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Concept

Transition matrix

A square table whose row for each state lists the chances of being found in each state one step later, so every entry is non-negative and every row sums to one. Multiplying it by itself gives the chances over several steps, and its eigenvalues say how fast the starting state is forgotten.

Named by 2 essays across 2 fields — 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 How many three-state chains constant rates can produce. s 0.05: 94.9%; s 0.1: 90.2%; s 0.2: 79.6%; s 0.3: 70.3%; s 0.5: 47.7%; s 0.7: 22.9%; s 1: 4.3%.

When a table of moves came from steady rates

A process that jumps between states at constant rates, watched once a year, produces a table of yearly moves that is the exponential of its rates. Most tables that anyone could write down are not — a random three-state table is only about one time in twenty-three — and the few that are can have two different sets of rates behind them, but only once the process has forgotten where it started.

analysis · The exponential

Named alongside it

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

Markov chainDetailed balanceDeterminantEigenvalueEigenvectorLogarithmMatrix exponentialMetropolis algorithmMonte CarloProbability densityReversibilitySampling

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