Tag: tech

Why Bayesian A/B testing can lead to misunderstandings, inflated false positive rates, introduce bias and complicate results

Over the past decade, I’ve engaged in countless discussions about Bayesian A/B testing versus Frequentist A/B testing. In nearly every conversation, I’ve maintained the same viewpoint: there’s a significant disconnect between the industry’s enthusiasm for Bayesian testing and its actual contribution, validity, and effectiveness. While the hype around Bayesian testing may have peaked, it remains widely popular.

My first exposure to Bayesian statistics was during my master’s studies, where my thesis focused on Thompson Sampling. Professionally, I encountered Bayesian A/B testing during my tenure at Wix.com, where I played a key role in transitioning from the classical method to the Bayesian method. My perspective, as described here, has been informed by both my academic background and my professional experience at Wix and beyond, where I’ve helped many companies enhance their A/B testing capabilities.

When referring to “Bayesian A/B testing”, I’m specifically talking about the methods promoted by VWO and similar approaches used in some current experimentation platforms as alternatives to the classic (Frequentist) method. There are other implementations of Bayesian statistics in A/B testing, such as Thompson sampling in Multi-armed-bandit experiments, which can be highly effective but are rare outside marketing platforms like Google Ads and Facebook Ads.

In this post, I’ll explain what Bayesian tests entail, outline the most common arguments in favor of Bayesian tests, and address each argument. I’ll then discuss the major drawbacks of the Bayesian method and, finally, cover when to use Bayesian methods in experiments.

So grab a cup of coffee, and let’s dive in.

What Do Bayesian Tests Mean?

Bayesian statistics and Frequentist statistics differ fundamentally. Bayesian statistics incorporates prior knowledge or beliefs, updating this prior information with new data to produce a posterior distribution. This allows for a dynamic and iterative process of probability assessment. In contrast, Frequentist statistics relies solely on the data at hand, using long-run frequency properties to make inferences without incorporating prior beliefs. Frequentist statistics focuses on the likelihood of observing the data given a null hypothesis and uses concepts like p-values and confidence intervals to make decisions.

In Bayesian A/B testing, we design the test in a way that after short time, and based on the data gathered so far, we could calculate the probability that the treatment variant (B) is better than the control variant (A), noted as P(B>A| Data). Another metric used is risk, or expected loss, which helps us understand the risk of making a decision based on the data collected.

Bayesian A/B testing typically involves running a test, computing P(B>A|Data) and/or the expected loss (Risk), and making a decision based on these metrics. The decision can be arbitrary or involve a stopping rule, such as:

1. The probability B is better than A is larger than X%. For example: P(B>A| Data) > 95%
2. The expected loss (Risk) is less than Y%. For example: expected loss < 1%

Arguments for Bayesian Tests

Throughout my career, I’ve encountered three common arguments in favor of Bayesian tests:

1. The early stopping argument — the ability to stop the experiment whenever you want (or based on a stopping rule), unlike the classic t-test / z-test that requires planning your sample size and analyzing the results only once the predefined sample size is reached. This is useful in cases where the sample size is small or when there is a very big effect and you would like to stop the test based on the results.
2. The prior argument — The use of prior knowledge or business knowledge to enrich data and make better decisions.
3. The language and terminology argument — bayesian metrics are more intuitive and suited to everyday business language compared to Frequentist metrics like p-value. Thus, “Probability B is better then A” is much more intuitive and well understood compared to “the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is true” — which is the p-value definition.

Let’s tackle each argument one by one.

You Can Stop Whenever You Want

In the online industry, data is collected automatically and often displayed in real-time dashboards that include various statistical metrics. Simple classical tests, like the t-test and z-test, do not permit peeking at the results, requiring a predefined sample size and only allowing analysis once that sample size is reached.

Anyone who has ever run an A/B test knows that this is not practical. The easy accessibility of information makes it hard to ignore, especially when a product manager notices significant results, whether positive or negative, and insists on stopping the experiment to move on to the next task. This highlights the clear need for a method that allows peeking at the data and stopping early. Thus, the argument for early stopping is perhaps the strongest for Bayesian A/B tests — if only it were true.

Bayesian statistics, when considered superficially as “subjective understanding incorporating prior beliefs to the data,” allows stopping whenever. However, if you expect guarantees like “controlling the false positive rate” (as in the Frequentist approach), this is problematic.

