Question about MCMC independent proposals

Question

I've a question re MCMC proposals, I was hoping you could help me with that. I need to implement for work an Independent Metropolis-Hastings algorithm to sample from a 10-dimensional posterior. I am setting up the proposals, and I was seeking confirmation that

• It is fine for the proposal for each of the posterior parameters to be all independent each other. E.g. proposal for parameter 1 is N(0,1), parameter 2 is Gamma(1,100) etc, all independent.
• I can either propose and accept/reject all these indipendently generated values at once, or one at a time in cycle.
• Even if there is a strong relationship between two parameters (e.g. because Parameter1 + Parameter2 = 1 always), it is fine (in theory) propose each of them independently, and at worst I will have non-very-efficient chain

Do this sounds correct? Thanks a lot for your help

Answer 1

Is it fine for the proposal for each of the parameters to be independent of one another?

Yes, the only formal constraint on the proposal distribution is that its support contains the support of the target distribution, i.e., that the target distribution is absolutely continuous wrt to the proposal distribution.

I can either propose and accept/reject all these independently generated values at once, or one at a time in a cycle.

Both versions are valid. The second one is a Metropolis-within-Gibbs algorithm in that each step in the cycle targets the corresponding conditional distribution rather than the joint target, all other parameters remaining fixed.

Even if there is a strong relationship between two parameters (e.g. because Parameter1 + Parameter2 = 1 always), it is fine (in theory) propose each of them independently, and at worst I will have non-very-efficient chain

No, this is incorrect as the target distribution is no longer absolutely continuous wrt to the proposal distribution. If there exists a deterministic relation between two parameters, as, e.g., $\theta_1+\theta_2=1$, the proposal must put (some of) its mass on this subset. In other words, one of the two parameters $\theta_1$ and $\theta_2$ must be removed from the simulation.