I'm coding a McMC algorithm for geophysical applications.
Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be clear: normally you have a current model $m$, you perturb one parameter (in my case velocity $v_i$) and in this way you get a trial model $m'$.
The acceptance probability for such a perturbation is:
$$\alpha(m'|m)=min\left[ 1, \frac{p(m')}{p(m)} \cdot \frac{p(d|m')}{p(d|m)} \cdot \frac{q(m|m')}{q(m|m)} \cdot |J|\right]$$
Where $p(m)$ are the priors, $p(d|m)$ are the likelihoods, $q(m|m')$ are the proposals and $J$ the jacobian.
So assuming that the priors are symmetric, and the jacobian is 1, I remain with:
$$\alpha(m'|m)=min\left[ 1,\frac{p(d|m')}{p(d|m)} \cdot \frac{q(m|m')}{q(m|m)} \right]$$
The likelihood ratio is not a problem, so now i wanna deal only with the proposal ratio:
Supposing that the way i make a perturbation is choosing one parameter, and perturb it with gaussian probability, so that the new velocity value will be $$v'_i=v_i + u \cdot \sigma$$ ($u$ is normal distributed with mean=0 and var.=1)
This perturbation can be described by a proposal in the form:
$$q(v'|v)= \frac{1}{\sigma \sqrt{2 \pi}} \cdot \exp \left[- \frac{(v'-v)^2}{2 \sigma ^2} \right]$$
It's easy to see that for this kind of perturbation the proposals are symmetric so the acceptance probability is just the ratio of the likelihoods.
Now the questions:
1) If instead of perturbing one parameter I want to change 3 of them at the same time, what happens to my proposals, and so to my acceptance probability $\alpha$? How should I compute them? ($v_1, v_2, v_3$ are not correlated)
2) If i want to include some deterministic constrains because I know that the 3 parameters i want to change are correlated, i.e. if I increase $v_1$ of $v_p$, then I want to decrease $v_2$ and $v_3$ of $v_p / 3$ (a sort of compensation)...how am I supposed to come up with a representation of the proposal ? To make it more clear: I am computing the resolution matrix for every accepted model, so I know how every parameter is correlated with the others. Suppose I want to perturb the $k^{th}$ parameter, $v_k$ of say $1km/s$. I compute the $k^{th}$ row of the resolution matrix, and I use its values to compute small perturbations that are aimed to balance the main perturbation of $v_k$.
I hope I have exposed my problem clearly, and not only in a lengthy way. Hoping that someone could help me...thanks in advance.
