# Likelihood Ratio Confidence Intervals and Credible Intervals for MCMC

I am fitting a 5 parameters model to some data using Maximum Likelihood and Non Informative Bayesian Inference using Metropolis-Hastings algorithm.

From the maximum likelihood fit, i calculated the expected value of the parameters and their 75% Wald confidence interval and the 75% likelihood ratio based confidence interval.

In the Bayesian framework, I used the Metropolis Hastings algorithm and I calculated the expected value for each chain (which is almost coincident with maximum likelihood estimate. To be honest, I would like to compare the uncertainties in this estimation with the uncertainties in Maximum Likelihood. So what I did is to calculate the 0.125 and 0.875 quantile of the stabilized part of the Markov Chain. The obtained values sometimes differ from the likelihood ratio confidence intervals, especially when the Wald CI and the likelihood ratio CI are not coincident. .

Is the procedure of evaluating quantiles of the converged Markov Chain correct in order to estimate Credible Intervals?

First: this link will be useful to you. The fact that your MLE estimate and posterior mean estimate coincides is likely because your posterior distribution is symmetric and unimodal.

Regarding credible intervals, there is really no reason why your credible intervals have to coincide with your confidence intervals. Credible intervals measure the variability in the posterior distribution. If you get the credible interval $(l_1, u_1)$, this means that the posterior has 75% of its mass between $l_1$ and $u_1$.

If the confidence interval is $(l_1, u_1)$, then if the experiment were to be exactly repeated many more times, than 75% of the resulting confidence intervals will contain the true parameter.

If you increase the data sample size, you can make the confidence intervals arbitrarily small. But when data sample size increases, the posterior distribution converges to a limiting distribution and the credible interval corresponds to the chosen quantiles of this limiting distribution. Whatever the limiting distribution may be, the important point is that the credible interval (in most cases), cannot be made arbitrarily small. Thus, credible intervals and confidence intervals cannot really be compared.