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Concept

Reversibility

The property of a random process that as much probability flows from one state to another as flows back. It is a condition on each pair of states separately, and it is sufficient for a given distribution to be the one the process settles into.

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

Rotating by √2 − 1 of a turn, 40 times. Points on a circle produced by repeatedly turning through the same angle.

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

dynamics · pigeonhole 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

Named alongside it

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

Detailed balanceEigenvectorEntropyIrrational rotationIterationMarkov chainMeasureMetropolis algorithmMonte CarloOrbitPigeonholeProbability density

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