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It is possible to do hypothesis testing, regression, classification, ect, all using Bayesian methods. Furthermore, these Bayesian methods are more flexible and easily interpret-able than Frequentist methods. Therefore, why solve a statistical problem in a more difficult manner? Really, the true reason why frequentist methods were standard is issues of computing. With lack of computer the Bayesian approach was not feasible. Now it is no longer an issue. Therefore, why would anyone still rely on frequenstist methods? Furthermore, still universities still teach frequenstist methods, or entirely abandon them? Look at it this way, the slide-ruler was once taught in universities, now nobody teaches it anymore since there is a better alternative. So why continue to use, and to teach frequentist statistics?

The only reason that I can think of is that frequentist methods are computationally faster (substantially) than Bayesian methods. For example, it is possible to do simple linear regression with least squares with thousands of predictor variables and a dataset in the trillions, very quickly, but that task might be too long to calculate for MCMC.

Edit: It says "question closed" because "it is not focused". This objection to close the question is non-sense, there is only one question being asked, namely, "why would anyone use a frequentist approach when it can be solved in a Bayesian manner?".

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    $\begingroup$ Not all scholars, e.g. Fisher, embrace the subjectivist view of probability. $\endgroup$
    – utobi
    Commented Dec 8, 2022 at 5:38
  • $\begingroup$ @utobi That is an "academic" objection. In practice, the answers are usually the same. One can replace any frequentist answer, by a Bayesian calculation which produces that same answer, and then interpret it the frequentist way. $\endgroup$
    – Nicolas Bourbaki
    Commented Dec 8, 2022 at 6:45
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    $\begingroup$ Related: stats.stackexchange.com/a/194152/76981 $\endgroup$
    – Cliff AB
    Commented Dec 8, 2022 at 6:51
  • $\begingroup$ @Nicolas Bourbaki I'm confused by your comments. If it were true that a prior exists that generates the same numerical value as a frequentist analysis, but that Bayesian analysis is more computationally burdensome then why do Bayesians bother with the Bayesian approach? $\endgroup$
    – Graham Bornholt
    Commented Dec 8, 2022 at 6:56
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    $\begingroup$ There are two topics being discussed: the model and the algorithm. The model is no simpler in a Bayesian approach than a frequentist as the likelihood is identical. So I think your claim is that MCMC can be applied to any model. While this is technically true (i.e. even a naive MH will cover the posterior eventually), in is not practically true due to how poorly MCMC methods cover posteriors that are oddly shaped (i.e. complex models). In my experience, more researcher time is spent specializing algorithms to models in Bayesian methods than Frequentists, although I will admit it can vary. $\endgroup$
    – Cliff AB
    Commented Dec 10, 2022 at 5:51

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Frequentist approaches make sense in the context of randomized control trials (RCTs).

In an RCT, an "exposure" is randomly assigned to a patient. If the exposure were nothing more than a placebo (i.e. we were just randomizing for the sake of randomizing) then the null hypothesis literally would be true.

So knowing that (and making some assumptions about the sampling distribution of the test statistic), frequentist statistics make a lot of sense. You can say "had the exposure had no effect, then the probability I would see results at least as extreme as what I saw is...". Of course, you have to make modelling and distributional assumptions when you make this statement. Bayesian methods are not immune to this.

Additionally, it isn't like the analysis of RCTs is a uniquely frequentist thing. Bayesian methods can be used for that too. But Frequentist methods can bound things like the false positive rate and the false negative rate (conditional on aforementioned assumptions, which I will repeat must be made in the Bayesian framework too).

Being able to control these error rates is very valuable in medicine and drug discovery. You're (ostensibly) able to ensure that 20% of all interventions which do have an effect fail to be discovered (or whatever your false negative rate is) $^\star$.

Bayesian's don't care about false positive/negative rates. And I think there is your answer: We still use frequentist methods when care about those error rates. If you care about those rates or not (or should care about them) is orthogonal in my opinion. IF you care about them, frequentism is a good tool.

$^\star$ Little bit of nuance here. Its 20% of all interventions which have the assumed minimal detectable effect. Here we see a really important point against frequentism; it isn't very intuitive and is filled with overloaded words like "confidence". I feel this is a more damning objection to the framework.

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  • $\begingroup$ I am not sure if I understand this answer. It is possible to get a p-value using a Bayesian method. I can run a t-test, get a p-value, and solve the same problem the Bayesian way, and instead get the probability that one mean is larger than another mean. I can then interpret that result as the "false-positive". If the answer that is obtained is the same as the frequenstist method, one can then give that answer the same interpretation. $\endgroup$
    – Nicolas Bourbaki
    Commented Dec 8, 2022 at 5:31
  • $\begingroup$ @NicolasBourbaki you could obtain a P value from a Bayesian method, but their calibration would be questionable due to the ability to specify a prior. If you use a non- informative prior in what sense are you really leveraging Bayesian methods? To be clear, I am a Bayesian myself. I’m just trying to argue from the other side. $\endgroup$
    – Demetri Pananos
    Commented Dec 8, 2022 at 12:37
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It’s familiar, so customers, bosses, and reviewers might be less likely to think you’re trying to “pull a fast one” on them.

It’s easy to implement.

There are no prior distributions for people to criticize.

The results returned from frequentist and Bayesian methods often (though certainly not always) lead to the same actions (e.g., “yes, we should use the new drug”).

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  • $\begingroup$ "There are no prior distributions for people to criticize." In my experience, by running simulations with few variables, it does not matter what the prior distribution even is, as long as it is not something pathological, and you have enough data. Indeed, just consider Bernoulli(p) estimation with a horrendous prior, but if you have enough data, the prior does not matter anymore. Therefore, the criticism of the prior distribution is academic, in practice, it seems to not even matter. $\endgroup$
    – Nicolas Bourbaki
    Commented Dec 8, 2022 at 5:25
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    $\begingroup$ @NicolasBourbaki That assumes you have enough data to overwhelm a bad prior. Not every field has large amounts of data. $\endgroup$
    – Dave
    Commented Dec 8, 2022 at 5:37
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    $\begingroup$ Once one says "the prior doesn't matter because the likelihood overwhelms it", then one might was well also say "Bayesian methods don't matter, because their results just approach those of MLE's". $\endgroup$
    – Cliff AB
    Commented Dec 8, 2022 at 6:46
  • $\begingroup$ @Dave If a problem (in science let us say) has a limited amount of data, then the conclusions that one draws from that are questionable. Indeed, there is a replication crisis. Part of the reason why there is this replication crisis is that people have small amounts of data in order to get favorable p-values. If one objects to Bayesian methods because "I have small amount of data", then one might have more serious problems in replication. $\endgroup$
    – Nicolas Bourbaki
    Commented Dec 8, 2022 at 6:51
  • $\begingroup$ @CliffAB Yes, you can just replace it by MLEs. However, how would one calculate an MLE for a complex model? Bayesian calculation runs on Markov chains which makes it possible to do those calculations. Hence, in general, the Bayesian approach would be superior because it allows one to work with many more models and is a lot more flexible. $\endgroup$
    – Nicolas Bourbaki
    Commented Dec 8, 2022 at 6:53

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