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I have a quick question regarding implementation of Metropolis-Hastings for a particular problem I'm dealing with.

Suppose that I have a probability density function $P(X)$ for a continuous random variable $X$. In Metropolis-Hastings, I am required to compute acceptance probability $\frac{ Pr(x') Q(x_{t}|x') } {Pr(x_{t} Q(x' | x)}$. However, the probability of any single event in a continuous space is zero. Do I just replace $Pr$ with $P$ and go on my merry way?

My primary concern is that the $P(x)$ is not necessarily less than or equal to 1.

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1 Answer 1

up vote 1 down vote accepted

This is usually the way to convert from discrete probabilities to continuous probabilities. You can think of this as promoting $x$ to an $\epsilon$-neighborhood of itself, whose probability is roughly $\epsilon P(x)$.

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Can you provide a reference for this equivalence? –  duckworthd May 17 '11 at 5:04
    
You can try looking in Machine Learning books. –  Yuval Filmus May 17 '11 at 14:43

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