Likelihood, P-values ​​and reproducibility crisis

Original author: Eliezer Yudkowsky
  • Transfer
Or: How the transition from the publication of P-values ​​to the publication of likelihood functions will help to cope with the crisis of reproducibility: the personal opinion of Eliezer Yudkovsky.

If Monroe drew comics about 75% of existing interesting problems, and a quarter of my articles are interesting, then what is the probability that sooner or later I will have to look for the KDPV somewhere else?

Translator's commentary: Yudkovsky, the author of HPMOR , the creator of Lesswrong and so on and so forth, stated his position on the use of Bayesian statistics in the natural sciences in the form of a dialogue. Directly classic such a dialogue from antiquity or the Renaissance, with characters setting forth ideas, exchanging barbs interspersed with convoluted arguments and inevitably blunt Simplicio. The dialogue is quite long, about twenty minutes of reading, but in my opinion, it is worth it.

  • Этот диалог был написан сторонником байесовского подхода. Реплики Учёного в нижеприведённом диалоге могут и не пройти идеологический тест Тьюринга на фреквентизм. Возможно, что они не отдают должное аргументам и контраргументам сторонников частотного подхода к вероятности.
  • Автор не рассчитывает, что описанные ниже предложения будут приняты широким научным сообществом в ближайшие десять лет. Тем не менее, это стоило написать.

Если вы ещё не знакомы с правилом Байеса, на сайте Arbital есть подробное введение.

Moderator: Good evening. Today in our studio: A scientist , a practicing specialist in the field of ... chemical psychology or something like that; his opponent is Bayesian , who intends to prove that the crisis of reproducibility in science can be somehow overcome by replacing the P-values ​​with something from Bayesian statistics ...
Student: Excuse me, how do you spell it?
Moderator: ... and, finally, not understanding anything Student to my right.

Moderator: Bayesian , could you start by telling you what the essence of your proposal is?
Bayesian: Roughly speaking, the essence is this. Suppose we have a coin. We throw it up six times and watch the series “OOOOOR” (note: here and hereinafter O - Orel, R - Reshka) . Should we suspect that something is wrong with the coin?
Scientist: No.
Bayesian: Coin here just for example. Suppose we offer a sample of volunteers a plate with two cookies: one with green dressing and one with red. The first five people take the green cookies, and the sixth takes the red. Is it true that people prefer cookies with green dressing, or is it better to consider such a result as random?
Student: I guess you cansuspect that maybe people prefer green topping. At least, psychology students who tend to volunteer to do strange experiments like green breading more. Even after six observations, one may suspect so much, although I suspect that there is some kind of trick here.
Scientist: I think this is not yet suspicious. Many which hypotheses look promising at N = 6, but not confirmed at N = 60.
Bayesian: Personally, I would suspect that our volunteers do not prefer red dressing , or at least prefer it not very much. But in general, I came up with these examples only to show how P-values ​​are considered in modern scientific statistics and what is wrong with them from the Bayesian point of view.
Scientist:Can't you think of a more realistic example with 30 volunteers?
Bayesian: Yes, but the Student doesn’t understand anything.
Student: That's for sure.
Bayesian: So, dear experts: Eagle, eagle, eagle, eagle, eagle, tails. Attention, question: will you call this result “statistically significant” or not?
Scientist: Mr. Leading, it is not significant. Under the null hypothesis that a coin is honest (or with a similar null hypothesis that the color of the dressing does not affect the choice of cookies), the same or more pronounced result can be obtained in 14 out of 64 cases.
Student: Yeah. I understand correctly: This is because we consider the outcomes of OOOOOOO and РРОРРР as “same or more pronounced,” there are a total of 14, and the total of possible outcomes with 6 shots 26 = 64. 14/64 is 22%, which is above 5%, so the result is not considered significant at the level of p <0.05. So?
Scientist: Right. I would also note that in practice, even at the outcome of OOOOOOO, it is not worth stopping the experiment and writing an article about the fact that the coin always falls out of an eagle.
Bayesian: The fact is that if you can stop throwing a coin at any time, you will have to ask yourself: "How likely is it that I will find a moment to stop the experiment in which the number of eagles will look public?" And this is in the P-value paradigm is a completely different story.
Scientist: I meant only that only six experiments - this is not serious, even if we study the color of the cookies. But yes, you are right too.
Student:And why is it important at all, can I stop throwing a coin or can't I?
Bayesian: What a wonderful question.
