Let's say I have two normally distributed variables: say height and body mass. I want to estimate the Pearson's correlation coefficient between them. I have a model that estimates the mean and SD of each variable as well as the correlation between them. I have no prior beliefs on any of these parameters so I use a diffuse prior for all of them. I use a Gibbs sampler to obtain a posterior distribution of the correlation coefficient rho. I have a few related questions:
- Is the MAP of rho for this type of analysis always equal to the MLE?
- Is the 95% density interval of the posterior distribution around the MAP equal to the 95% CI of the MLE?
- Is there any difference between doing a maximum likelihood analysis and a Bayesian analysis in this situation?