Frequentist vs Bayesian for common tests
Hi, hoping that this isn't too theoretical of a question but would really appreciate help (or a pointer in the right direction).
I'm trying to learn a bit about how Bayesian testing works and feel like I have made some progress but I am a bit lost about the likelihood function in practice. As I understand it, we specify a prior, compute a likelihood from observations, and then compute the posterior. I've done some exercises that use a binomial likelihood function to demonstrate the concept but I fail to see how this applies to a practical scenario using a t-test/correlation/ANOVA etc., and there doesn't seem to be a lot of information about how this is done in practice.
What is the nature of the likelihood function when we have, for e.g., 2 arrays that we want to compare means for as with a t-test? Does it come from a t distribution on the difference, so we then combine the t distribution (with the observed mean?) with our cauchy prior to get the posterior? I have the same question for Pearson correlation, ANOVA, regresison, etc. All I see in Jasp is the Cauchy prior and the posterior, but no likelihood.
Apologies if this isn't clear as I don't have a very strong background in probability theory/statistics. Hopefully someone can point me in the right direction.
Comments
The more complicated the test, the more involved all of this becomes. For instance, in the Pearson correlation we really have 5 parameters (correlation, plus two means and two variances). What also complicates matters is that we often conduct a test on effect size, so on delta = mu/sigma. At any rate, some background is in this paper: http://www.ejwagenmakers.com/2016/LyEtAl2016JMP.pdf and in this one: https://www.tandfonline.com/doi/full/10.1080/00031305.2018.1562983
Cheers,
E.J.