Scientist: The thing is, P-values ​​are tricky stuff. You can not just take the numbers, throw them into the program and publish what this program will issue. If you decide to flip a coin exactly six times in advance, and then stop regardless of the result, then the result of OOOOOO or PPPPPP will be obtained on average 2 times out of 64, or 3.1% of cases. This is significant at p <0.05. But suppose that in fact you are a false and shameless forger. Or just an incompetent student who himself does not understand what he is doing. Instead of choosing the number of shots in advance, you throw and throw a coin until you get a result that looks statistically significant. TheyIt would be statistically significant if you decided to throw a coin exactly the same amount in advance. But in fact, you did not decide in advance. You decided to stop only after you got the results. So you can not do.
Student: Okay, I read about it somewhere, but I didn’t understand what was wrong with that. This is my research, and I should know better if there is enough data or not.
Scientist:The whole point of P-values ​​is to create a test that the null hypothesis cannot pass. To make sure, in other words, that smoke without fire happens not too often. To do this, you need to organize research in such a way as not to generate “statistically significant” discoveries in the absence of the desired phenomenon. If you flip a coin exactly six times (and decide on this number in advance), then the probability to receive from an honest coin six eagles or six tails is less than 5%. If you throw a coin as many times as you like , and after each throw, recalculate the P-value ( pretending that the number of throws was known in advance), then the chance to get less than p <0.05 is much more than 5% sooner or later . Therefore, such an experiment detects smoke without fire much more often than in 1 out of 20 cases.
Bayesian: Personally, I like to formulate this problem like this: let's say you throw a coin and get OOOOOR. If at the same time you are in the slave only to Allah (for Allah is wise, knowing) the depths of your heart have determined the number of shots in advance , then the result is not significant; p = 0.22. If, after a three-month fast, you brought a vow to Saint Francis to throw a coin until it was all done , then the same result is statistically significant with a quite good p = 0.03. Because the chance that with probabilities of 1: 1 tails will have to wait six or more shots, 1/32.
Student: What?
Scientist:It is rather a parody, of course. In practice, no one will throw a coin until a single tail falls and then stop. But in general, Bayesian is right, P-values ​​do exactly that. Strictly speaking, we are trying to find out how rare is the result obtained among those that we couldreceive. A person who throws a coin before the first tail can get the results {R, OR, OOR, OOOR, OOOOR, OOOOOR ...} and so on. The class of results in which six or more shots are made is {OOOOOR, OOOOOOR, OOOOOOOOO ...}, the total probability of which is 1/64 + 1/128 + 1/256 ... = 1/32. A person throwing a coin exactly six times gets one of the results of the class {РРРРРР, OOOOOR, OOOORO, OOOOORR ...}, in which there are 64 elements. For the purposes of our experiment, OOOOOR is equivalent to OOOORO, OOOROO and others the same. So yes, this is all pretty counter-intuitive. If we really carried out the first experiment - LLCOOR would be a significant result, which is unlikely with an honest coin. And if we carried out the second experiment - OOOOOR would not be significant, because even with an honest coin it happens from time to timesomething like that .
Bayesian: You do not accidentally worry that the results of the experiment depend on what you think?
Scientist: This is a matter of conscience. Any kind of research will cost little if you lie about their results, that is, literally telling the truth about which side the coin fell out. If you lie about what kind of experiment was conducted- the effect will be the same. So you just have to honestly say what rules were used for throwing. Of course, the contents of the scientist’s head are less obvious than the side of the coin. Therefore, it is always possible to tweak the analysis parameters, not to write how the number of subjects was determined, choose the statistical test that confirms your favorite hypothesis ... You can think of a lot of things if you wish. And it will be easier than falsifying the original data. In English, this is called p-hacking. And in practice, of course, much less obvious ways of creating smoke without fire are used than the stupid null hypothesis invented after the fact. This is a serious problem, and to some extent a crisis of reproducibility is associated with it, although it is not clear to which one.
Student:Does this ... sound reasonable? Probably, this is one of those things that you need to deal with and search through a bunch of examples for a long time, and then everything will become clear?
Bayesian: No.
Student: I mean?
Bayesian: In the sense of "Student, you were right from the very beginning." If what the experimenter thinks doesn’t affect the side by which the coin falls, then his thoughts should not influence the fact that the results of the throw inform us about the universe. My dear Student, the statistics you are taught is nothing more than a heap of crooked crutches that you don’t even bother to do internally consistent. For God's sake, she gives out differentwrong results depending on what is going on in your head! And this is a much more serious problem than the tendency of some scientists to slightly lie in the “Materials and Methods”.
Scientist: This is ... a serious statement, to say the least. But tell me, I ask you: what are we, the unfortunate, to do?
