Is the MAP estimate always identical to the ML estimate when a diffuse prior is used?

Question

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:

1. Is the MAP of rho for this type of analysis always equal to the MLE?
2. Is the 95% density interval of the posterior distribution around the MAP equal to the 95% CI of the MLE?
3. Is there any difference between doing a maximum likelihood analysis and a Bayesian analysis in this situation?

Answer 1

1. Yes, the two should be equal, although this is dependent on exactly what you mean by a "diffuse prior". If it is truly a prior that does not affect the posterior density, then the posterior distribution is exactly equal to the likelihood function. Therefore, maximizing the likelihood function is equivalent to maximizing posterior distribution. On the other hand, if you just mean a prior that is very diffuse, i.e. $\mu \sim N(0, \sigma = 10,000)$, the maximum likelihood function is not exactly equal to the posterior distribution, and so the MAP will not be exactly equal to the MLE.

2. So even if the diffuse prior is meant in the first sense, the corresponding CI's are still not necessarily equivalent, although this is mostly due to approximations typically used by each method. Often times, MLE-based confidence intervals are based on a quadratic approximation of the log-likelihood function (i.e. the fact the parameter estimates typically approach a normal distribution in large samples). But in the Bayesian world, credible intervals are typically done with MCMC integration of the full posterior distribution rather than quadratic approximations. In smaller samples, this may well make a difference that favors the direct integration techniques rather than the quadratic approximation. In large samples, this difference should be quite small.

3. Not really beyond what was mentioned in 2.