Two refinements that do not multiply

The rise at which half a set of starting angles still settle onto a lattice is the wall, and the interval of rises consistent with that crossing is its bracket. The design that went and located it changed two things at once: it doubled the starting angles from forty to eighty, and it cut four extra rises into the stretch the wall crosses.

The reasoning written down for that design was that each would buy about a factor of two. Doubling the sample halves the error on a share; halving the step between rises halves the interval a crossing can hide in. Two halvings, and a bracket four times narrower.

Both halvings happened. The bracket did not narrow four times, and the reason is not that the estimate of either was optimistic. It is that the two refinements are not two things.

One table, four corners #

The eighty starting angles contain the forty, and the nine rises contain the five published ones the refinement kept — 0.03, 0.02, 0.013, 0.008 and 0.005. That was a rule of the design rather than a convenience, and it is what makes this decomposition free.

Read the whole table and the answer is the corner with both refinements in it. Read it on the forty angles only and the answer is the corner with the finer rises alone. Read it on the published rises only and the answer is the corner with the doubled angles alone. Read it on both restrictions and the published reading comes back exactly.

Four answers, no run grown twice, and no arithmetic joining them.

Which is the only reason the question is answerable #

A two-by-two normally costs four experiments. This one costs none, because the containment rule this thread has followed at every doubling makes each smaller design a sub-table of the larger.

Without it the four corners would be four sweeps of a stochastic quantity, and the difference between two of them would carry the noise of both. Here the published corner is not an estimate of the published reading; it is the published reading, cell for cell.

What is being measured across the corners #

The geometric mean of the four brackets, in the ratio of rises. A bracket runs from the finest rise whose settling share is confidently above the level to the coarsest whose share is confidently below it, so its natural size is a ratio and its natural average is geometric.

At a half only two of the four exponents have a closed bracket at every corner — exponents 2 and 3 — so the mean is taken over those two. Averaging four brackets at one corner and two at another would measure which exponents each corner can close rather than how wide its brackets are, and that is a different quantity with the same units.

The four numbers #

On forty angles and the published rises the mean bracket is 3.062 times. Doubling the angles alone gives 1.961. Halving the rise spacing alone gives 1.458. Both together give 1.445.

The count of closed brackets moves with them: two of four at the published corner, three of four when the angles double, and four of four at either corner with the finer rises.

Read those four numbers in the order the prediction expects and the prediction is already gone. Both together is barely better than the rises alone, and the rises alone are far better than the angles alone.

The narrowing, in the logarithm #

Brackets are ratios, so their effects add in the logarithm rather than multiplying in the number. The joint narrowing is −0.751; the angles alone give −0.445 and the rises alone −0.742.

Those are the whole of the result. Everything below is what they mean.

Ninety-nine and fifty-nine #

As a share of the joint narrowing, the finer rises account for 98.8 per cent and the doubled angles for 59.3 per cent.

The rises did essentially all of it. The angles, taken on their own, did rather more than half of it — which is not nothing, and is the part that makes the arithmetic worth writing out rather than dismissing.

A sum of one hundred and fifty-eight #

Add the two shares and they come to 158 per cent.

Two causes that each explain most of an effect and sum to half again as much as the effect are not two independent factors. They are two descriptions of the same narrowing, arriving by different routes and overlapping almost completely.

If the two had multiplied, the shares would have summed to a hundred by construction. If they had been unrelated, they would have summed to a hundred as well. A sum well above a hundred is the signature of substitutes: either one alone gets most of the way there, and the second has almost nothing left to buy.

What the angles bought on top of the rises #

0.9 per cent of the narrowing.

That is the number the design was really bought for, and it is the honest way to price the expensive half. Two thousand and eighty stems were grown, half an hour of machine time went into them, and their contribution to locating the wall — given that the rises were being refined anyway — is under one part in a hundred.

The floor a rise list imposes #

Here is the mechanism, and it is one sentence. A bracket’s ends are rises. The interval is not a computed range around a point estimate; it is a pair of entries from the list, chosen by which of them the shares put confidently on either side of the level.

