# Using MCMC to sample from a posterior, are our posterior beliefs on parameters independent?

I've been given a classification problem in which MCMC (slice-sampling) is used to sample from a hierarchical posterior. After getting $n$ samples, the Monte Carlo method can be used to give an estimate of parameters of the model.

My question is: Are our posterior beliefs about individual parameters independent? I would argue they are not, as each sample in slice-sampling is auto-correlated...

• I've rolled your edit back, leaving it as only saying "I've been given a classification problem" made the question unanswerable. Since you had a question which had an answer please only change it to clarify your question, not to remove context necessary for the answer to apply. – Glen_b Dec 14 '15 at 23:34
• This feels somewhat unclear - at first reading I thought "Are our posterior beliefs about individual parameters independent" asks whether, say, parameter $\mu$ and parameter $\sigma^2$ are a posteriori independent, however, the answer does not talk about this and it's unclear how the slice-sampler would be related. Is the question actually concerned with the approximation of the posterior of parameter $\mu$ and the approximation of the posterior of parameter $\sigma$ (data and thus the underlying true posterior distribution treated fixed)? – Juho Kokkala May 4 '18 at 16:34

The posterior beliefs about the parameters are represented by the posterior distribution $\pi(\theta|\mathfrak{D})$ if $\mathfrak{D}$ denotes the data. This is a distribution conditional or dependent on the data.
Since this distribution is too complex to envision as such, we use simulation techniques to understand it. For instance, MCMC simulation produces an outcome $(\theta^{(t)})_t$ that is approximately distributed from $\pi(\theta|\mathfrak{D})$ and with correlated entries. They are therefore dependent.
Hence the answer to your question is that the approximation technique creates dependence, but the posterior beliefs themselves are not associated with any sort of dependence, except on the data $\mathfrak{D}$. It is important to distinguish the inference part from the simulation part as they partake of fundamentally different concepts.