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Concept

Probability density

A non-negative function whose integral over a region gives the chance of landing in it, and whose integral over everything is one. It is the continuous counterpart of a list of probabilities, and its value at a point is a rate rather than a chance.

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

Two unbiased estimates of one integral, and their spread. The sharply peaked integrand with the proposal density that follows it, above a strip plot of 200 estimates from each of two methods; the weighted estimates cluster 4.2 times more tightly about the same value.

Sampling where the answer lives

Monte Carlo error cannot be made to fall faster than the square root, so the only thing left to attack is the constant in front of it. Drawing points where the integrand is large, and dividing by how often they were drawn, leaves the answer alone and can shrink the noise many times over.

probability · monte carlo 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.

Monte CarloSamplingConvergence rateDetailed balanceEigenvectorEstimator biasExpectationImportance samplingIndependenceIntegralMarkov chainMetropolis algorithm

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