However, few if any care to offer a technical explanation of what they mean by the term “probability”. However, the issue is that credible intervals (typically highest probability density intervals (HPDI)), coincide with frequentist intervals under conditions encountered in A/B testing. “Statistical tests give indisputable results.” This is certainly what I was ready to argue as a budding scientist. That would be an extreme form of this argument, but it is far from unheard of. It only has possibilities: it could be true or false, or maybe just partially or conditionally true. There again, the generality of Bayes does make it easier to extend it to arbitrary problems without introducing a lot of new theory. No known good statistic would be expected to show an increased probability with an increase in the sample size of an A/A test. Pierre Simon Laplace. The bandwagon of the 2000's (model selection, small n large p, machine learning, false discovery rate, etc.) For posterior odds to make sense, prior odds must make sense first, since the posterior odds are just the product of the prior odds and the likelihood. You can connect with me via Twitter, LinkedIn, GitHub, and email. With Bayes' rule, we get the probability that the coin is fair is $$\frac{\frac{1}{3} \cdot \frac{1}{2}}{\frac{5}{6}}$$. Absence of evidence vs evidence of absence Background. Bayesian and non-Bayesian approaches to statistical inference and decision-making are discussed and compared. The framing of the question does not refer to any particular tool or methodology, and purposefully has no stated probability for day one, as stating a probability might bias the outcome depending on the value. In the Bayesian view, a probability It should be noted that the supposedly intuitive nature of Bayesian estimates is the basis on which it is argued that Bayesian statistical results are easier to interpret and are less prone to erroneous interpretations. In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; the model with the lowest BIC is preferred. One to one prior odds means that one starts from the position that it is just as likely that the current control, which was probably improved over hundreds of iterations, is worse than whatever new version is being proposed as it is that it is better. (Conveniently, that $$p(y)$$ in the denominator there, which is often difficult to calculate or otherwise know, can often be ignored since any probability that we calculate this way will have that same denominator.) The Bayesian looks at the P(parameter|data) the … In fact Bayesian statistics is all about probability calculations! Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown. At a magic show or gambling with a shady character on a street corner, you might quickly doubt the balance of the coin or the flipping mechanism. Bayesian statistics rely heavily on Monte-Carlo methods. A statistical software says there is some ‘probability’ that the variant is better than the control, where ‘probability’ means whatever you intuitively understand it to mean (there is no technical documentation about the statistical machinery). Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event.The degree of belief may be based on prior knowledge about the event, such as the results of previous … I argue that if it were so intuitive, the majority of above average users of statistics in an experimental setting would not have had the exact opposite expectation about the outcomes of this hypothetical A/A test. Namely a uniform distribution, usually Beta(1, 1). 's Bayesian Data Analysis, which is perhaps the most beautiful and brilliant book I've seen in quite some time. Are equal prior odds reasonable in all situations (as these tools assume)? Option C is the one which corresponds to what a Bayesian would call posterior probability. Option B is the answer one would expect from someone who considers the hypothesis to be either true or false which corresponds to the frequentist rendering of the problem. 2. Georgi Georgiev is a managing owner of digital consultancy agency Web Focus and the creator of Analytics-toolkit.com. This does not stop at least one vendor from using informative prior odds based on unknown estimates from past tests on their platform. The qualitative nature of the sample means it is more likely that the respondents have been exposed to Bayesian logic and the Bayes rule itself, or that they have been using Bayesian tools such as Optimize for assessing A/B tests (online controlled experiments). I don’t mind modeling my uncertainty about parameters as probability, even if this uncertainty doesn’t arise from sampling. Here is my why, briefly. Machine learning is a broad field that uses statistical models and algorithms to automatically learn about a system, typically in the service of making predictions about that system in the future. Any apparent advantages of credible intervals over confidence intervals (such as unaccounted for peeking) rest on the notion of the superiority of the Bayesian concept of probability. A pragmatic criterion, success in practice, as well as logical consistency are emphasized in comparing alternative approaches. Do these odds make any sense to you in practice? That original belief about the world is often called the "null hypothesis". The scale for these was from 1 to 10 ranging from “Minimal or no experience” to “I’m an expert”. Again, in an A/A test, the true value of such a ‘probability’ would be zero. is entirely non-Bayesian so the prospects for a Bayesian utopia seem problematic until someone figures out how to make Bayesianism scale to big data (at least as well as the existing competion does). 