# Metropolis-Hastings algorithm for a completely specified distribution

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 $$c$$?

• This does not bring any information on how to simulate $f$. Dec 20, 2020 at 14:15
• 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. Dec 20, 2020 at 14:24
• However, the normalization constant will not cancel out on the M-H? Dec 20, 2020 at 14:30
• Since the question is about M-H, my answer remains that knowing the constant $c$ is not helping. Dec 20, 2020 at 14:30
• 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. Dec 20, 2020 at 14:39