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?