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I was just recently using the Student t-Test to check whether values from two samples could have an identical mean or not. I was wondering whether there is a complementary technique in bayesian statistics that does a similar thing.

Especially I am wondering about the $P$ values that I obtain as a result (http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html). With a sample size of 2 in my two samples, I figure that the resulting p value would probably fluctuate quite a bit if I added some more measurements or take some away.

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  • $\begingroup$ A typical very simple Bayesian approach would be to form an interval for the difference in means, based off the posterior for it. One might make all the same kinds of assumptions as in the t-test or one might make a variety of changes toward more robustness/wider applicability (e.g. considering possible contamination of the data or heavier tailed distributions, putting hyperiors in and so on). Of course there's also a more formal Bayesian decision-theoretic approach and a variety of more nearly hypothesis-testing style things one might do. $\endgroup$ – Glen_b -Reinstate Monica Jun 27 '14 at 0:18
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Two come to mind.

Morey and Rouder present "Bayesian t tests for accepting and rejecting the null hypothesis". The reference paper is here, there is a handy web interface, and an R package. There are also equivalent programs for ANOVA and correlation.

John Kruschke claims that "Bayesian estimation supersedes the t test" ("BEST"). The reference paper is here, a web app by Rasmus Bååth here, and of course an R package. For more, see the web page, including a Python implementation. Rasmus Bååth has implemented a range of further tests in this tradition, but also provides a very readable explanation of BEST.

The primary difference between the two, BEST and Rouder/Morey's test, is that Morey and Rouder present their approach explicitly in the tradition of hypothesis testing, whereas BEST is parameter estimation. Under the hood, the two are not too dissimilar - default priors, MCMC sampling - but the outpood is quite different; the first gives you the Bayes Factors for or against your hypothesis, the other focuses the graphical presentation on the credible interval for the estimated parameter.

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