Generator

sample

A generator in the probability library, called 17 times across 4 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

sample is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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.

`show: "metropolis"`

`show: "quasi"`

`show: "importance"`

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.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

row 0 of the transition matrix adds to one ×20
the grid's error in 4 dimensions falls like N^(−2/4) ×5
in 2 dimensions the grid is the faster ×2
in 8 dimensions the grid is the slower ×2
and a run of the chain reproduces it to within a twentieth ×1
and at four dimensions the two rates meet ×1
and holds no negative probability ×1
and is between 5 and 400 sharp ×1
and is positive everywhere on the interval, so no weight is unbounded ×1
and is repeated between 40 and 800 times ×1
and is run for between 2,000 and 200,000 steps ×1
and so is the weighted one — neither is biased ×1
and the Halton points' falls faster than that ×1
and the random one falls like N^(−1/2), whatever the dimension ×1
and the weighted one is appreciably quieter ×1
as much probability flows from i to j as back again ×1
between 32 and 2,048 points are drawn ×1
each dimension is a whole number between 1 and 12 ×1
each estimate uses between 50 and 5,000 draws ×1
in 1 dimension the grid is the faster ×1
so a step leaves the target distribution where it was ×1
the chain has between 5 and 24 states ×1
the discarded start is shorter than the run ×1
the discrepancy is probed on a grid of between 40 and 300 boxes a side ×1
the grid's error in 1 dimension falls like N^(−2/1) ×1
the Halton points are more evenly spread than the random ones on this measure ×1
the peak sits inside the interval ×1
the plain average is centred on the answer ×1
the points to fit a slope through are not all at one place ×1
the proposal is a probability density ×1
the random points' discrepancy falls like one over the square root ×1
the seed is a whole number the drawing can be made from ×1
the spread is measured over between 4 and 200 runs ×1
the sweep runs over between 3 and 8 sizes ×1
the sweep runs to between 2^8 and 2^18 points ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Probability

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

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

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

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.

The whole library · What the figures prove