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Consider a random variable $X\sim f(x)$, such that $$ f(x)=\frac{1}{c}\times K(x)\propto K(x), $$ where c: normalizing constant, K(x): the kernel of the distribution (ie the part which involves $x$). $f$ is unknown and complicated, in the sense that it does not resemble any known distribution.

The M-H algorithm is designed to simulate from $f$ based only on the kernel $K(x)$, which makes its application rather complicated. I wonder how easy the M-H algorithm would become if we know the normalizing constant $c$?

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  • $\begingroup$ This does not bring any information on how to simulate $f$. $\endgroup$
    – Xi'an
    Dec 20, 2020 at 14:15
  • $\begingroup$ Thanks for your comment. Well, it depends on the actual kernel. For instance, provided I know $c$, I could use inverse transform sampling (en.wikipedia.org/wiki/Inverse_transform_sampling) instead of M-H. This can be achieved by some numerical integration method. But if I still need to use MH, how that information would help me? that was my question. $\endgroup$
    – Ludwig
    Dec 20, 2020 at 14:24
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    $\begingroup$ However, the normalization constant will not cancel out on the M-H? $\endgroup$
    – Fiodor1234
    Dec 20, 2020 at 14:30
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    $\begingroup$ Since the question is about M-H, my answer remains that knowing the constant $c$ is not helping. $\endgroup$
    – Xi'an
    Dec 20, 2020 at 14:30
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    $\begingroup$ Fiodor, I am afraid so, so we shall conclude that MH is not the best choice in this case, since it does not make any difference to know the constant. Thanks all. $\endgroup$
    – Ludwig
    Dec 20, 2020 at 14:39

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