Bayesian:Analyze as follows: this particular result of OOOOOP can be obtained with six shots of a perfectly balanced coin with a probability of 1/64, or approximately 1.6%. Suppose we already suspected that our coin was balanced imperfectly. And not just imperfectly, but in such a way that it fell out on average five out of six times. This, of course, is a wild simplification, but I will move on to realistic hypotheses a bit later. And so, this hypothetical shulersky coin gives out OOOOOR sequence with probability (5/6) 5 * (1/6) 1 . This is about 6.7%. So we have two hypotheses: "The coin is the most common" and "The coin falls out of an eagle in 5/6 cases." This particular result is 4.3 times more likely in the second case.than in the first. The probability of the sequence of OOOOOR for another hypothetical cheat coin, which in 5 cases out of six falls on a tail, is 0.01%. So if someone suddenly thought that this second coin is in front of us, then we now have a good argument against his hypothesis. This particular result is 146 times more likely for an honest coin than for a coin that falls out of an eagle only once out of six. Similarly, our hypothetical lovers of red cookies would be much less likely to eat green.
Student: Okay, I seem to understand math. But, frankly, I do not catch, what is its meaning.
Bayesovets: Let me explain, but first pay attention to the following: the results of my calculations diddo not depend on why the coin was planted exactly six times. Maybe after the sixth shot you decided that there was already enough data. Maybe after a series of five shots, Namagiri Tayyar appeared to you in a dream and advised you to throw the coin again. Coin anyway. The fact remains: this particular series of OOOOOR for an honest coin is four times less likely than for a coin that falls out of an eagle five times out of six.
Scientist: I agree, your calculations have at least one useful property. What's next?
Bayesian:And then you publish the results in the journal. It is desirable along with the raw data, because then anyone can calculate the plausibility of any hypothesis. Suppose someone unexpectedly became interested in the hypothesis “A coin falls out of an eagle 9 times out of 10, not 5 times out of 6”. In this case, the series of observations of OOOOOR has a probability of 5.9%, which is slightly less than our hypothesis about five eagles out of six throws (6 , 7%), but 3.7 times more than the hypothesis that the coin is perfectly balanced (1.6%). It is impossible, and not necessary, to come up with all possible hypotheses in advance. It is enough to publish complete data - then anyone who has a hypothesis can easily calculate the likelihood he needs. The Bayesian paradigm requires the publication of raw data, because it focuses precisely on a specific result., and not on some class of supposedly identical outcomes.
Scientist: In this I agree with you, the publication of complete data sets is one of the most important steps to overcome the reproducibility crisis. But personally, I don’t understand what I should do with all these “And so much more likely than B”.
Student: Me too.
Bayesian: It's not entirely trivial ... did you read our introduction to Bayes' rule ?
Student: Great. Here are just another three-page statistical textbook, and I did not have enough.
Bayesian: You can actually read it in an hour . It's just that it's literally not trivial., that is, requires an explanation. But okay, in the absence of a full-fledged introduction, I will try to think of something. Most likely, it will sound reasonable - and the logic is indeed correct - but not a fact that is self-evident. Go. There is a theorem that proves the correctness of the following reasoning:
(Bayesian gains air)
Bayesian:Suppose Professor Plume and Miss Scarlet are suspected of murder. After examining the biographies of both, we assume that it would be twice as easy for a professor to kill a man than Miss Scarlet. With this assumption, let's start. It turns out, however, that the deceased was poisoned. We know that if Professor Plume is about to kill someone, he will use poison with a probability of 10% (and in 9 cases out of 10 he will prefer, for example, a revolver). Miss Scarlet, if she decides to kill, uses poison with a probability of 60%. In other words, professor’s use of poison is six times less likely than Miss Scarlet’s use of poison. Since we have new information, namely the method of murder, we must update our assumption and assume that Plume is about three times less likely killer: 2 * 1/6 = 1/3.
Student:Not sure I understood that. What does the phrase "Professor Plume mean three times less likely murderer than Miss Scarlet"?
Bayesian: It means that if we do not have other suspects, then the probability that the victim was killed by Plume is 1/4. The remaining 3/4 make up the probability that the killer is Miss Scarlet. Therefore, the probability of the professor’s guilt is three times lower than that of Miss Scarlet.
Scientist: And now I want to know what you mean by "probability of guilt." Plume either committed the murder, or he did not commit it. We cannot look at the assassination sample and find that Plume is indeed guilty of a quarter of them.
Bayesian:I was hoping not to get into it, but oh well. My good Scientist, I mean that if you offered me a bet with 1: 1 bets on whether Plume killed the victim or not, I would bet that he did not. But if under the terms of the betting, I would pay you $ 1 in case of his innocence, and you pay me $ 5 in the event of his guilt, I would gladly bet on the guilt. The 2012 presidential election was held only once, and Obama's Probability of Victory is the same conceptually vague thing as Plume's Guilt Probability. But if on November 7 you were offered to put $ 10 on Obama and promised $ 1000 in case of his victory, then you would hardly refuse such a bet. In general, when prediction markets and large liquid pools of bets accept 6: 4 bets on an event, this event occurs in about 60% of cases. Markets and poolswell calibrated by probabilities in this range. If they were calibrated poorly, that is, if the events that are being bid at 6-4, happened 80% of the time, then someone would have noticed and enriched themselves at the expense of such rates. At the same time, he would raise the price of the rate until the market becomes well calibrated. And since events with a market estimate of a probability of 70% do occur about 7 times out of 10, I don’t understand why to insist that such a probability does not make sense.
