A proposal distribution generates candidate values when drawing directly from a target distribution p(x) is difficult. It must be possible to sample from the proposal, and the sampling method must account for the difference between proposal and target. Two common roles are proposing the next state of a Markov chain and supplying samples for a weighted estimate of an integral. section 2.6 section 2.7
The proposal has a different form and correction in each method:
- In Metropolis–Hastings, q(x′|x) proposes a new state x′ given the current state x. Accepting or rejecting the candidate according to the Metropolis–Hastings ratio corrects for the proposal so that the target p(x) is stationary; the ratio is derived from detailed balance. Proposal choice affects acceptance. section 2.6
- In importance sampling, q(x) supplies samples for estimating an expectation under p(x). Each sample receives an importance weight w(x)=p(x)/q(x), so the estimate accounts for sampling from q rather than p. A useful proposal draws samples where both p(x) and |f(x)| are large. section 2.7
Two proposals in practice
For a two-dimensional Gaussian-mixture target, a multivariate Gaussian proposal generates candidate states in a Metropolis–Hastings sampler; a sample routine computes the acceptance ratio and keeps or rejects each candidate. For an importance-sampling estimate involving f(x)=2 sin(πx/1.5) on x≥0 and an unnormalized chi-shaped target, a Gaussian proposal supplies samples restricted to positive values. The routine evaluates target and proposal densities, weights f(x) by their ratio, and averages the results. section 2.6 section 2.7
