▶Proposal
A symmetric Gaussian proposal suggests a candidate near the chain's current position.
Markov Chain Monte Carlo: The Metropolis Algorithm
Sampling a target through accept-reject moves
MCMC constructs a dependent sequence whose long-run distribution is the target. The Metropolis rule accepts every move to higher density and some moves to lower density.
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The essential idea: relative target density is enough; the unknown normalization constant cancels in the acceptance ratio.
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What to notice
- The next state depends only on the current state.
- Rejected proposals repeat the current state.
- Proposal scale controls how efficiently the chain moves.
Propose, compare, accept or stay
Acceptance rule
The target-density ratio determines whether the candidate becomes the next state.
How should a Metropolis chain move?
Change the proposal step size. Small steps accept often but move slowly; large steps explore farther but are rejected more often.
The rug below the axis shows the first chain states; repeated positions reveal rejected proposals.
Metropolis sampling uses an accept-reject Markov chain to explore a target distribution using only relative probabilities.