our propensity to act impulsively. There are many scientific studies on this topic that show how meditation reduces our propensity to multitasking (hyperactivity) and all those irrational and emotional behaviors. This is an important fact because the iterative approach is based on random (irrational) and continuous actions with a strong emotional component.

Meditation in this way modifies our problem-solving skills making us more reflective, consequently increasing our propensity towards the use of the logical approach in solving problems. This result is significant because when we talk about problem-solving we are talking, in most cases, about the ability to solve complex problems. In fact, in a society where technological progress has an exponential trend, our ability to solve problems of this type becomes an increasingly important and requested skill.

Another fundamental aspect to keep in mind, regarding the importance of training the mind to a more reflective approach, is to understand the impact that new technologies have on our minds. To answer this question, we need to understand how most of the applications that are used on smartphones, tablets, etc. are developed.

The main purpose of these applications is to create an addiction, and to do this they take advantage of the iterative approach that the mind uses to solve simple problems. This is done because, in this situation, the person is forced to perform a continuous series of actions, which will correspond to a series of results, the consequence of which is a stimulation of the reward system present in our brain. With this technique, the user of the application will compulsively experience a succession of emotions, the result of which is to create a real addiction.

In conclusion, this type of technology is changing the approach to solving the problems of the new generations, favoring the iterative approach over the logical one. For this reason, it is essential to counteract the conditioning caused by these applications with techniques such as meditation, which inhibit our propensity to solve problems iteratively.

Conclusion

In this article, I have used a simple mathematical logical analysis to relate our problem-solving skills and the practice of meditation. In this way, we find a similarity between two very different realities. On the one hand, we have a scientific formalism, in which through the analysis of a mathematical data the optimal approach is found to solve a class of problems. On the other hand, we have meditation, which represents a topic mainly studied in the philosophical field. So we have two extremely distant points of view which, however, as we have seen, tend to have incredibly similar elements of convergence. Indeed, the practice of meditation represents a way of acting contrary to the iterative approach. Consequently, meditation acts by inhibiting our propensity to act iteratively, making us prefer the logical approach, fundamental in solving complex problems. We have also seen how modern technologies are influencing new generations to hyperactive and compulsive approaches. So, it becomes essential to contrast this type of mental conditioning, with something that goes to act in the opposite direction leading us to act more thoughtful. Meditation, from this point of view, can be seen as a practice that acts on some primary aspects used by the mind in many of its processes, such as problem-solving. In this way, we can partly understand why something so simple has such profound implications in many areas of the brain. For these reasons, I believe that meditation will become an increasingly important formative practice.

How meditation can improve our problem solving skills was originally published in Age of Awareness on Medium, where people are continuing the conversation by highlighting and responding to this story.