Metropolis–Hastings invariance
Theorem taught in Markov chains, Gibbs sampling, and Metropolis–Hastings, part of Bayesian statistics
Metropolis–Hastings invariance
| Type | Theorem |
| Taught in | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Part | Bayesian statistics |
| Worked examples | 1 |
The Metropolis–Hastings transition defined by Equation (5) in Markov chains, Gibbs sampling, and Metropolis–Hastings satisfies detailed balance with respect to \pi. Consequently, \pi is stationary.
Examples
C14. Metropolis–Hastings cancellations
Question page: C14. Metropolis–Hastings cancellations
Why can Metropolis–Hastings use an unnormalised target? When do its proposal-density terms cancel?
Definitions/notation. Let \pi(z)=k(z)/Z be the target and q(y\mid x) the proposal density from current state x to candidate y. The acceptance probability is \min\{1,k(y)q(x\mid y)/[k(x)q(y\mid x)]\} because Z cancels; the proposal terms cancel only when q(y\mid x)=q(x\mid y), as for a fixed-scale symmetric random walk.
Used in
- Taught in Markov chains, Gibbs sampling, and Metropolis–Hastings, where the derivation and examples are.
- All 31 questions on Markov chains, Gibbs sampling, and Metropolis–Hastings
See also
Metropolis–Hastings update · Stationarity and detailed balance · Markov chain · Detailed balance implies stationarity · Systematic-scan Gibbs sampler · Warm-up is not a convergence proof