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Concept

Detailed balance

The condition that a Markov chain's traffic along each edge is equal in the two directions, one edge at a time. It is strictly stronger than being stationary, and where it holds the long-run shares can be read off the edge weights with no system to solve.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A walk that visits each state as often as its weight says. The target distribution over 12 states with the share of a 40,000-step Metropolis run beside each bar, above the first 300 steps of the walk itself; the two distributions differ by 0.5 per cent in total.

A walk that samples a distribution

When a distribution can be evaluated but not drawn from, a wandering point can be arranged to visit each state as often as its weight says. The rule needs no normalising constant, compares two weights and steps or stays.

probability · Monte Carlo A walk on a weighted graph, and a cycle whose traffic goes one way. A weighted graph with the long-run share of each state read off its total weight, beside a three-state cycle whose shares are equal and whose traffic circulates.

The chain that runs the same backwards

Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.

probability · Markov chains

Named alongside it

The objects these essays reach for when they reach for this one.

Markov chainStationary distributionEigenvectorGraphInvariantMetropolis algorithmMonte CarloProbabilityProbability densityRandom walkReversibilitySampling

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