Metropolis-Hastings Method

In Importance Sampling and Rejection Sampling, the proposal distribution $q (x)$ needs to be similar to $p (x)$ . The Metropolis-Hastings method uses a proposal density $q (x; x^{(} t))$ that is dependent on the current state $x^{(} t)$ . A simple distribution such as a Gaussian centered on $x^{(} t)$ can be used.

Method

Evaluate $p^{⋆} (x)$ for any $x$ .
A tentative new state $x^{'}$ is generated from the proposal density $q (x^{'}; x^{(t)})$ .
Compute $a = \frac{p^{⋆} (x^{'})}{p^{⋆} (x^{(t)})} \frac{q (x^{(t)}; x^{'})}{q (x^{'}; x^{(t)})}$
If $a \geq 1$ , accept new state and set $x^{(t + 1)} = x^{'}$ , else set $x^{(t + 1)} = x^{(t)}$

Pros and Cons

Will still give answers in high-dimensional settings
Lengthy simulations may be needed for convergence, because of the quadratic dependence on the lengthscale-ratio. A random walk is extremely slow, and should try to be suppressed.

Suppressing Random Walks

Hamiltonian Monte-Carlo methods make use of gradient information to reduce random-walk behaviour.

Jethro's Braindump

Metropolis-Hastings Method

Method

Pros and Cons

Suppressing Random Walks

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