Student: I admit, it sounds convincing. But for sure it only seems to me, and in fact there are a whole bunch of tricky arguments for and against.
Bayesian: There really is a bunch of argumentsBut the general conclusion from it is that your intuitive understanding is pretty close to the truth.
Scientist: Well, we'll come back to this. But what if there are two agents, both in your terms “well calibrated”, but one of them claims “60%” and the other says “70%”?
Bayesian:Suppose I throw a coin and do not see which side it fell. In this case, my ignorance is not information about a coin, it is information about me. It exists in the head, not in the surrounding world, just as the white spots on the map do not mean that there is no territory in this place. If you looked at the coin, but I didn’t, it’s quite reasonable that you and I are in different states of uncertainty about it. Considering that I am not one hundred percent certain, it makes sense for me to express my insecurity in terms of probability. There are three hundred theorems that claim that if someone’s expression of uncertainty is notin fact, the distribution of probability is what he needs in general. For some reason, it always happens that if an agent’s thinking under conditions of uncertainty violates any of the standard axioms of probability theory, the earth opens up, water turns into blood, and dominated strategies and obviously losing stakes spill from heaven.
Scientist: Well, here I was wrong. We will come back to this too, but first of all, answer my question: what should we do with credibility after we received them?
Bayesian: According to the laws of probability theory, these likelihoods areproof of. It is they who force us to change our prior probabilities from 2: 1 in favor of Plume to 3: 1 in favor of Scarlet. If I have two hypotheses and the likelihood of data for both, then I should change my opinion in the manner described above. If I change it somehow differently - then the heavens open, strategies pour in, and so on. Bayes theorem: this is not just a statistical method, it is a LAW.
Student: I apologize, but I still do not understand. Suppose we are conducting an experiment. And, let's say, the results obtained six times more likely if Herr Troopa was killed by Professor Plume than they would have been if Miss Scarlet was the killer . The student obviously confused the likelihood of using poison by two killers.. Arrest us professors or not?
Scientist: I think, first we need to come up with a more or less realistic a priori probability, for example, " a priori I believe that the probability of killing the Troupe by Plume is 20%." Then it must be multiplied by a likelihood ratio of 6: 1, and the ratio of a posteriori probabilities of 3: 2, that Plume did kill the Troupe. After that, you can declare that Plume is guilty with a probability of 60%, and then let the prosecutor's office understand.
Bayesovets: None . For heaven's sake! Do you really think Bayesian statistics work that way?
Scientist:Does she work wrong? I have always believed that its main advantage is that it gives us a posteriori probabilities, which P-values ​​do not really give, and the main drawback is that it requires a prior probabilities. Since they have to be taken more or less from the ceiling, the correctness of a posteriori probabilities can be challenged to the end of time.
Bayesian: Articles need to publish likelihood . More precisely, it is necessary to publish raw data and calculate for them a few likelihoods of interest. But certainly not a posteriori probabilities.
Student: I'm confused again. What is a posteriori probabilities?
Bayesian: A posteriori probability- This is a statement like "With a probability of 60% of Herr. The troupe was killed by Professor Plume." As my colleague has already noted, such statements do not follow from P-values. And, in my opinion, they have no place in experimental articles, because these are not the results of an experiment .
Student: But ... ok, Scientist, a question for you: let's say we got results with p <0.01, that is, something with a probability of less than 1% with the null hypothesis "Professor Plume did not kill Herr Troupe". Arrest us or not?
Scientist:First, it is not a realistic null hypothesis. Most likely, the null hypothesis will be something like "Nobody killed Herr Troupe" or "all suspects are equally guilty." But even if the null hypothesis described by you worked, even if we could reject Plume's innocence with p <0.01, it would still be impossible to say that Plume is guilty with a probability of 99%. P-values ​​of this are not reported to us.
Student: And what do they report then?
Scientist: They report that the observed data are included in a certain class of possible outcomes, and that the results of this class are observed in less than 1% of cases if the null hypothesis is correct. More p-value means nothing. You can not just go and go from p <0.01 to "Professor Plume is guilty with a probability of 99%." A Bayesian is more likely to better explain why. In general, in science one cannot interpret one thing as something else. Figures denote exactly what they denote, no more and no less.
