The split at the critical value

Two effects in one number found the missing term in the field’s reading of two searches on one sample. The excess — what a rule charging the two searches separately over-charges — is the net of two things, and one extra supremum per draw separates them exactly. Run the first search alone, pin it at its answer, and search the second: the gap between that sequential run and the sum of the two separate searches is overlap, ground both of them find; the gap between the joint search and the sequential run is interaction, configurations only the joint search reaches. The excess is overlap minus interaction, on every draw, as an identity.

Every number in that field is an average, and so is every number in the sequence it splits, from the control to the pair the first field on two searches measured. The overlap at the critical value then read the excess where a test uses it — at the joint statistic’s 95% point — and found it larger on every pair, including two searches over independent columns whose excess on average is nothing. It left the field’s own question unanswered: is the over-charge at the critical value ground found twice, or reach only the joint search has? This essay splits it.

The same identity, read at a quantile #

On the doubled log-likelihood scale a critical value is quoted on, with $A$ and $B$ the two searches alone, $C$ the joint search and $S$ the sequential run, the split at a quantile level $p$ is

$$q_p(A) + q_p(B) - q_p(C) \;=\; \bigl[q_p(A) + q_p(B) - q_p(S)\bigr] \;-\; \bigl[q_p(C) - q_p(S)\bigr].$$

It is the same cancellation as the average split, so it holds at every level exactly. And the second bracket, the interaction, is still never negative: the joint search’s feasible set contains the sequential run’s answer, so $C \ge S$ on every draw, and the quantiles of a quantity that is never smaller sit at or above the other’s at every level. Both properties were checked on all five pairs at all five levels: the split adds to the quantile excess to machine precision, and no interaction is below zero.

The overlap climbs and the interaction does not #

For the two independent dictionaries the average split is an overlap of −0.02 and an interaction of 0.06 — two small effects nearly cancelling, which was the field’s finding about its own control. At the 95% point the overlap is 2.44 and the interaction 0.06; at the 99% point 5.72 and 0.00. The interaction did not move. The whole of the control’s over-charge at the critical value is in the overlap term.

For the two step dictionaries, cut a few rows apart, the average split is an overlap of 4.76 and an interaction of 4.59, nearly cancelling to an excess of 0.16. Read at the median, the 75%, 90% and 95% points, the interaction is 4.87, 4.55, 4.90 and 4.75: it barely moves across the whole body of the distribution. The overlap goes −0.73, 8.19, 22.53, 31.02. At the 99% point the interaction finally falls, to 1.10, while the overlap reaches 47.75.

For the break and the window, the average split is 21.68 of overlap and 1.46 of interaction. At the 95% point it is 53.79 and 3.22. The interaction has roughly doubled and the overlap has more than doubled from a much larger base; in units, the overlap has grown by thirty-two and the interaction by under two.

So the pattern holds on all three: the interaction travels into the tail more or less as it is, and the overlap takes the tail.

What the overlap means at a quantile #

On average the overlap is ground both searches find — the share of what the second search finds that the first search has already taken. That reading depended on the average being linear: the mean of a sum is the sum of the means, so $\bar A + \bar B - \bar S$ is the shortfall of the sequential run against the two searches done separately, and nothing else.

At a quantile that linearity is gone. The sequential statistic is a sum — the first search’s statistic plus what the second adds once the first is pinned — and a quantile of a sum is not the sum of quantiles. So $q(A) + q(B) - q(S)$ contains the arithmetic of adding tails, whatever the two searches share, and for two right-skewed statistics that arithmetic is positive at every upper level.

The control makes this concrete. Its two searches are over independent noise columns, correlated draw by draw at 0.025; there is no ground for them to share, and on average the split says so. At the 95% point the split’s overlap is 2.44 — nearly two thirds of a single parameter’s 95% critical value — on a pair with nothing in common. At the critical value the split’s overlap is no longer a measure of shared ground, and the identity that defines it gives no sign of the change. A reader carrying the field’s average reading into the tail would conclude that two independent dictionaries overlap substantially, and that conclusion is refused here with the control’s own numbers beside it.

The question the convolution left #

The previous essay explained the control’s tail over-charge by convolving the two searches’ null laws: pairing every value of one search with values of the other at random gives the joint statistic two searches would have if they shared nothing, and its 95% point, 12.16, followed the joint search’s 12.57 rather than the summed 14.96. It left a few tenths unexplained, and the obvious candidate was interaction — configurations of two columns that fit together better than either alone, which a random pairing cannot reproduce.

