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 \(p^\star(x)\) for any \(x\).
- A tentative new state \(x’\) is generated from the proposal density \(q(x’;x^{(t)})\).
- Compute \(a = \frac{p^\star(x’)}{p^\star(x^{(t)})} \frac{q(x^{(t)};x’)}{q(x’;x^{(t)})}\)
- If \(a \ge 1\), 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.