# Gibbs Sampler for mixture models: shall I skip some samples to avoid to use correlated samples? [duplicate]

I am implementing a Gibbs sampler in order to estimate the parameters of a mixture model.

Assuming that the parameters are contained in a vector $$\theta$$ what I will do is:

• Implement and run the sampler.
• Wait to reach a stationary distribution (so to remove the impact of my initialization); say that we reach the stationary distribution at iteration $$n$$.
• After iteration $$n$$, I start collecting the sampled parameters into a list $$[\theta_n,...\theta_{n+m}]$$
• Last, I evaluate the expected value of $$\theta$$ doing $$\theta_{m,estimated} = \sum_{i=1}^{m} \theta_{n+i}/m$$

Now, I have some questions:

• Two subsequent samples $$\theta_{n+i}, \theta_{n+i+1}$$ will obviosly be correlated. Should I skip some samples in order to ensure independence between the samples I use to calculate my estimator? If yes, there is some standard heuristic to do this?
• Can I say that (if the sampler is properly implemented) $$\theta_{m,estimated}$$ is a consistent estimator of $$\theta$$?
• Your questions seem to be already answered on this site: for "skipping" samples (thinning) see this answer (thinning is not useful), and for conditions for consistency see this answer. Please look at those answers and re-post with more specific questions if any remain.
– EdM
Jan 26 '19 at 21:44