Student: Generally excellent. At first I did not understand what to do with plausibility, and now I still do not understand what to do with P-values. What experiment is required to finally send Plume to prison?
Scientist: In practice? If another pair of experiments in other laboratories confirms his guilt with p <0.01, then most likely he is really guilty.
Bayesian:A “reproducibility crisis” is when the matter is later raised and it turns out that he did not commit murder.
Scientist: In general, yes.
Student: Somehow it turns out unpleasant.
Scientist: Life is generally an unpleasant thing.
Student: So ... Bayesian, you probably have a similar answer? Something like the fact that if the likelihood ratio is large enough, say, 100: 1, then in practice can we assume that the corresponding hypothesis is true?
Bayesian:Yes, but it is somewhat more complicated. Suppose I throw a coin 20 times and get OOOROOOROROROROOOOOOORROR. The catch is that the plausibility of the hypothesis “The coin is guaranteed to issue a sequence of OOOROOOROROROOOOOOORROR” above the plausibility of the hypothesis “Coin equiprobably drops out like an eagle or tails” is about a million times. In practice, if you did not hand me this hypothesis in a sealed envelope prior to the start of the experiment, I will consider it strongly retrained. I will have to give this hypothesis a penalty for complexity of at least 2 : 20 : 1, because the sequence description alone takes 20 bits. In other words, lower the prior probability to such a degree that it more than compensates for the likelihood advantage. And this is not the only underwater rock. But nevertheless, if you understand how and why the Bayes rule works - then in each case you can understand along the way. If the ratio of credibility for Plume versus any other suspect is 1000: 1, and there are only six suspects in general, then it can be assumed that the a priori probability was hardly much more than 10: 1 against the fact that he was a murderer. If so, then we can assume that he is guilty with a probability of 99%.
Scientist: But still, the article is not worth writing ?
Bayesian: Right. How to formulate ... The key condition for Bayesian analysis is that the wholerelevant information. You can’t exclude data from analysis just because you don’t like it. This is actually the key condition of science as such, regardless of the statistics used. There are a lot of articles, the conclusions of which turned out only because some factor was not taken into account or the sample was unrepresentative in some parameter. I'm talking about what? And besides, how do I (as an experimenter) know what “all relevant information” is? Who am I to calculate a posteriori probabilities? Maybe someone published an article in which there are additional data and additional likelihoods that I should have taken into account, but I have not read it yet. So I just publish my data and my likelihood functions - and that’s it! I can not say that I considered everythingArguments and now I can offer reliable a posteriori probabilities. And even if I could, then in a week another article may come out, and these probabilities will become obsolete.
Student: Roughly speaking, the experimenter just has to publish his data, calculate some likelihood for them and that's all? And then someone else will decide how to deal with them?
Bayesian: Someone will have to choose a priori probabilities — equal, or with maximum entropy, or with fines for complexity, or some other — then try to collect all possible data, calculate likelihoods, make sure that the result is not crazy , and so on. other And they still have to recount if a new article comes out in a week.
Student:It sounds quite laborious .
Bayesian: It would be much worse if we took up the meta-analysis of P-values. Updating Bayesian probabilities is much easier. It is enough to simply multiply the old a posteriori probabilities by the new likelihood functions and normalize them. Everything. If experiment 1 gives a likelihood ratio of 4: 1 for hypotheses A and B, and experiment 2 gives a likelihood ratio of 9: 1 for them, then together they give a ratio of 36: 1. That's all.
Student: And you can't do that with P-values? One experiment with p = 0.05 and another experiment with p = 0.01 does not mean that actually p <0.0005?
Scientist: No .
Bayesian:Dear viewers, please pay attention to my arrogant smile.
Scientist: But I still worry about the need to invent a priori probabilities.
Bayesian: Why does it bother you more than the fact that everyone decided to consider one experiment and two replications with p <0.01 criterion of Truth?
Scientist: You want to say that the choice of a priori values ​​is no more subjective than the interpretation of P-values? Hm I wanted to state that a requirement, say, p <0.001 should guarantee objectivity. But then you will answer that the figure 0.001 (instead of 0.1 or 1e-10) is also sucked from the finger.
Bayesian:And I will add to this that it is less effective to require any arbitrary P-value than to suck a prior probability from the same finger. One of the first theorems that threatened violators of probability axioms with Egyptian punishments was proved by Abraham Wald in 1947. He tried to describe all the acceptable strategies , calling the strategy some way to react to what you are observing. Of course, different strategies under different circumstances can be more or less profitable. Acceptable strategyHe called one that is not dominated by any other strategy under all possible conditions. So, Wald discovered that the class of acceptable strategies coincides with the class of strategies that contain a probability distribution, update it based on observations using Bayes' rule, and optimize the utility function.
Student: Excuse me, is it possible in Russian?
