4
$\begingroup$

Say I want to use Metropolis Hastings algorithm to get posterior draws of multivariate parameters.

In the one variable case, you could manipulate delta until you found something that worked (gave 40% acceptance).

However how would that work in the 2 variable case? Am I manipulating a vector of deltas that are the variance of a multivariate normal proposal distribution? How do I know if I am moving enough in both parameter spaces? Each parameter might require a different delta to move enough in their own probability space a a vector delta may not be adequate.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

This is a very natural question! Rather than trying to scale one direction at a time, a possible resolution is to first run a default algorithm, e.g. Gibbs or Metropolis-within-Gibbs or even maximum likelihood estimation to get a rough idea of the variance-covariance matrix of the targeted posterior. Using this first step estimate $\mathbf{M}$, one can then implement a random walk Metropolis with variance $\delta\mathbf{M}$ towards calibrating $\delta$. If this does not cover the acceptance rate of interest, it may be necessary to return to [block] Gibbs.

Improve this answer
$\endgroup$
1
  • $\begingroup$ So does this mean that if you have the conditional posterior available you use that to get an idea of $M$, then switch from Gibbs to MH? Why would you make that switch at all? Because of slow mixing using Gibbs? In some sense this would kind of be like a two-step version of the adaptive Metropolis-Hastings algorithm, I suppose. $\endgroup$
    – hejseb
    Commented Oct 27, 2018 at 10:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.