Seeds, Streams, and Reproducibility

Why the same code gives different answers, and how a seed turns a random study into a repeatable one.

Ch 1 1.4

Explorer

Foundations & Data Quality

12 mindifficulty 1/5

Controls

Seed mode

Random seed

Every result on this page is a deterministic function of the seed and the controls.

Draws per runn2000

Parallel workersk4

Naive seeding (seed = worker index)

Off uses a properly spaced substream offset per worker.

Same seed, same answer

Two independent replays of the identical seed produce bit-identical estimates.

Replay A

3.14200

Replay B

3.14200

Identical?

yes

as guaranteed by determinism

A different seed (43) gives a different, equally valid estimate: 3.09800. Neither run is "more random" than the other — reproducibility is about the seed being recorded, not about the draws being predictable in advance.

Running Monte Carlo estimate of π

Grey band: 10th–90th percentile across 60 independent seeds, showing simulation error shrink with n.

this run (current seed)true π10th–90th pct across seeds

Parallel streams: k workers

Naive seeding (seed = worker index) — nearby small seeds can start highly correlated streams.

worker	seed used	first draws	π estimate
0	1	0.627, 0.003, 0.527, 0.981, 0.968	3.1580
1	2	0.734, 0.325, 0.285, 0.538, 0.875	3.1700
2	3	0.720, 0.039, 0.456, 0.075, 0.763	3.1700
3	4	0.924, 0.333, 0.222, 0.082, 0.234	3.1300

Provenance

What a reviewer or collaborator needs to exactly reproduce this run.

seed=42; n=2000; generator=mulberry32; hash=877b89b9

Report the seed, n, and generator alongside the estimate; the hash lets anyone verify they replayed the exact same draw sequence.

This run's estimate

3.14200

Effective seed

Spread (sd of estimate across seeds)

0.0390

n = 2000

Generator

mulberry32

deterministic 32-bit PRNG

“Random” does not mean “unreproducible”

A pseudo-random generator is a deterministic function of its seed: same seed, same stream of numbers, every time, on every machine. What is random is the relationship between the seed and the outcome you'd get for a different seed — not whether the computation can be repeated. Recording the seed converts a random-looking study into a repeatable one.

Predict before you look

You run a Monte Carlo simulation twice with the exact same seed and the same code. What happens?

Scenarios

Teacher-ready starting points.

Documentation

What this shows

A Monte Carlo estimate of π, computed from a seeded pseudo-random generator, replayed to show that identical seeds give identical answers, that a clock-seeded run gives a new answer every click with no seed to record, and that naively seeding parallel workers by their index can silently correlate their supposedly independent streams.

Determinism, streams, and simulation error

A pseudo-random generator is an iterated deterministic map xₜ₊₁ = f(xₜ) started from a seed x₀. Every draw thereafter is a fixed function of that seed, so the entire sequence — and any statistic computed from it — is exactly reproducible given the seed, the generator, and the number of draws taken.

Running k workers in parallel means running k independent copies of this map from k different starting states. If those starting states are not well separated in the generator's state space, the resulting streams can overlap or be strongly correlated, which silently destroys the independence a parallel Monte Carlo study assumes.

Even with a fixed seed, a single run's estimate is only one draw from the sampling distribution of the estimator. The spread across independent seeds — not any single run — is the honest measure of simulation error, and by the law of large numbers it shrinks like 1/√n.

x_{t+1} = f(x_t; seed = x_0)

π̂_n = 4 · (1/n) Σᵢ 1[Xᵢ² + Yᵢ² ≤ 1],   Xᵢ, Yᵢ ~ U(0,1)

SD(π̂_n) = O(1/√n)

Try this

Enter seed 42, note the π estimate, switch to a different seed, then switch back to 42 — the exact same estimate returns.
Switch to clock mode and click 'New run' five times. Try to write down which run produced which number before clicking again.
Turn on naive seeding with 6 workers and inspect the first-draws column for near-duplicate rows, then turn it off.
Drag n from 50 to 20000 and watch the grey spread band around π narrow.
Copy the provenance string, change nothing, reload the idea in your head: could a classmate reproduce your exact number from that string alone?

Common misconception

“Randomness means a computation can't be reproduced, and any seed is as good as any other.”

A pseudo-random generator is deterministic given its seed: the same seed always produces the same stream, so recording the seed makes a 'random' study exactly repeatable. But not all seeds are interchangeable in practice — seeding parallel workers naively by index (1, 2, 3, ...) can start highly correlated streams, which breaks the independence a parallel simulation relies on even though every individual worker is running the identical deterministic algorithm.

Ch. 1.4 — Simulation and Reproducibility