PYTHON · NUMPY

What a pass certifies

Sixteen chains per superchain, one draw each, and every superchain started from the same cleared register. The rule passes at R-hat of 1.0356 or less. At a warmup of zero sweeps it reads:

Compute the convergence statistic a many-chain sampler reports, exactly, and name what its pass certifies and what it cannot see.

01Challenge

A p-bit array runs thousands of short chains and reports that a standard convergence check has passed. Compute what that check would read, without running a chain, and find the conditions under which it passes on an answer that is still wrong.

Predict firstSixteen chains per superchain, one draw each, and every superchain started from the same cleared register. The rule passes at R-hat of 1.0356 or less. At a warmup of zero sweeps it reads:

02Model

A p-bit array runs thousands of chains at once, each only a few sweeps long, and it needs a rule for when to stop. The standard rule, R-hat, asks whether chains started in different places have come to agree, and it was built for a few long chains. Nested R-hat, from Margossian, Gelman and colleagues in 2024, was built for the many-short-chains regime. It groups chains into superchains that share a starting point and watches how much the superchains still disagree, which the authors call the nonstationary variance, and so, by proxy, the squared bias. Every part of it can be computed exactly from the machine's one-step law, with no chain run at all, and that exactness shows where the proxy holds and where it does not. From starting points spread across the space it holds: on ten-spin glasses, when the rule first passes, the squared bias is at most half the tolerance. Three things it cannot see. A start every superchain shares, such as a cleared register, leaves nothing to disagree about, so the statistic sits exactly at its floor from the first sweep and passes on an answer that has not moved. A bias every start shares, which a cold ferromagnet's energy has, passes three tolerances early. And a machine that converges to the wrong law, as a synchronous sweep does, passes as soon as it has converged, with its energy here nine standard deviations from the Boltzmann value. A convergence check certifies that the machine has settled, into whatever it settles into. Whether that is the distribution you asked for is a second question, and it needs a second measurement.

03Guided practice

Write the population nested R-hat from the kernel, check that a shared start reads exactly its floor at every warmup, then find where the rule first passes in three cases and measure the squared bias against the Boltzmann law at that point.

04Feedback

PASS WHEN You computed a convergence diagnostic from the machine itself and found where its pass means what it is taken to mean. From dispersed starts on a kernel whose law is the one you want, it does. From a shared start, or on a machine that samples the wrong law, it passes anyway.

If your run fails, check:

Advance both f and f*f by warmup + 1 steps: the one draw each chain makes comes after the warmup sweeps and one more.
Each start's variance is its second moment minus its mean squared; the nonstationary variance is the variance of the per-start means under the start law.
With one draw per chain there is no within-chain variance, so W is the expected per-start variance and the persistent part of B is that divided by M.

05Retrieve & space

Nested R-hat groups short chains into superchains that share a start and measures how much the superchains still disagree.
It can be computed exactly from a machine's one-step law, which shows the value the statistic converges to without running a chain.
From a start every superchain shares, the statistic reads exactly its floor, the square root of one plus one over the chains per superchain, at every warmup, and passes at once.
A bias that every start shares, and a machine that converges to the wrong law, both pass the rule.
A convergence check certifies that the machine has settled; whether it settled into the distribution you asked for needs a separate measurement.

06Mastery & project

Given a sampler's convergence report, state the conditions under which its pass implies a small bias, and name the separate measurement that checks the distribution it converged to.

Take a sampler you run with many short chains, compute or estimate its nested R-hat from dispersed and from shared starts, and report the squared bias at the warmup where the rule first passes in each case.

← Every spin at once Conditioning bandwidth, and what is still open →

PYTHON · NUMPY

What a pass certifies

The triangle's exact one-step laws, and three ways of starting and running it. Nothing is sampled: every number is the value the statistic converges to.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you