Field

What makes it checkable

A stated seed for every random draw, so a simulated figure comes out the same every time it is drawn. A closed form beside every simulation, so there are two routes to each number. And the simulation's own error counted as carefully as the method's: a table of correct cells that flags one of them most of the time, two methods whose comparison is sharper or noisier for sharing their draws, a run that stops when it looks done and reports what it was looking for, and draws aimed at a tail that look settled while their variance is infinite.

Twenty samples of 40, every one of them genuinely normal. Each panel is a quantile-quantile plot of 40 draws from a normal distribution. The worst point in the worst panel sits 0.87 standard deviations off the line. Anything a reader would reject here would be a false alarm.

The seed is part of the figure

Every other site in this fleet draws from a deterministic rule, so a figure either is or is not what it claims. Here the figures are samples, and a sample can be right by luck. That changes what a figure has to carry.

Coverage of four nominal 95% intervals, n = 30. Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Two routes to every number

A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree.

Twenty cells of an interval that is exactly 95%, 1,000 replications each. The t interval covers exactly 95% in every cell. Estimated at 1,000 replications its cells read 93.9% to 96.5%, and 2 of the twenty are flagged by their own ±1.96 standard errors.

A coverage table with its own error

Twenty cells estimating the coverage of an interval that is exactly 95%, at a thousand replications each, read from 93.9% to 96.5% — and a table like that flags at least one of its correct cells on 69.9% of honest runs. Ten times the replications does not repair it: at ten thousand the same table still flags one 63.3% of the time.

Which samples Wilson and Clopper–Pearson each cover, n = 50, p = 0.2. Each bar is the probability of one count, shaded by which interval built on that count contains 0.2. Both cover 95.1% of samples, only Wilson 0.0%, only Clopper–Pearson 1.6%, neither 3.3%. The correlation between their hits is 0.810, so on shared draws the variance of their difference is 4.891 times smaller than on independent ones.

The same draws for both methods

Two intervals computed on the same simulated datasets give a difference in coverage whose variance can be 4.891 times smaller than on separate datasets — or, for a pair that covers different samples, 1.164 times larger. Which one a comparison gets is an exact sum over the counts each interval covers, and a standard error that ignores the sharing covers 100.00% for one pair and 93.07% for the other.

Twenty runs simulating an exactly 95% interval, checked every 250 replications. Each line is one run's running estimate; the dashed band is where the Wilson interval of the running estimate still contains 95%, and a run stops, marked, the first time it leaves the band. 8 of these twenty stop before 10,000 replications. The exact probability of stopping, from the recursion over the count, is 29.54%.

A simulation that stops when it looks settled

A simulation of an interval that covers exactly 95%, checked every 250 replications for a significant departure and stopped when it finds one, flags that correct interval on 29.54% of runs. Stopped instead as soon as its estimate reaches 95%, it reports an interval that covers 94% as meeting its level on 37.21% of runs. Stopped when the estimate stops moving, it reports the right number — and has quietly chosen to run about fifteen hundred replications.

Estimating P(Z > 5) = 2.8665×10⁻⁷ with plain draws and with four proposals. One seed each. Plain simulation draws nothing past 5 in 100,000 and estimates zero throughout. At 100,000 draws the proposal N(5, 1) reads 1.009 of the truth, N(9, 1) 1.041, N(4.5, 0.25²) 1.039 and N(5, 0.3²) 1.008. Values above 2.2 are drawn at the top edge.

The draws aimed at the tail

The chance a standard normal exceeds 5 is 2.8665×10⁻⁷, and a plain simulation needs 349 million draws to estimate it to within ten per cent. Draws aimed at the tail and weighted back need 565. Aimed slightly too narrowly, the same method has an infinite variance, an interval that covers 86.0% and gets worse with more draws, and an effective sample size that reads healthier than a proposal that works.

Two companions on one simulation, two hundredfold apart. How many times as many draws each companion is worth, on the same 4,000 simulated samples of 40 observations. The coverage of the interval is estimated with the observed count as its companion, whose expectation is 12 exactly; they correlate at 0.2665 and the companion is worth 1.08 times the draws. The expected width is estimated with p̂(1 − p̂) as its companion, whose expectation is 0.20475 exactly; they correlate at 0.9977 because the width is a monotone function of it, and the companion is worth 214 times the draws — 856 thousand simulated samples' worth of precision from four thousand.

The check worth more than the check

The same exactly known companion that verifies a simulation can sharpen it. On one set of four thousand draws, one companion is worth 1.08 times the draws and another is worth 214 times them, and the factor is 1 − ρ² with nothing else in it.

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