Bayesian: If you do something because of what you observe and get more or less money, for example, depending on what the real world is, then one of two things is true. Either your strategy in some sense contains a probability distribution and updates it according to Bayes rule, orthere is some other strategy that is never inferior to yours and sometimes surpasses it. That is, for example, you say: “I will not quit smoking until I see an article proving the connection between smoking and cancer at p <0.0001”. At least theoretically, there is a way to say “In my opinion, the link between smoking and cancer exists with a probability of 0.01%. What are your likelihoods? ”, Which will be no worse than the first formulation, no matter what a priori probabilities of the existence of such a connection.
Scientist: Really?
Bayesian:Yeah. The Bayesian revolution began with this theorem; since then it has been slowly gaining momentum. It is worth noting that Wald proved his theorem a couple of decades after the invention of P-values. This, in my opinion, explains how it happened that all modern science was tied up with obviously ineffective statistics.
Scientist: So you propose to throw out P-values ​​and instead publish only likelihood ratios?
Bayesian: In short, yes.
Scientist: Something I do not really believe in ideal solutions that are suitable for any conditions. I suspect - please do not consider it an insult - that you are an idealist. In my experience, in different situations different tools are needed and it would be unwise to throw out all but one.
Bayesian:Well, I am ready to explain what I am an idealist and what is not. Likelihood functions alone will not resolve the reproducibility crisis. It cannot be completely resolved by simply ordering everyone to use more efficient statistics. The popularity of open access journals does not depend on the choice between plausibility and P-values. Problems with the review system also do not depend on it.
Scientist: And everything else, therefore, depends?
Bayesovets: Not everything, but they have a lot what to help . Let's count.
Bayesian:First of all. Likelihood functions do not force us to draw a line between “statistically significant” and “insignificant” results. An experiment cannot have a “positive” or “negative” outcome. What is called the null hypothesis is now just one of the hypotheses that is not fundamentally different from all the others. If you throw a coin and get an OORRRROOO - one cannot say that the experiment could not "reject the null hypothesis at p <0.05" or "reproduce the previously obtained result." He merely added data that supports the hypothesis of an honest coin against the 5/6 eagles hypothesis with a likelihood ratio of 3.78: 1. So with the massive acceptance of Bayesian statistics, the results of such experiments will be less likely to go to the table. Not at all, because the editors of magazines have unexpected results that are more interesting than honest coins, and this must be dealt with directly. But P-values ​​do not just do not struggle with this approach, they are hisstimulate ! It is because of him that p-hacking exists at all. So the transition to the likelihood will not bring happiness to all and a gift, but it will definitely help .
Bayesian: Secondly. The likelihood system emphasizes the importance of the source data much more and will stimulate their publication wherever possible, because Bayesian analysis is based on how likely these particular results are in a particular model. The system of P-values, on the contrary, forces the researcher to consider the data as just one of the members of the class of “equally extreme” results. Some scientists like to keep all their precious data with them; it's not just statistics. But P-values stimulateand this, because for the article, it’s not the data itself that is important, but whether they belong to a particular class. After this is established, all the information contained in them as if collapses into a single bit of "significance" or "insignificance."
Bayesian: Thirdly. From the point of view of probability theory, from the Bayes point of view, different magnitudes of effects are different hypotheses. This is logical, because they correspond to different likelihood functions and, accordingly, different probabilities of the observed data. If one experiment found an effect size of 0.4, and another experiment found a “statistically significant” value of the same effect of 0.1, then the experiment did not reproduce.and we don't know what the effect really is. This will avoid a fairly common situation where the magnitude of the “statistically significant” effect all decreases and decreases with increasing sample size.
Bayesian: Fourth. Likelihood functions greatly simplify data integration and meta-analysis. They may even help us notice that data is collected in heterogeneous conditions or that we do not consider the true hypothesis. In this case, either all the functions will be close to zero for all possible parameters, or the best hypothesis will give a much lower likelihood on the combined data than it itself predicts . A more rigorous approach to reproducibility allows you to quickly understand whether such an experiment can be considered as a repetition of such and such.
Bayesian: Fifth. Likelihood functions do not depend on what they think about them. These are objective statements about the data. If you publish likelihood values, then there is only one way to deceive the reader - to falsify the data itself. P-hacking will not work.
Scientist: This is what I strongly doubt. Suppose I decide to convince you that the coin often drops out of an eagle, although in fact it is honest. I will take a coin, I will throw it until I accidentally get a little more eagles, and then I’ll stop. What then?
Bayesian: Go ahead. If you do not falsify the data, you will not deceive me.
Scientist:The question was about what would happen if I checked the likelihood ratio after each roll and stopped as soon as it supported my favorite theory.
