# Do frequentists and Bayesians use different hypothesis tests?

In a basic refresher course on stats, they covered Z-tests, T-tests from a practical perspective: they described the assumptions for each test, and "let the software packages do their job" - as an engineer this fits me fine.

A. However, one question did pop into my head - are these standard tests based on the frequentist philosophy or the bayesian philosophy?

B. Also, as of now I view frequentist and bayesian views as mutually exclusive with their own sets of non-overlapping theories. Is this correct? Or is there an overlap in certain problems where one is clearly better than the other (one uses more known values than the other OR has lesser assumptions OR does more fine grained calculations, for example). (in other words: bayesian theory looks daunting - why would I learn it if frequentist works fine?)

A. However, one question did pop into my head - are these standard tests based on the frequentist philosophy or the bayesian philosophy?

They are grounded in frequentist reasoning. Nonetheless, while Bayesians do not focus that much on hypothesis tests, as already noted by kjetil b halvorsen, but often there are Bayesian alternatives to the tests, for example BEST as an alternative to $t$-test. This is covered to some extent in Doing Bayesian Data Analysis by John K. Kruschke.

B. Also, as of now I view frequentist and bayesian views as mutually exclusive with their own sets of non-overlapping theories. Is this correct? Or is there an overlap in certain problems where one is clearly better than the other (one uses more known values than the other OR has lesser assumptions OR does more fine grained calculations, for example). (in other words: bayesian theory looks daunting - why would I learn it if frequentist works fine?)

They differ. This does not directly mean that one is better then another. First of all, they differ in their interpretation of probability and this has consequences in other areas. TLDR; Bayesian interpretation of probability is less restrictive and more general, what enables Bayesians to think in probabilistic terms of things that are usually not interpreted in such fashion. There are arguments on which interpretation of probability is more "intuitive", but at least in some cases Bayesian approach gives more intuitive interpretations, e.g. in case of confidence intervals vs credible intervals. You can check other questions tagged as to see extended discussion on this subject. As about assumptions, both approaches can make multiple assumptions. One of the differences in here is that in Bayesian case you are forced to state your a priori assumptions about parameters directly in the model, while in frequentist case they may not be stated directly but follow from the used procedure.

The Z-test (and probably the other tests you were reviewing) are based on frequentist theory. Hypothesis tests do fit badly with bayesian statistics, while they might be constructed. Using bayesian statistics one is mostly focusing on other inference methods, such as uncertainty intervals (credible intervals).

EDIT

AS of your new question B, which is close to being to broad! And I am probably not the right person to answer it ... I think it would be wrong to view as "two essentially non-overlapping theories", in practice there is overlap, especially in the field of regression models. Have a look at http://andrewgelman.com/2015/03/29/bayesian-frequentist-regression-methods/ for one thoughtful view, and then you can search this site for "bayes frequentist", for instance Who Are The Bayesians? Why does frequentist hypothesis testing become biased towards rejecting the null hypothesis with sufficiently large samples? Is it possible to interpret the bootstrap from a Bayesian perspective? Is there any *mathematical* basis for the Bayesian vs frequentist debate?

• thanks, that helps! could you also help me with part (B)? – N K Feb 6 '17 at 8:12

I'd add Gelman and Hill 2007 to your reading. They address many of these issues, too. And I second the Kruschke BEST article.

Also keep in mind the Bayesian approach is the probability of your MODEL given the data. The data are assumed correct. By it's very function bayesian inference is a model test in an of itself, in away. Particularly if you do regression analysis. The frequentist examine the inverse-- meaning the probability of the data give you model. The model is assumed to be correct. This is something you should always keep in the back of your head while doing bayesian analysis.

Sometimes the line between the two can be blurry....