I have a Metropolis-Hastings scheme implemented, where I am currently inferring a number of parameters using Gaussian proposal distributions.
However, I would now like to assume I don't know the values of one of my discrete variables, and instead infer these from the scheme as well. I think they would be well described by a $n$-dimensional multinomial distribution, and have a reasonable prior I can use for this. However, I'm not sure what kind of proposal distribution I would use for this.
For the Gaussians, you can update the mean and variance based on whether a proposal was accepted. In my case, I want to generate $N$ samples from my discrete proposal. Do I then just simply calculate $p_1,\ldots, p_n$ of my generated sample if my proposal is accepted (after changing the acceptance probability accordingly), and use that as my new input vector for the multinomial proposal? Or is there something smarter I should be doing?
Would I simply be better off just generating a new sat of values at each iteration, with the proposal distribution remaining the same, rather than trying to update it (so that I sample the space of that variable more accurately, rather than picking one sample and sticking with it).
Edit: Another question. Thinking about it, I'm not sure I do want a multinomial distribution. If I want the sample for each of the $N$ entries (where each entry takes an integer from $0$ to $n$), then should I just generate a random number between 0 and 1, and see where each entry lies in my $p$ vector? Because generating samples from a multinomial distribution just gives me the number of entries taking each value, without assigning them to a specific row.
Edit 2: The other possibility would be that I have a different proposal distribution for each entry in my dataset. Obviously, this means there's a lot more parameters I'd have to infer, and so I'm not sure if that would be sensible?