So a bracket can never be narrower than the step between two adjacent rises. The published list steps by 1.538 and 1.625 across the stretch the wall crosses, and the narrower of those is 1.54. No number of starting angles can produce a bracket below it, for the same reason a nine-rise sample cannot see a feature one rise wide: a list is an instrument, and its step is a floor on everything read off it.

Which is why one corner is stuck #

The corner with eighty angles and the published rises comes out at 1.961. It cannot do much better: the floor is 1.54, and 1.961 is one step above it in a list whose steps are half again.

Doubling to a hundred and sixty angles, or to three hundred and twenty, would move that corner towards 1.54 and stop there. The rise list, not the sample, is what that corner is measuring.

The other corner is not #

Forty angles on the refined rises gives 1.458 — already below the floor the published list imposes, on half the runs.

That is the comparison that settles it. The cheap half of the design, applied alone to the sample that was already on disk, beats the expensive half applied alone by a wide margin, and beats the floor the expensive half can never get under.

Exponent 2, which is the argument in one bar #

Its bracket ran from 0.02 down to 0.008 — 2.50 times — and now runs from 0.02 to 0.013, at 1.54.

Both ends of both intervals are published rises. Nothing was gained by locating a crossing between two rises; what happened is that the shares at the four added rises decided which pair of entries the bracket takes, and the pair it now takes is the closest the published list holds across the stretch the wall crosses. Exponent 2’s bracket, after a design that doubled the sampling and cut four new rises, is sitting exactly on the floor the old list imposes there.

The error did exactly what it was supposed to #

The binomial standard error on a share falls from 0.0791 at forty starting angles to 0.0559 at eighty. That is a factor of 0.707, which is the reciprocal square root of two, which is what doubling a sample does to a proportion’s error and the only thing it does.

Nothing about that number went wrong. It was quoted beside the wall for three readings across which the wall’s own spread refused to shrink, and it fell precisely as arithmetic requires each time.

And it was the wrong error bar #

It is the error of the share, and the wall is not a share. The wall is the rise at which the share crosses a level, and the error on a crossing depends on how quickly the share moves with the rise as well as on how well each share is known.

Here the share moves slowly through the level. A column that falls from about three quarters to about a tenth across the whole table is nearly flat where the crossing sits, so halving the uncertainty on each point moves the crossing hardly at all. The reading that first measured the brackets said so before any stem was grown for this. The two-by-two is the confirmation.

Why the prediction was reasonable anyway #

Because the alternative — that a bracket’s width is set by the list it is read from — is obvious once stated and easy to miss when the two refinements are packaged as one design.

It is the same shape as a grid that decides an answer while nobody varies it: a setting inside the instrument does the work, the quantity that everybody watches moves correctly, and the two are not connected. The error bar behind the wall halved four times across this thread and the wall’s own spread never did.

A bracket is not a confidence interval #

That distinction is the whole of what this essay has to sell. A confidence interval is continuous, it shrinks as the root of the sample size, and it has no floor.

A bracket is discrete. Its ends are drawn from a finite list, it shrinks by jumping from one pair of list entries to a closer pair, and it has a floor equal to the finest step in the list. Below that step it does not shrink slowly; it stops.

Sampling improves a bracket only by deciding, more confidently, which pair of rises to take. Once the confident pair is the adjacent pair, more sampling buys nothing at all.

Which is what a share of 59.3 per cent is made of #

Not a partial improvement of a continuous interval. It is the doubled angles moving one of the two open brackets to a closed one and tightening another by one step of the list — and then having nowhere left to go.

Counting closed brackets says the same thing more bluntly: two of four before, three of four with the angles alone, four of four as soon as the rises are refined.

The decomposition belongs to a level #

Everything above is read at a half, because that is the level the four walls were published at. The decomposition changes when the level does, and at one level it changes sign.

At two fifths, all four exponents have a closed bracket at every corner, so the mean is over four rather than two. The published corner reads 2.267, the finer rises alone read 1.604 — and the doubled angles alone read 2.812, which is wider than the corner they started from.

