# Question about MCMC independent proposals

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

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.