Bayesian A/B testing is not inherently immune to the pitfalls of peeking at the data. For those looking for a good statistical explanation, please take a look at Georgry’s excellent blog post. For now, let’s address Greorgry’s point, but from a different perspective:

In the case of two variants, control and treatment, and when the number of users is large enough, the one-tailed p-value is almost identical to the Bayesian probability the control is better than the treatment, noted as P(A>B| Data) =1-P(B>A| Data). In an A/B test, a low one-tailed p-value and low P(A>B| Data) (which is equivalent to high P(B>A| Data)) indicates that the treatment is better than the control. The fact that these two measures are almost identical means that technically, early stopping based on P(B>A | Data) is equivalent to early stopping based on the p-value failing to maintain the type I error rate (false positive rate).

Although the Bayesian method does not commit to maintaining the false positive rate (aka type I error), practitioners would likely not want to see false “significant” results frequently. The notion of “stop whenever you want” is usually interpreted by practitioners as “we’re safe to draw valid conclusions at any point because we’re doing Bayesian analysis” rather than “we’re safe to draw conclusions at any point because Bayesian A/B testing doesn’t guarantee to maintain something similar to false positive rate”. We now understand that Bayesian A/B testing, in the popular way it is practiced, means the latter.

Sequential testing in the Frequentist approach, on the other hand, allows for peeking and early stopping while maintaining control over the false positive rate. Various frameworks, such as Group Sequential Testing (GSP) and the Sequential Probability Ratio Test (SPRT), enable this and are widely implemented in experimentation platforms like Optimizely, Statsig, Eppo, and A/B Smartly.

In summary, both Frequentist and Bayesian methods are not immune to the issues of peeking, but sequential testing frameworks can help mitigate these issues while making sure they do not inflate the false positive rate.

Use of Prior

The second argument in favor of Bayesian A/B testing is the use of prior knowledge. Throughout the web and conversations with practitioners, I’ve encountered comments regarding prior such as “Using prior allows you to incorporate existing and relevant business knowledge into the experiment and thereby improve performance”. These statements sound very appealing because they play on a very correct sentiment — usually using additional data is better. The more, the merrier. But anyone who understands a bit how the concept of priors in Bayesian probability works will understand that the use of priors in A/B testing is at least risky, and can lead to incorrect results.

The basic idea in Bayesian statistics is to combine any prior knowledge we have, aka prior, with the data to produce posterior distributions — knowledge that combines our prior knowledge with the data. Seemingly, there is something here that does not exist in the classical method. We are not just using the data; we are also adding more knowledge and business information that exists in our organization!

In the case of comparing two proportions — the meaning of prior is actually very simple. It is simply an addition of a virtual # of success and # of users to the data. Suppose we did such a test, and out of 1000 users in the control group, and we have 100 conversions.

Assuming my prior is “10 successes out of 100 users”, it means that my posterior knowledge is the sum of successes and users of the prior and the data. In our example: 110 “conversions” out of 1100 “users”. This is not the exact statistical definition, but it captures the idea very well.

A prior can be weak (1 success out of 10 users) or strong (1000 successes out of 10000 users for example). Both represent a knowledge that the conversion rate is 10%. In any case, when we accumulate a lot of data, the prior weight naturally decreases.

How should we incorporate prior knowledge in a two proportions A/B test? There are two options:

1. We incorporate, based on historical data, the general conversion rate in the population and add it to each variant. This is common practice.
2. We incorporate, based on historical data, which variant, control or treatment, usually show better results and give that variant an advantage based on this knowledge.

How will the prior manifest in the first option? Let’s stick to the example of 1000 users in each variant, 100 conversions to control variant and 120 conversions to treatment variant.

Suppose we know that the CVR is 10%, so an appropriate prior could be to add 100 successes and 1000 users to the existing data and then perform a statistical test as if we have 2000 users in each group, 200 conversions in control and 220 conversions in treatment. What’s described here is exactly what happens; it’s not approximately or as if — that’s the technical meaning of the prior in the case of two proportions bayesian test (assuming beta prior, for the statisticians reading this article).

A simple calculation shows that using a stronger prior in our example will increase P(A>B| Data), which means less indication for difference between variants — compared to the weak prior. That is what happens when you add the same amount of successes and users to each variant. This practice goes against our motivation to stop as early as possible, so why on earth would we want to do such a thing?

A common argument is that the Bayesian method is very liberal in choosing a winner, and the priors are a restraining factor. That’s true, the Bayesian method as I represented is very liberal, and priors are a restraining factor. So why not choose a more conservative approach (hmmm hmmm Frequentist) to begin with?

Moreover, if that is the argument, then it is clear to everyone that the glorified claim about priors that “add business information to the experiment” is misleading. If the business information is just a restraining factor, then the idea of using strong prior does not seem appealing at all.

