# How can I calculate a probability from a likelihood, e.g. in the Metropolis-Hastings algorithm?

This is a follow-up to my previous question, how can I compute a posterior density estimate from a prior and a likelihood

I am having difficulty understanding how it is possible to calculate the probability of a parameter set given its likelihood.

For example, in the answer to the previous question, the likelihood is used to weight the probability of sampling from the prior distributions.

However, the answer suggested I use a sequential sampling algorithm, so I have chosen to try a Metropolis-Hastings approach. However, it is not clear to me how I can use likelihoods to determine the probability that I will accept a proposed parameter set.

I can empirically estimate (using the R fitdistr function) that the likelihood is approximately $L\sim\textrm{log-}N(8, 0.7)$, I suppose that I could calculate the probability as plnorm(L, 8, 0.7) and the probability of acceptance as $P(L_\textrm{new})/P(L_\textrm{old})$. But is there a way to do this without assuming that the likelihood has some particular distribution? Or is it standard practice to make such an assumption?

• I should clarify that I am actually assuming the functional form of the likelihood is double-exponential I didn't realize that $p(\textrm{accept}) = L_\textrm{new}/L_\textrm{old}$ was the correct calculation; I thought that I needed to calculate $p(L_\textrm{new})/p(L_\textrm{old})$. And to estimate $p(L)$ I used AIC(fitdistr()) to find the bset fit to the distribution of $L$ and so that I could determine the $p(L|\theta)$ for each value of $L$. – Abe Jul 27 '11 at 22:54