Bayesian: As an idealist, seduced by the deceptive beauty of the theory of probability, I answer you: as long as you give me honest raw data, I can and must do only one thing - multiply according to Bayes' rule.
Scientist: Really?
Bayesian: Seriously.
Scientist: So you don't care that I can check the likelihood ratio until I like it?
Bayesian: Go ahead.
Scientist: Okay. Then I will write a script on Python, which simulates a throw of an honest coin to, let's say, 300 times, and see how often I manage to get a 20: 1 ratio in favor of the “coin eagle drops out in 55% of cases” ... What?
Bayesian: Just a funny coincidence. When I first found out about this and doubted that the likelihood relationship could not be deceived in any tricky way, I wrote the same program on Python. Later, one friend of mine also learned about the likelihood relationship and also wrote the same program, also for some reason on Python . He launched it and found that the 20: 1 ratio for the hypothesis "55% of the eagles" was found at least once in 1.4% of the series of shots. If you require, for example, 30: 1 or 50: 1, their frequency drops even faster.
Scientist:If you count your one and a half percent P-value, it looks good. But this is a very rude way to trick analysis; Perhaps there are more complex and effective?
Bayesian: I was ... about five years old, probably, if not less, when I first learned about addition. One of my earliest memories. I sat, added 3 to 5 and tried to think of some way not to get 8. Which, of course, is very nice and generally an important step towards understanding what addition is (and mathematics in general). But now this is exactly what is nice, because we are adults and we understand that 5 plus 3 is inevitably equal to 8. The script, which constantly tests the likelihood ratio, does the same thing that I did in childhood. Having understood the theory, I realized that attempts to deceive Bayes' rule are obviouslydoomed. It's like trying to decompose 3 in some tricky way into 2 and 1 and add them separately to 5, or try to add first 1, and only then 2. Neither that nor 7 or 9 will work. The result of the addition is a theorem , and it doesn't matter what sequence of operations we perform. If it is really equivalent to adding 3 to 5, then nothing can be obtained at the output, except 8. Theorem of probability theory is also a theorem. If the script could really work, it would mean a contradiction in probability theory, which means a contradiction in Peano arithmetic, on which the analysis of probabilities is constructed using rational numbers. What you and I tried to do was exactly as difficult as adding 3 and 5 in standard arithmetic axioms and getting 7.
Student: Uh, why?
Scientist: I did not understand either.
Bayesian: Let e denote observations, H denote the hypothesis ,! X denotes "not X", P ( H ) denotes the probability of the hypothesis, and P ( X | Y ) denotes the conditional probability of X provided that Y is true. There is a theorem showing that

P (H) = P ( H | e ) * P ( e )) + (P ( H |! e ) * P ( ! e )

Therefore, for the probability functions there is noarbitrarily complex analogue of p-hacking, apart from data falsification, because no procedure known to a Bayesian agent will force it to update its a priori probabilities in a deliberately incorrect direction. For every change that we can get from watching an e , there is a inverse variation that can be expected from the observation ! An e .
Student: What?
Scientist: I did not understand either.
Bayesian: Okay, let's put it off until math and see ... yes, to the crisis of reproducibility. The scientist said that he is suspicious of ideal universal solutions. But in my opinion, the transition to likelihood functions should reallysolve many problems at once. Suppose ... now come up with. Suppose a certain corporation has major accounting problems. These problems are due to the fact that all accounting uses floating point numbers; and it would be still half the problem, but three different implementations are used (approximately in one third of the corporation each), so it turns out that God knows all. Someone, for example, takes 1.0, adds 0.0001 a thousand times, then subtracts 0.1 and gets 0.999999999999989. Then he travels to another floor, repeats the calculations on their computers and gets 1.000000000000004. And everyone thinks that this is necessary. And the error, suppose, is really HUGE, all three implementations are the fruit of an unnatural union of cave paintings and Roman numerals. So due to the differences between them, it is possible to get quite tangible differences in the results. Of course everyone picks up the sales in such a way that quarterly reports come together. Therefore, it is considered a good result if the budget of the department does not contradict even to itself, and the department of cognitive priming is likely to go bankrupt 20 years ago. And here I go out, all in white, and I say: “Good afternoon. And what if, instead of your three implementations, you will use this cool thing that cannot be manipulated in this way and that will solve half of your problems. ”
( Bayesian , in the voice of the Scientist ) : “I am suspicious of such universal solutions,” the chief accountant replies to me. “Do not consider it an insult, but you, my friend, are an idealist. In my experience, different entries of floating point numbers are well suited for different operations, so you shouldn’t immediately throw out all the tools except one. ”
Bayesian: To which I answer him:“ Maybe it sounds too bold, but I'm going to demonstrate you are perfecta representation of fractions in which the results do not depend on the order in which you add numbers or on whose computer the calculations take place. Maybe in 1920, when your system was just being created, it required too much memory. But now is not the year 1920, you can afford not to save computing resources. Especially since you have there how many, 30 million bank accounts? This is really nonsense. Yes, my presentation has its flaws. For example, square roots are taken much more difficult. But how often, honestly, do you need to take the square root of someone's salary? For most real-world tasks, this system is not inferior to yours, and besides, it cannot be fooled without faking the input values. ”Then I explain to them, how to represent an integer of arbitrary length in memory and how to represent a rational number in the form of a ratio of two integers. That is what we would now call a self-evident way of representingreal rational numbers in computer memory. The only and unique system of theorems about rational numbers, for which floating-point numbers are just an approximation. And if you handle an unfortunate 30 million bills; if in practice your approximations do not converge with each other, nor with themselves; if they also allow everyone to steal your money; if, finally, the yard is not 1920 and you can afford normal computers, then the need to transfer accounting to real rational numbers is pretty obvious. In the same way, Bayes' rule and its corollaries are the only system of theorems about probabilities based on axioms and rigorously proved. And so p-hacking does not work in it.
