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Concept

Monte Carlo

Computing a deterministic quantity by averaging a random process designed to have that quantity as its mean. Its error falls like one over the square root of the sample size whatever the dimension, which makes it slow everywhere and the only workable method in high dimensions.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Also named here as sampling — the same set of essays touches all of them, so they are one junction rather than several.

120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Getting pi by dropping needles on the floor

Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.

probability · monte carlo How fast each way of averaging closes in, as the dimension grows. Relative error against the number of points, both on logarithmic scales, for a regular grid in 1, 4, 8 dimensions and for random points in 8; the grid's lines steepen or flatten with the dimension and the random one does not move from a slope of a half.

The error that does not care how many dimensions

A grid gets rapidly better in one dimension and hopelessly worse in twenty. Random points get better at the same slow rate whatever the dimension, which is why a method that is bad everywhere ends up being the only one that works.

probability · monte carlo 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 Points too even to be random. 256 independent random points beside 256 points of a Halton sequence, with the largest mismatch between a box's share of points and its area plotted against the number of points for both.

Points too even to be random

Independent random points clump, and the clumping is what makes the error fall only as the square root. Points chosen to be evenly spread rather than independently beat that rate, and the price is that nothing about them is random at all.

probability · monte carlo

Named alongside it

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

SamplingConvergence rateEstimator biasIntegralIndependenceProbability densityVarianceBuffon's needleCentral limit theoremConvergenceCurse of dimensionalityDetailed balance

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