Normal Distribution
nothing is called 95% until it has been counted
Concept

Detailed balance — where it appears

The condition that makes a Markov chain's stationary distribution the one intended. A symmetric proposal that stands still when it is refused satisfies it against the uniform distribution; one that re-proposes until it succeeds does not, and samples proportionally to how many admissible neighbours a state has.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

The test, checked where the answer is known. A fourteen-unit trial at eight tolerances. At each one the admissible set is enumerated — 1534, 886, 304, 158, 126, 116, 102, 84 assignments — and its components counted, which is only possible because 3432 equal splits of fourteen units can be walked. The dots are the test, which walks none of them: two chains, one started at an assignment and one at its complement, compared on a statistic the rule was not handed. Filled marks are tolerances the enumeration says leave the set in more than one piece. The test fires on every one of them and on none of the others, 0 misses and 0 false alarms.

A test rather than a survey

A thin admissible set falls into an arrangement and its mirror image, and the walk that samples it is uniform on half the reference distribution for ever. That was found by enumerating fourteen units, and enumeration stops at twenty-four.

reach · Randomisation What the rule blocks is not where it splits. How much of the separating direction each probe carries, over 100 designs of 14 units whose admissible set is enumerated and split into two pieces. The deferral this field answers proposed the constraint's active set — which exchanges the tolerance box actually blocks — as a better probe than the design's own leverage, on the ground that leverage is a heuristic and the active set is the quantity. Modelled from the design and the tolerance, it reads 0.4272 against leverage's 0.5395, at 4.43 paired standard errors the wrong way. Counted exactly over the enumerated set — at a cost no trial can pay — it reads 0.3854, worse again. Both beat a random direction at 0.2622, so they are probes; neither beats the two the earlier field already had.

What the rule blocks

A balancing rule breaks the admissible set into pieces by refusing exchanges. Which exchanges it refuses is computable from the design and the tolerance alone, before any assignment exists — and it makes a probe.

blocked · Randomisation Three readings, one verdict. Every tolerance of a fourteen-unit trial, with three comparisons on each. The first is between a chain started at an assignment and a chain started at its complement, which is what a mirror split separates. The second is between two chains started at the same assignment on different streams, which nothing about the set can separate — so a large reading there says the run is too short and not that the set is in pieces. The third is the same comparison on the statistic's absolute value, which is symmetric under the complement and therefore blind to the split by construction. The verdict is the pattern rather than any one line: the split is called only where the first fires and the other two do not, which happens at exactly the tolerances the enumeration calls disconnected — 0.8, 0.75, 0.7.

The statistic that changes sign

A test for an unreachable half needs a quantity that tells one half from the other. Every symmetric reading of a mirror pair is identical, and a magnitude is the natural thing to reach for.

reach · Randomisation How long a walk has to be given. Every equal split of twelve units is enumerated, the 410 admissible ones are found, the transition matrix is built, and the distance from uniform is computed exactly at each step — no simulation anywhere. The walk is started at the least balanced admissible assignment, which is the state a rejection sampler is least likely to have handed it and the one a burn-in has to cover. It is 0.0849 away after twenty steps and 0.00008 after ninety. A real cost, and a small one, and naming it is what stops it being assumed to be zero.

Walking the admissible set

A rerandomisation test hunts for admissible assignments and throws away the rest. A walk visits them instead — and it is exactly uniform only because it stands still when a proposal fails, which is the step that looks like waste.

joint · Randomisation Where a walk is cheaper than a hunt. Both costs in the same unit. A rejection sampler evaluates 1/p assignments per independent draw and does not care how large the trial is; a walk evaluates one per step and yields an effective draw every τ steps, and τ is a property of the constraint and the statistic together. They cross at a tolerance of 0.194 standard deviations, where about one assignment in 396 is admissible — far tighter than any trial is designed at. And the walk does not remove the acceptance cost; it pays it once, hunting for somewhere to start.

Draws that repeat each other

A hunt costs 1/p evaluations per independent draw. A walk costs one per step and yields an effective draw every τ steps. Both are counted in the same unit, and the walk is dearer at every tolerance a trial is designed at.

joint · Reference A thin enough set is not one set. Every admissible set of 14 units this table can enumerate, by how much of the assignment space it admits and how many pieces it falls into under single swaps. A walk is uniform on the piece it starts in and never leaves it. The pieces are not fragments: at 522 admissible assignments the set splits into 3 halves of exactly 520 each, and every assignment's complement is in the other half — no sequence of admissible single swaps takes an assignment to its own mirror image. Two-swap proposals reconnect four of the six disconnected sets here — the two they do not are the thinnest, where a two-unit move rarely lands anywhere admissible either — which makes a bigger proposal a correctness repair rather than the speed dial it was measured as.

The walk that cannot cross

A thin enough admissible set is not one set. It splits into an assignment and its mirror image, no sequence of admissible single swaps joins them, and the walk that samples it is uniform on half the reference distribution for ever.

dict · Randomisation

Named alongside it

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

Markov chain Monte CarloRandomisation testBurn-inCovariate balanceReference distributionRerandomisationAssignment mechanismExact enumerationImbalanceCombinatorial searchConnected componentCritical value

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