Scientist: This is ... bold. Even if everything you say is true, there are still practical difficulties. The statistics that we use now has been forming for more than a decade; she proved her worth. How did your bright Bayes path prove to be in practice?
Bayesian: It is almost never used in the natural sciences. In machine learning, where, as it is more modest to say, it is quite easy to notice that the model is incorrect - because the AI ​​based on it does not work - so, in machine learning I last saw the frequency approach to probability ten years ago. And I can't remember a single in which the AI ​​would consider the P-value of some hypothesis. If probability at all somehow appears in the study, then it is almost certainly Bayesian. If something is classified by unitary codes, then the cross entropy is minimized, but not ... I do not even know what could be analogous to the P-values ​​in AI. I would venture to suggest that this is what it is. The statistics in machine learning either works, or does not, and it is immediately obvious: the AI ​​either does what it should or tupit. And in the natural sciences, all are primarily needed publications. Since it so happened that it is customary to specify P-values ​​in the articles, and for non-reproducible results we don’t punish - we have what we have.
Scientist:So you are a mathematician or programmer rather than a natural scientist? For some reason, this does not surprise me. I have no doubt that a more successful statistical apparatus may exist, but the experience of using P-values ​​is also worth something. Yes, now they are often twisted in one way or another, but we know how to do it, and begin to understand how to deal with it. The pitfalls of the P-values ​​are at least known. In any new system they will be too. But that's exactly where - it turns out only after decades. Perhaps they will be even more dangerous than the present.
Bayesian:Yes, thievish accountants will probably come up with some new exciting manipulations with rational numbers. Especially in those cases when the exact operations will still be too computationally expensive and will have to be somehow approximated. But I still believe that if the same experimental psychology right now breaks the crisis of reproducibility, and if this crisis is clearly associated with the use of P-values, which, frankly, are nothing more than a bunch of conflicting crutches - then you should at least try to use more rational method. Although I also do not call for all to demolish and rebuild again. In practice, you can start to abandon the P-values ​​in any one area (at least in psychology) and see what happens.
Scientist:And how are you going to persuade psychologists to such an experiment?
Bayesian: I have no idea. Frankly, I do not really expect anyone to change anything. Most likely, people will simply use the P-values ​​until the end of the ages. So it goes. But there is a chance that the idea will still be popular. I was pleasantly surprised by how quickly Open Access took root. I was pleasantly surprised by the fact that the crisis of reproducibility was generally noticed, and moreover, people care about it. Perhaps, the P-values ​​will still be pulled out onto the market square and upturned with a large crowd of people ( note: at least one psychological journal in 2015 refused to test null hypotheses). If so, I will be pleasantly surprised. In this case, it turns out that my work on popularizing Bayesian rules and likelihood was not in vain.
Scientist: It may also turn out that like no one likes experimental science , and P-values ​​are all considered convenient and useful.
Bayesian: If the university course of statistics was so monstrous that when one thinks about the theory of probability, they become shaky, then yes, the changes will have to come from the outside. I personally hope that our dear Student will read a short and rather fascinating introduction to Bayesian probability theory., compare it with his awesome textbook on statistics and will be begging you for the next six months, "Well, please, can I just consider the likelihood and everything, please, well, allow me."
Student: Uh ... well, I read him first, okay?
Bayesian: Dear Student, think about your choice. Some changes in science occur only because students grow up surrounded by different ideas and choose the right ones. This is Max Planck's famous aphorism, and Max Planck will not say nonsense. Ergo, the ability of science to distinguish bad ideas from good ones depends solely on the students' intelligence.
Scientist: Well, this is already ...
Moderator: And this is where we complete our transmission. Thanks for attention!

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