3. On the flip side, if a lot of qualitative and quantitative research was performed to arrive at the new version, is it really just as likely that it is worse than the current version as it is that it is an actual improvement over the control? It is evident that Bayesian probability is not “exactly what it sounds like”, despite the cheerful statements made by Google Optimize and many other supporters of Bayesian methods in online A/B testing and beyond. Section 1 and 2: These two sections cover the concepts that are crucial to understand the basics of Bayesian Statistics- An overview on Statistical Inference/Inferential Statistics. For example, the probability of a coin coming up heads is the proportion of heads in an infinite set of coin tosses. I invite you to read it in full. Notice that when you're flipping a coin you think is probably fair, five flips seems too soon to question the coin. The results from 60 real-world A/A tests ran with Optimize on three different websites are shown above. The poll consisted of asking the following question: “On day one an A/A test has 1000 users in each test group. a probability of 50% on day one might bias respondents to replace ‘probability’ with ‘odds’ in their mind for the context of the poll and such priming would be undesirable given that the meaning of ‘probability’ is the subject of the question. Tests 1-20 and 60-80 had hundreds of thousands of users and their estimates are closer to 50% whereas tests 120-140 had around 10,000 users per arm hence the wider disparity in the outcomes. I’m simply trying to get an estimate of the intuitive understanding of ‘probability’ in relation to a piece I’m working on.”, except on Twitter (where it got least noticed). This post was originally hosted elsewhere. This video provides an intuitive explanation of the difference between Bayesian and classical frequentist statistics. There were also two optional questions serving to qualitatively describe the respondents. This website is owned and operated by Web Focus LLC. The interpretation of the posterior probability will depend on the interpretation of the prior that went into the computation, and the priors are to be construed as conventions for obtaining the default posteriors. In any particular one? “probability of B beating A”, etc.. The issue above does not stop Bayesians as they simply replace the technical definition of ‘probability’ with their own definition in which it reflects an “expectation”, “state of knowledge”, or “degree of belief”. With 1,000 users the odds are likely to remain roughly the same as the prior odds. The non-Bayesian approach somehow ignores what we know about the situation and just gives you a yes or no answer about trusting the null hypothesis, based on a fairly arbitrary cutoff. However, this does not seem to be a deterrent to Bayesians. As explained above, this corresponds to the logic of a frequentist consistent estimator if one presumes an estimator can be constructed for “‘probability’ that the variant is better than the control”. Various arguments are put forth explaining how posteri… Bayesian statistics has a single tool, Bayes’ theorem, which is used in all situations. Back with the "classical" technique, the probability of that happening if the coin is fair is 50%, so we have no idea if this coin is the fair coin or not. Consider the following statements. They would have been surprised that a 10-fold increase in the amount of data does not nudge the ‘probability’ estimate closer to the true probability and that it is in fact expected to behave in that same way with any amount of data. Perhaps this is the logical way out which would preserve the Bayesian logic and mathematical tooling? All Bayesian methods are subjective, but so are the non-Bayesian ones as well. Whereas I’ve argued against some of the above in articles like “Bayesian vs Frequentist Inference” and “5 Reasons to Go Bayesian in AB Testing – Debunked”, this article will take the intuitiveness of the Bayesian approach head on. Going in this direction would result in mixing of the highest paid person’s opinion (HiPPO) with the data in producing the posterior odds. All 61 respondents also responded to the optional questions for which I am most grateful. His 16 years of experience with online marketing, data analysis & website measurement, statistics and design of business experiments include owning and operating over a dozen websites and hundreds of consulting clients. The above definition makes sense superficially. Required fields are marked *. So, ‘probability of a hypothesis’ is a term without a technical definition which makes it impossible to discuss with any precision. Can the ‘probability to be best’ estimator be salvaged in its current form by simply replacing ‘probability’ with ‘odds’? 