Must the proposal equal the target distribution?

No. The algorithm corrects their difference through an appropriate acceptance or weighting mechanism.

Is a normal random walk always suitable?

No. Constraints, scale differences, dependence, and multiple modes can make it inefficient or inappropriate.

What happens after rejection in Metropolis - Hastings?

The next state remains the current state, and that repeated value is part of the recorded chain.

AI Glossary / Statistics and probability / Intermediate

Proposal distribution

Definition

A proposal distribution is a distribution used to generate candidate values in a sampling algorithm. In Metropolis - Hastings, it specifies the probability or density of proposing a new state given the current state, commonly written q(y given x). A separate acceptance rule decides whether the candidate becomes the next state. The proposal distribution is generally different from the target distribution that the algorithm is intended to sample.

What it means

Sampling directly from a complicated target can be difficult, even when its density can be evaluated up to a constant. A proposal supplies a convenient way to suggest possible moves. The acceptance mechanism corrects for differences between those suggestions and the desired target.

Proposal choice affects computational efficiency. Very small moves can lead to highly correlated samples, while unsuitable large moves can be rejected frequently. A useful proposal also needs access to the relevant parts of the target's support. Good-looking acceptance rates alone do not establish that a sampler explores all important regions or has produced reliable estimates.

How it works

At current state x, draw a candidate y from q(y given x). In the ordinary Metropolis - Hastings construction, accept it with probability min(1, pi(y) q(x given y)/(pi(x) q(y given x))), where pi denotes the target density or mass and the ratio is considered where defined.

If the candidate is rejected, retain x as the next chain state. Repeated states are part of the algorithm and must not be silently removed. If the proposal is symmetric, the forward and reverse proposal terms cancel. Otherwise, dropping them generally changes the stationary distribution.

Choose a proposal appropriate to the geometry and support of the problem, then assess mixing and effective sample size using suitable diagnostics. Independent proposals, random-walk proposals, and more structured transitions have different behavior. Adaptive changes require conditions that preserve valid sampling; repeatedly tuning a proposal during collection is not automatically harmless.

Three practical examples

A Bayesian computation routine proposes nearby parameter values while exploring a posterior distribution.
An importance sampler draws from a convenient distribution and corrects the mismatch with target-to-proposal weights.
A rejection sampler uses an accessible proposal together with a valid acceptance construction to produce target-distributed draws.

Think of it this way

Think of the proposal as a guide suggesting where to visit next, while an acceptance rule checks whether following that suggestion is consistent with the intended sampling itinerary. A guide can make poor suggestions even when the checking rule is correct, slowing exploration without necessarily changing the target of a valid algorithm.

Worked example

Consider an original two-state target with pi(A) = 0.70 and pi(B) = 0.30. Let the proposal probabilities be q(B given A) = 0.80 and q(A given B) = 0.40. The remaining proposal probabilities are q(A given A) = 0.20 and q(B given B) = 0.60.

For a proposed move from A to B, the acceptance probability is min(1, (0.30 × 0.40)/(0.70 × 0.80)) = 3/14, approximately 0.21429. The reverse move is accepted with probability one because the reciprocal ratio exceeds one.

The target-weighted flow from A to B is 0.70 × 0.80 × 3/14 = 0.12. The reverse flow is 0.30 × 0.40 × 1 = 0.12. They balance.

Ignoring the asymmetric proposal terms would give a different acceptance probability and break this calculation. The example checks the transition logic exactly; it does not rely on a simulated chain or claim that every proposal with a valid correction is efficient.

Illustrative case study

Fictional case study: Cedar Modeling builds a sampler for a custom probabilistic model. Its first proposal changes several tightly related parameters independently, producing many poor candidates and a slowly moving chain.

The analysts review the parameter geometry, compare better-scaled proposals, and inspect effective sample sizes and behavior across runs. They preserve the acceptance correction and keep repeated states. This fictional case illustrates the distinction between sampler validity and efficiency, not a recommendation for a universal proposal or acceptance-rate target.

Common misconceptions and mistakes

A proposal is not itself a draw from the target merely because the target density appears in an acceptance formula. Accepted moves and retained states together form the chain, whose convergence and sampling properties require appropriate conditions.

Do not omit the reverse-proposal term unless symmetry or another valid simplification justifies it. Removing rejected iterations also changes the empirical distribution of recorded states. A proposal that cannot reach relevant target regions cannot be rescued by longer runtime alone. Finally, a high acceptance rate can coexist with tiny, highly correlated moves and poor exploration.

Business use

Proposal distributions matter when businesses use custom Monte Carlo methods for uncertainty estimation, simulation, or Bayesian analysis. Their design can strongly affect how much useful information is obtained for a given computational budget.

Prefer established, tested sampling implementations where suitable. For a custom method, document the target, proposal, acceptance or weighting rule, support, and diagnostics. Evaluate numerical stability and whether important regions are explored. Communicate Monte Carlo uncertainty separately from uncertainty inherent in the model and data; more iterations do not correct an inappropriate substantive model.

Check your understanding

A proposal has q(y given x) different from q(x given y). Can its terms generally be canceled from the Metropolis - Hastings ratio?

Answer: No. The asymmetry must be included unless a separate valid argument provides the required simplification. Otherwise, the transition rule may target the wrong distribution.

Related terms

Learn next

Start with discrete Markov chains and calculate transition probabilities by hand. Then study Metropolis - Hastings acceptance ratios for symmetric and asymmetric proposals. Compare proposal scales using convergence and efficiency diagnostics, and learn support requirements before implementing a custom sampler.

Sources

Timo Koski's KTH notes, On the Discrete Metropolis–Hastings Algorithm, explain proposal distributions, acceptance correction, and the target stationary distribution: https://www.math.kth.se/matstat/gru/sf2955/metropolis3.pdf

The asymmetric two-state construction, balanced-flow calculation, and fictional case are original. The worked example uses discrete masses; continuous algorithms use corresponding densities and require their own regularity conditions.

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