Ising model
Topic of Markov chains, Gibbs sampling, and Metropolis–Hastings, part of Bayesian statistics
Ising model
| Type | Topic |
| Taught in | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Part | Bayesian statistics |
A model for binary spins on a lattice, x_i\in\{-1,+1\}, whose joint law favours neighbouring sites that agree, p(\mathbf{x})\propto\exp(\beta\sum_{i\sim j}x_ix_j). Its normalising constant sums over 2^m configurations, but ratios of unnormalised weights are easy, which is why Metropolis methods came from statistical physics.
In Markov chains, Gibbs sampling, and Metropolis–Hastings
This cancellation is why Metropolis methods became central in statistical physics. An Ising model on m binary sites has 2^m configurations, making its partition function impractical to sum, but ratios of unnormalised target weights are immediate. The original Metropolis algorithm used symmetric proposals; Hastings’ generalisation supplied the proposal-density correction in Equation (5) in Markov chains, Gibbs sampling, and Metropolis–Hastings.
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See also
Markov chain · Stationarity and detailed balance · Detailed balance implies stationarity · Systematic-scan Gibbs sampler · Warm-up is not a convergence proof · Metropolis–Hastings update · Metropolis–Hastings invariance · Markov chains and equilibrium