An improvement that widens an interval #

That is not a contradiction and it is worth being clear about why. A confident share is a share whose error bar clears the level, and halving the error bar makes more rises confident in both directions.

If a rise that was undecided becomes confidently on the far side of the level, the bracket grows to include it. The interval got more honest, not worse: it now covers rises the coarser sample had no right to exclude. As a share of the joint narrowing at two fifths, the angles alone score −68.5 per cent and the rises alone 110.0 per cent.

So the number is not one number #

The share the rises account for is 98.8 per cent at a half and 110.0 per cent at two fifths. The share the angles account for is 59.3 and −68.5.

That is the same dependence the ordering of the four walls turned out to have, arriving in a different quantity. A statistic read at one level, on curves whose ceilings differ, is a statistic about the level as much as about the subject — and the decomposition inherits it, because it is a statistic about brackets and a bracket is read at a level.

What survives every level tried is the sign and the ordering: the rises account for more than the angles, everywhere.

What this does not say about the runs #

Nothing at all. The 2,080 stems that were grown are grown, their shares are the most precise quantity in the table at an error of 0.0185 over 720 runs an exponent, and the pooled settling share has converged — which is a positive the doubled angles bought and this decomposition does not touch.

The claim is narrow: for the purpose of narrowing a bracket, at the levels this table can be read at, the doubled angles were nearly redundant with a change that costs no runs at all.

What it does not say about the wall #

It does not say the wall is badly measured. Every bracket is closed, the four walls sit inside a factor of 1.111 of one another, and the reading is much better than the one it replaced.

Nor does it say the four exponents differ. They do not separate at a half by any pair, and the decomposition is silent on whether they ever would. What the four added rises did buy is of a different kind altogether: a maximum in the settling share that the published list stepped over, which is a feature of the curve rather than a threshold on it.

What would have made them multiply #

Two refinements multiply when they attack independent limits. Doubling the angles attacks the error on a share; halving the rise spacing attacks the resolution of the list. Those are independent limits, and they would have multiplied — if the binding one had been the share error.

It was not. The list was binding at every corner, both before the refinement and after it, so one refinement was working on the limit and the other was working on slack. Doubling a sample against a limit it does not touch is a sample that is confidently answering the wrong question, and the tell is the same: the number that comes back is precise and the precision is about something else.

What a design should have done instead #

Spent the whole budget on rises. The refined list uses steps of 1.155 and 1.176; the same half hour spent entirely on rises at forty angles would have bought a list finer still, and the corner at forty angles with refined rises is already the second-best of the four.

That is not a criticism available before the fact, which is the reason to write it down after it. Nothing in this thread had ever decomposed a refinement, and the assumption that two halvings multiply had never been examined because no design had done two at once.

Which is the general lesson and it is cheap #

Any design that changes two things at once, where one design’s inputs contain the other’s, can be decomposed for free. The cost is arithmetic on a table already on disk, and the return is knowing which half was paid for.

This thread has now run that check twice on the same table — once to show a level was doing the ordering and once here — and both times the expensive, obvious refinement turned out to be working on the wrong variable.

What is claimed #

That the eighty starting angles contain the forty and the nine rises contain the published five, so all four corners of the two-by-two are readings of one table with no run grown twice.

That the geometric mean bracket over the exponents closed at every corner runs 3.062 times at forty angles on the published rises, 1.961 with the angles doubled alone, 1.458 with the rises refined alone and 1.445 with both — so in the logarithm the rises account for 98.8 per cent of the joint narrowing, the angles for 59.3, the two summing to 158 per cent, and the angles adding 0.9 per cent on top of the rises.

That the mechanism is that a bracket’s ends are rises, so the published list floors any bracket at 1.54 times whatever sampling is thrown at it — and that at two fifths the doubled angles alone leave the bracket wider than they found it, at −68.5 per cent of the narrowing, because a smaller error bar makes more rises confidently excluded and confidently included alike.

And that the binomial error fell from 0.0791 to 0.0559, a factor of 0.707 exactly as arithmetic requires, while being the error of a share rather than of the rise at which that share crosses a level.