40 participants out of 61 (65.6%, one-sided 95%CI bound is 55.6%) favored an interpretation according to which the probability, however defined, should decline as sample size increases. Wouldn’t it generally be expected to have a much higher probability of being better than the new version proposed? A hypothesis is, by definition, a hypothetical, therefore not an event, and therefore it cannot be assigned a probability (frequency). And the Bayesian approach is much more sensible in its interpretation: it gives us a probability that the coin is the fair coin. So it seems the only way to justify any odds is if they reflect personal belief. Turning it around, Mayo’s take is most delightful. However, to … This is why classical statistics is sometimes called frequentist. In our case here, the answer reduces to just $$\frac{1}{5}$$ or 20%. A public safety announcement is due: past performance is not indicative of future performance, as is well known where it shows the most clearly – the financial sector. For some of these distinct concepts the definition can be made sense of. The Bayesian next takes into account the data observed and updates the prior beliefs to form a "posterior" distribution that reports probabilities in light of the data. I'm kinda new to Bayesian Statistics and I'd like to try to fit Bayesian Logistic Regression but I don't have prior knowledge about my dataset. Bear #1: I have had enough please go away now. The statistic seems fairly straightforward – the number is the probability that a given variant will continue to perform better than the control on the chosen metric if one were to end the test now and implemented it for all users of a website or application*. One would expect only a small fraction of respondents to choose this option if they correctly understand Options B and C below so it serves as a measure of the level of possible misinterpretation of the other two options. The average of the reported probabilities is 48%. Jeﬀreys, de Finetti, Good, Savage, Lindley, Zellner. This is the behavior of a consistent estimator – one which converges on the true value as the sample size goes to infinity. While this might be acceptable in a scenario of personal decision-making, in a corporate, scientific, or other such setting, these personal beliefs are hardly a good justification for using any specific prior odds. Others argue that proper decision-making is inherently Bayesian and therefore the answers practitioners want to get by studying an intervention through an experiment can only be answered in a Bayesian framework. The probability of an event is equal to the long-term frequency of the event occurring when the same process is repeated multiple times. So, I guess I have to use non-informative prior for . This was written by Prof. D. Mayo as a rejoinder to a short clip in which proponents of Bayesian methods argued against p-values due to them being counterintuitive and hard to grasp. This site also has RSS. Apparently “to be the best performing” refers to a future period, so it is a predictive statement rather than a statement about the performance solely during the test duration. That's 3.125% of the time, or just 0.03125, and this sort of probability is sometimes called a "p-value". A coin is flipped and comes up heads five times in a row. Many adherents of Bayesian methods put forth claims of superiority of Bayesian statistics and inference over the established frequentist approach based mainly on the supposedly intuitive nature of the Bayesian approach. Perhaps Bayesians strive so hard to claim the term ‘probability’ through a linguistic trick because they want to break out of decision-making and make it into statistical inference. As per this definition, the probability of a coin toss resulting in heads is 0.5 because rolling the die many times over a long period results roughly in those odds. In the case of the coins, we understand that there's a $$\frac{1}{3}$$ chance we have a normal coin, and a $$\frac{2}{3}$$ chance it's a two-headed coin. I'm thinking about Bayesian statistics as I'm reading the newly released third edition of Gelman et al. A world divided (mainly over prac-ticality). Bayesian and frequentist statistics don't really ask the same questions, and it is typically impossible to answer Bayesian questions with frequentist statistics and vice versa. Why use it? So there is a big question – to what extent can prior data be used to inform a particular judgement of the data? Odds of 1 to 1 do not seem to make sense here either. Bayesian vs. Frequentist Statements About Treatment Efficacy. The bread and butter of science is statistical testing. 1 Bayesian vs frequentist statistics In Bayesian statistics, probability is interpreted as representingthe degree of belief in a proposition, such as “the mean of X is 0.44”, or “the polar ice cap will melt in 2020”, or “the pola r ice cap would have melted in 2000 if we had A probability in the technical sense must necessarily be tied to an event to be definable as the frequency with which it occurs or is expected to occur if given an opportunity. Frequentist vs Bayesian statistics — a non-statisticians view Maarten H. P. Ambaum Department of Meteorology, University of Reading, UK July 2012 People who by training end up dealing with proba-bilities (“statisticians”) roughly fall into one of two camps. If a tails is flipped, then you know for sure it isn't a coin with two heads, of course. It should also be pointed out that unlike frequentist confidence intervals and p-values, Bayesian intervals and Bayesian probability estimates such as Bayes factors may disagree…. At first glance, this definition seems reasonable. This contrasts to frequentist procedures, which require many different tools. One of these is an imposter and isn’t valid. Similarly, an initial value of 1% or 99% might skew results towards the other answers. On day ten the same A/A test has 10,000 users in each test group. In the frequentist world, statistics typically output some statistical measures (t, F, Z values… depending on your test), and the almighty p-value. Those who criticize Bayes for having to choose a prior must remember that the frequentist approach leads to different p-values on the same data depending on how intentions are handled (e.g., observing 6 heads out of 10 tosses vs. having to toss 10 times to observe 6 heads; accounting for earlier inconsequential data