Implement the collapsed Gibbs sampler for the Gaussian mixture model. At each iteration in the sampler, sample the means \(\mu_{k}\) even though they are not needed (sample them from the posterior you computed to marginalise them). Compare the autocorrelation and \(\hat{R}\) for the \(\mu_{k}\) with that obtained in Exercise 10.2 .

Data from Exercise 10.2

Implement the Gibbs sampler for the mixture model. Compute \(\hat{R}\) and the autocorrelation of the component means \(\mu_{k}\). How many samples are required for convergence? How much should the samples be thinned?