I was wondering what the counterpart to Pearson product-moment correlation would be in a Bayesian framework. Or if there are many alternatives, what would be the most convenient or conventional method. I know that I could do Bayesian linear regression analysis but then I have to assume that I have one dependent and one independent variable, which I don't do when calculating a correlation.
Any suggestion for how to implement the model in BUGS/JAGS is also appreciated!
Edit:
My question is what a Pearson product-moment correlation in an Bayesian framework would be as opposed to a classical framework including p-values and confidence intervals of the correlation. Sometimes the transition from classical statistics to Bayesian statistics is not straight forward, for example, some Bayesian ANOVAs that are proposed actually are not analyses of variance. Correlation analysis is a really common statistical analysis but I haven't been able to find any Bayesian example implementation in, for example, JAGS or BUGS.
Edit:
I now found the following pdf presentation that describes using a multivariate normal distribution with a Wishart distribution as the prior on the precision to calculate the correlation coefficient. This was, sort of, the answer I was looking for.
Best Answer
There is no essential Bayesian / frequentist divide with a correlation any more than there is a Bayesian equivalent of a mean or median. A correlation is just an arithmetic calculation. The need for specific Bayesian techniques only arises when you do inference with it, so the appropriate Bayesian approach would depend on what your actual question is. But there's no fundamental reason why it wouldn't invove Pearson's product-moment correlation.