looks in sequential testing). The difference is that Bayesian methods make the subjectivity open and available for criticism. So the frequentist statistician says that it's very unlikely to see five heads in a row if the coin is fair, so we don't believe it's a fair coin - whether we're flipping nickels at the national reserve or betting a stranger at the bar. Some numbers are available to show that the argument from intuitiveness is very common. In the Optimize technical documentation  under “What is “probability to be best”?” one sees the cheerful sounding: Probability to be best tells you which variant is likely to be the best performing overall. The History of Bayesian Statistics–Milestones Reverend Thomas Bayes (1702-1761). These are probably representative since adding [-“bayesian”] to the search query reduces the results to a mere 30,500. ** As some of those who voted would read this article, I would be happy to hear of cases where one chose a given answer yet would not subscribe to the notion of probability which I assign to it. The Bayesian approach to such a question starts from what we think we know about the situation. Bayesian's use probability more widely to model both sampling and other kinds of uncertainty. The same behavior can be replicated in all other Bayesian A/B testing tools. All Bayesian A/B testing tools report some kind of “probability” or “chance”. From the poll results it is evident that the majority of respondents would have been surprised to see that the average “probability to be best” from the 60 A/A tests is not close to zero percent, but to fifty percent instead. It is therefore a claim about some kind of uncertainty regarding the true state of the world. I also do not think any currently available Bayesian A/B testing software does a good job at presenting reasonable odds as its output. To I think the characterization is largely correct in outline, and I welcome all comments! Option A does not correspond to the expected behavior of a statistic under any framing of ‘probability’. But the wisdom of time (and trial and error) has drille… Bayes Theorem and its application in Bayesian Statistics As a final line of defense a Bayesian proponent might point to the intervals produced by the tools and state that they exhibit a behavior which should be intuitive – they get narrower with increasing amounts of data and they tend to center on the true effect which is, indeed, zero percent lift. The reasoning here is that if there is such a probability estimate, it should converge on zero. I will show that the Bayesian interpretation of probability is in fact counter-intuitive and will discuss some corollaries that result in nonsensical Bayesian statistics and inferences. I think users of statistics would do best to retain the exact meaning of terms and continue applying frequentist and Bayesian methods in the scenarios for which they were designed. The following clarifier was added to the announcements: “No answer is ‘right’ or ‘wrong’. In Gelman's notation, this is: $\displaystyle p(\theta|y) = \frac{p(\theta)p(y|\theta )}{p(y)}$. I will end this article with a quote from one of my favorite critiques of Bayesian probabilities. The important question is: can any prior odds be justified at all, and based on what would one do that in each particular case? The latter are being employed in all Bayesian A/B testing software I’ve seen to date. I’m not satisfied with either, but overall the Bayesian approach makes more sense to me. It's tempting at this point to say that non-Bayesian statistics is statistics that doesn't understand the Monty Hall problem. You can see, for example, that of the five ways to get heads on the first flip, four of them are with double-heads coins. The possible answers were presented in random order to each participant through an anonymous Google Forms survey advertised on my LinkedIN, Twitter, and Facebook profiles, as well as on the #measure Slack channel. Just 4 chose the third option, which seems to confirm that the majority of the others understood the question and possible answers as intended.**. I devised a simple poll to determine how intuitive the meaning and usage of Bayesian probability versus the frequentist alternative is among an audience with higher than average proficiency in stats and experimentation. •Non-parametric models are a way of getting very ﬂexible models. In other words, I don’t see them fulfilling the role many proponents ascribe to them. Whether you trust a coin to come up heads 50% of the time depends a good deal on who's flipping the coin. The Bayesian approach to such a question starts from what we think we know about the situation. First, the self-qualifying questions that describe the respondents’ experience with A/B testing and statistics. It exposes the non-intuitive nature of posterior probabilities in a brilliant way: Bear #2: The default posteriors are numerical constructs arrived at by means of conventional computations based on a prior which may in some sense be regarded as either primitive or as selected by a combination of pragmatic considerations and background knowledge, together with mathematical likelihoods given by a stipulated statistical model. Want to take your A/B tests to the next level? When would you say that you're confident it's a coin with two heads? In general this is not possible, of course, but here it could be helpful to see and understand that the results we get from Bayes' rule are correct, verified diagrammatically: Here tails are in grey, heads are in black, and paths of all heads are in bold. If the value is very small, the data you observed was not a likely thing to see, and you'll "reject the null hypothesis". This results in prior odds of 1 to 1, 50% / 50%. ... My research interests include Bayesian statistics, predictive modeling and model validation, statistical computing and graphics, biomedical research, clinical trials, health services research, cardiology, and COVID-19 therapeutics. B: Non-Bayesians are just doing Bayesian statistics with uninformative priors, which may be equally unjustiﬁable. 