This is the second in a set of posts about Monte Carlo Markov Chain (MCMC, or MC2) methods in Bayesian econometrics. The background was provided in this first post, where the Gibbs sampler was introduced.
The main objective of the present post is to convince you that this MCMC stuff actually works!
To achieve this, what we're going to do is work through a simple example - one for which we actually know the answer in advance. That way, we'll be able to check our results from applying the Gibbs sampler with the facts. Hopefully, we'll then be able to see that this technique works - at least for this example!
I'll be using some R script that I've written to take students through this, and it's available on the code page for this blog. I should mention in advance that this code is not especially elegant. It's been written, quite deliberately, in a step-by-step manner to make it relatively transparent to non-users of R. Hopefully, the comments that are embedded in the code will also help.
It's also important to note that this first illustration of the Gibbs sampler in action does not involve the posterior distribution for the parameters in a Bayesian analysis of some model. Instead, we're going to look at the problem of obtaining the marginals of a bivariate normal distribution, when we know the form of the conditional distributions.
In other words - let's proceed one step at a time. The subsequent posts on this topic will be dealing with Bayesian posterior analysis.
Let's take a look at the set-up, and the analysis that we're going to undertake.