The second option for incorporating a prior, giving one version an advantage over the other version based on historical data, is even worse. Why would anyone want to do this? Why should one experiment be influenced by the successes or failures of previous experiments? Each experiment should be a clean slate, a new opportunity to try something new without bias. Adding 200 successes to one version and 100 to the other sounds absurd and unreasonable in any way.

Language and Terminology

The third argument in favor of Bayesian A/B testing is the more intuitive language and terminology. A/B testing results are often consumed by people without strong statistical backgrounds. Frequentist metrics like p-values and confidence intervals can be unintuitive and misunderstood, even by statisticians. Many articles have been written about people’s misunderstanding of these metrics, even people with a background in statistics. I admit that it was only a considerable time after my master’s degree in statistics that I understood the exact definition of a classical CI. There is no doubt that this is a real pain point and an important one.

If you ask someone without a background in statistics to compare two versions with partial performance data for each version and ask them to formulate a question, they are likely to ask, “What is the probability that this version is better than the other version?” The same is true for confidence intervals. Most likely, when you explain the definition of a Frequentist confidence interval to someone, they will understand it in a Bayesian way.

This argument is actually true. I agree that Bayesian statistical metrics are much more intuitive to the common practitioner, and I agree that it is preferred that the statistical language will be as simple as possible and well understood, since A/B testing is mostly being conducted and consumed by non-statisticians. However, I don’t think it’s a disaster that practitioners don’t fully understand the statistical terms and results. Most of them are thinking in terms of “winning” and “losing” and it’s okay.

I recall, when I was at Wix, showing our new Bayesian A/B testing dashboard to a product manager as part of a usability test, to learn how he reads it and what he understands. His approach was very simple — searching for “greens” and “reds” KPIs and ignoring the “grays” KPIs. He didn’t really care if it was a p-value or probability B is better than A, a confidence interval or a credible interval. I bet that if he knew, it would rarely change his decision about the test.

Major Drawbacks of the Bayesian Method

So far, we have discussed the alleged advantages of using the popular Bayesian method for A/B testing and why some of them are not correct or meaningful enough. There are also very considerable disadvantages to using the Bayesian method:

1. The lack of maximum sample size
2. The lack of guidelines and framework to make a decision regarding the test when the results are inconclusive.

These drawbacks are significant, especially since most experiments do not show a significant effect.

Let’s assume we run an experiment which does not affect the KPI we are interested in at all. In most cases, the data will indicate indecision, and we will not be sure what to do next. Should we continue the experiment and collect more data? Or go with the more probable variant even if the results are not conclusive?

One can argue that predefined sample size is a limiting factor, but it also provides an important framework for decision-making. We decide upon a sample size, and we know that we will be able, with high probability (known as statistical power), detect a predefined effect size. If we are smart enough, we will use a sequential testing method that will allow us to stop before we reach the maximum predefined sample size.

It is true that when using one of the Bayesian stopping rules mentioned before, the test will eventually end even if there is no effect. For example, the risk will gradually, and slowly, decrease and eventually will reach the predefined threshold. The problem is it will take a very long time when there is no difference between the variants. So long that in reality practitioners will likely won’t have the patience to wait. They will stop the experiment once they feel there is no point in continuing.

When to Use Bayesian Methods in Experiments

In Multi-Armed Bandit (MAB) experiments, Bayesian statistics flourish and are considered best practice. In these types of experiments, there are usually several variants (for example several ads creative) and we want to quickly decide which ads are performing the best. When the experiment begins, users are allocated equally to all variants, but after some data is gathered, the allocation changes and more users are allocated to the better performing variant (ad). Eventually, (almost) all users are allocated to the best performing variant (ad).

I also came across an interesting Bayesian A/B testing framework in an article published by Microsoft, but I never met any organization using the suggested methodology, and it still lacks a maximum sample size which should be very important to practitioners.

Conclusion

While Bayesian A/B testing offers a more intuitive framework and the ability to incorporate prior knowledge, it falls short in critical areas. The promises of early stopping and better decision-making are not inherently guaranteed by Bayesian methods and can lead to misunderstandings and inflated false positive rates if not carefully managed. Additionally, the use of priors can introduce bias and complicate results rather than clarify them. The Frequentist approach, with its structured methodology and sequential testing options, provides more reliable and transparent results, especially in environments where rigorous decision-making is essential.

Bayesian A/B Testing Falls Short was originally published in Towards Data Science on Medium, where people are continuing the conversation by highlighting and responding to this story.