I am working on a bayesian sampling algorithm and I am particulary puzzled about the following:
Using a Gibbs sampler for the joint distribution $p(X,Y)$, we sample from $p(X|Y)$ and $p(Y|X)$ and so on and so forth. Now, the standard approach is to take $N_s = 1$ samples of say $X|Y$ and then move on to sample the other variable $Y|X$.
Here comes the question: Can I sample more then once $N_s > 1$ and use e.g. the last sampled $x_i$ to move on? Intuitively, I feel like that this should work, as all of the $N_s$ samples are draws from the conditional target distribution. Of course the mixing of the chain might not be optimal, but that is different issue..
Thanks a lot.