2. But what if it comes up heads several times in a row? This is further clarified in “What is “probability to beat baseline”? B: Bayesian results ≈ non-Bayesian results as n gets larger (the data overwhelm the prior). The example with the coins is discrete and simple enough that we can actually just list every possibility. That claim in itself is usually substantiated by either blurring the line between technical and laymen usage of the term ‘probability’, or by convoluted cognitive science examples which have mostly been shown to not hold or are under severe scrutiny. It isn’t science unless it’s supported by data and results at an adequate alpha level. * It should be noted that whatever “Probability to be Best” actually means, it should not be interpreted as the probability that one will see the improvement observed during the test after implementing the variant. The Bayesian formulation is more concerned with all possible permutations of things, and it can be more difficult to calculate results, as I understand it - especially difficult to come up with closed forms for things. If you stick to hypothesis testing, this is the same question and the answer is the same: reject the null hypothesis after five heads. Is it a fair coin? ), there was no experiment design or reasoning about that side of things, and so on. There are currently 9,930,000 results in Google Search for [“bayesian” “intuitive”] with most of the top ones arguing in favor of the intuitive nature of Bayesian inference and estimation. One is either a frequentist or a Bayesian. The median is 8 out of 10 for A/B testing proficiency and 7 for statistical proficiency with means slightly below those numbers at 7.77 and 6.43 out of 10, respectively. https://www.quantstart.com/articles/Bayesian-Statistics-A-Beginners-Guide  Optimize Help Center > Methodology (and subtopics) [accessed Oct 27, 2020], currently accessible via https://support.google.com/optimize/topic/9127922?hl=en Wikipedia article on “Bayesian probability” [accessed Oct 27, 2020], currently accessible via https://en.wikipedia.org/wiki/Bayesian_probability. After four heads in a row, there's 3% chance that we're dealing with the normal coin. To the extent that it is based on a supposed advantage in intuitiveness, these do not hold. All but one of the tools I’m aware of use default priors / noninformative priors / minimally informative priors. The updating is done via Bayes' rule, hence the name. I'll also note that I may have over-simplified the hypothesis testing side of things, especially since the coin-flipping example has no clear idea of what is more extreme (all tails is as unlikely as all heads, etc. Say a trustworthy friend chooses randomly from a bag containing one normal coin and two double-headed coins, and then proceeds to flip the chosen coin five times and tell you the results. Interpreted in layman terms ‘probability’ is synonymous with several technically very distinct concepts such as ‘probability’, ‘chance’, ‘likelihood’, ‘frequency’, ‘odds’, and might even be confused with ‘possibility’ by some. Frequentist/Classical Inference vs Bayesian Inference. A: It all depends on your prior! For other reasons to not use credible intervals see my other posts from the “Frequentist vs Bayesian Inference” series. A common question that arises is “isn’t there an easier, analytical solution?” This post explores a bit more why this is by breaking down the analysis of a Bayesian A/B test and showing how tricky the analytical path is and exploring more of the mathematical logic of even trivial MC methods. Bayesian statistics gives you access to tools like predictive distributions, decision theory, and a … But of course this example is contrived, and in general hypothesis testing generally does make it possible to compute a result quickly, with some mathematical sophistication producing elegant structures that can simplify problems - and one is generally only concerned with the null hypothesis anyway, so there's in some sense only one thing to check. They would expect any measure of so-called ‘probability’ to converge to zero with increasing amounts of data since the true ‘probability’ for a variant to be superior to the control in an A/A is exactly zero. If you're flipping your own quarter at home, five heads in a row will almost certainly not lead you to suspect wrongdoing. For a Bayesian account to be sensible, it would need to stick to terms like ‘degrees of belief’ or ‘subjective odds’ and stay away from ‘probability’. In order to keep this piece manageable, I will only refer to documentation of the most prominent example – Google Optimize, which has a market share of between 20% and 40% according to two technology usage trackers. However, even among such an audience, the results turned out decidedly in favor of the frequentist interpretation in which there is no such thing as a ‘probability of a hypothesis’ as there are only mutually exclusive possibilities. Does one really believe, prior to seeing any data, that a +90% lift is just as likely as +150%, +5%, +0.1%, -50%, and -100%, in any test, ever? 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