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I have a really basic question. But I am a little bit confused about that. How do I calculate the acceptance ratio within a Metropolis-Hasting step?

I have something like $min\left\{1,\frac{p(new)}{p(old)}\times \frac{h(old)}{h(new)}\right\}$.

Is $p(x)$ the DENSITY at point $x$ and $h(x)$ the PROBABILITY at point $x$? Or is it true that $h(x)$ is also the DENSITY at point $x$? That means I have to calculate the DENSITIES only and never the PROBABILITIES?

Sorry for this basic question, but some people/books mesh up this terms and no I'm really confused.

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    $\begingroup$ I actually disagree that the content of this question is a duplicate, although the title makes it appear as such. Perhaps this post should have a title that more closely reflects its content. $\endgroup$
    – Silverfish
    Oct 13, 2015 at 17:56

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Both $p(\cdot)$ and $h(\cdot)$ are probability densities: $p(\cdot)$ stands for the distribution you want to simulate from (target) but cannot, while $h(\cdot)$ stands for the distribution you can simulate from (proposal) and thus did simulate $new$ from.

In the special case where the distributions are on a finite or countably infinite set, $\mathcal{X}$, both $p(\cdot)$ and $h(\cdot)$ are probability mass functions (which are densities against the counting measure), that is, $p(new)$ is the probability to generate $new$ for the target distribution and $h(new)$ is the probability to generate $new$ for the proposal distribution that is used in the first step of the Metropolis-Hastings algorithm.

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    $\begingroup$ Thank you for your quick reply. But I have an additional question. Let's assume that $new\sim N(\mu,\sigma)$ and probability density $p(\cdot)$ is something like $\hat{p}(\cdot)\times N(\mu,\sigma)$. In this special case the distribution I draw from is a part of my target distribution. And $N(\mu,\sigma)$ does not depend on previous samples. Does that mean that for calculation only $min\{1,\frac{\hat{p}(new)}{\hat{p}(old)}\}$ is left? $\endgroup$
    – JohnScott
    Oct 13, 2015 at 20:49
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    $\begingroup$ Yes, indeed, there is a simplification in that case because the proposal is a component of the target. An example is when the prior is used as proposal. In that case, the Metropolis-Hastings ratio is the likelihood ratio. $\endgroup$
    – Xi'an
    Oct 13, 2015 at 20:53

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