# proposal distribution for metropolis algorithm

All,

I'm wondering whether it is possible to use an asymmetric distribution, eg the exponential distribution as the proposal dist'n for a metropolis algorithm (wiki) (not the metropolis-hastings). The reason I ask is because it is asymetric. For example, if I have the support of my desired probabilty distribution is (.2,4) could I use $$f(t,\lambda)=\lambda^{-\lambda t}$$ with $$\lambda=.5$$ could I us this to simulate eg the line y=x from 0http://www.mas.ncl.ac.uk/~ndjw1/teaching/sim/metrop/metrop.html I'm not sure you can just replace the line

innov <- runif(1, -alpha, alpha)


with the exponential distribution...

## 1 Answer

As I understand it, the main difference between Metropolis and Metropolis-Hastings is the the former requires a symmetric proposal distribution and the latter does not. The acceptance criterion used by M-H 'corrects' for the 'bias' inherent in using an asymmetric distribution for moves to the next proposal.

In particular, I doubt that exponential moves you propose would work well even for M-H. You need to be able to move relatively freely among possible proposals, Even if the exponential is offset so that movement in both directions is possible, I think the M-H acceptance criterion would essentially have to reject for a very high proportion of jumps resulting from the long exponential tail.

Addendum (7/19): For archival purposes, I should note that the main applications of both Metropolis and M-H methods is to simulate from multivariate distributions, where choices of methods are often quite limited. Based on what I saw in the reference you provide, this is a drill problem to sample from a univariate distribution. There are much simpler acceptance-rejection methods for univariate cases that produce iid sequences (for example, see my answer to question 1358708 on this site). Also, of course, sampling from the distribution of your reference is most efficiently done by a direct method that does not use acceptance-rejection at all. (I am not at all objecting to this kind of drill exercise as a way towards understanding Metropolis and M-H; I have assigned similar problems myself. But no one should mistake Metropolis or M-H as an optimal sampling method in the current context.)