The split says otherwise. At the 95% point the control’s interaction is 0.06, the gap between the joint statistic’s 12.57 and the sequential run’s 12.51. The few tenths sit between the sequential run and the convolution: the second search, run on residuals the first search has already improved, does slightly better in the tail than an independent draw of the same search would. That is a property of fitting the same response twice, not of the joint search’s extra reach, and in the split it lands in the overlap term, as a small negative contribution to the arithmetic of tails.

Component by component #

The fourth live pair, a break and an independent column, is the one the field found complementary: on average its joint search finds more than its two parts, an excess of −2.50 made of an overlap of −2.16 and an interaction of 0.33. At the 95% point its overlap is 1.54 and its interaction exactly 0.00 — the two statistics’ 95% points coincide. The complementarity that defined this pair on average has gone from its critical value: at the point a test uses, the joint search is not reaching anything the sequential run does not.

That is the sharpest form of the essay’s finding. The field’s average split told its five pairs apart — a control of two small effects cancelling, step dictionaries whose two large components nearly cancel, a complementary pair with a negative overlap, a break and a window mostly overlap, a containment that is nothing else — and at the critical value every live pair is dominated by overlap and the distinctions are gone. What survives is the containment, whose interaction is zero on every draw and therefore at every level, and whose overlap is its whole excess, because its joint search is its first search.

Two readings of the interaction in the tail #

There is a second way to ask whether the interaction grows in the tail, and it gives a different answer, so the two have to be kept apart.

Read draw by draw, the interaction $C - S$ is a random quantity of its own. On the control it is exactly zero on 85.4% of draws — the joint search’s best pair of columns is the sequential run’s — and its own 95% point is 0.49. On the step dictionaries it is zero on 44.9% of draws and its 95% point is 17.65. On the break and the window, zero on 26.7%, 95% point 5.62.

So on a bad draw the interaction can be large: nearly four times its mean, on the step dictionaries and on the break and the window alike. That is not what the split at the critical value measures. The gap $q(C) - q(S)$ asks how far apart the two statistics’ 95% points are, and the draws where the interaction is large are not the draws where the joint statistic is in its tail; on the step dictionaries the gap at the 95% point is 4.75 while the per-draw interaction’s own 95% point is 17.65. A test cares about the gap. The per-draw tail says the joint search sometimes finds something the sequential run misses badly; it does not say the critical value moves by that much.

What this changes about the field’s sequence #

What a zero is made of used the split to read the field’s sequence: a control whose zero was two cancelling effects, a pair that was what it looked like, a pair whose net hid a third of its larger component. Every one of those readings is about averages and remains true of averages. The averages are still the right summary of what two searches find together; they are the wrong summary of what a test has to charge for letting them look.

For a test the sequence needs a different reading. Every pair over-charges at the critical value, mostly through a term that carries the arithmetic of adding tails whatever the searches share, plus a small interaction that sits close to its average. The practical rule this supports is the one the charge that is not a sum already reached for one pair: a joint critical value has to be simulated as a joint critical value, because neither component of the split, read at the tail, is the quantity its average name suggests.

The other order says the same #

A split that depends on the order showed that pinning the second search instead of the first moves the average overlap and interaction by the same amount, and on one pair puts the overlap on the other side of zero. Everything above pins the first search. Pinning the second is one more statistic per draw, and it answers whether the interaction’s steadiness is a property of the pair or of the order.

It is a property of the pair. With the second search pinned, the control’s interaction is 0.06 on average and 0.05 at the 95% point, and its overlap goes from −0.02 to 2.44. The step dictionaries’ interaction is 4.61 on average and 3.68 at the 95% point while the overlap goes from 4.77 to 29.95. The break and the window’s interaction is 0.94 on average and 2.55 at the 95% point while the overlap goes from 21.16 to 53.11.

The one pair that behaves differently between the orders is the one that did on average. The break and a column, run with the column pinned and the break searched, has an average interaction of 1.55 rather than 0.33 — the break search, run second, finds configurations the joint search shares and the sequential run with the break pinned does not — and at the 95% point that interaction is 2.42 rather than exactly zero. Its overlap goes from −0.95 to 3.96. So on this pair the interaction at the critical value is a matter of order, as its average was; but in both orders it stays within about a unit of its own average while the overlap moves by four or five.

The finding therefore does not depend on a choice the field showed to be arbitrary. Whichever search is pinned, the tail over-charge lands in the overlap term, and the interaction arrives at the critical value roughly as it left the mean.

How far into the tail twelve hundred draws can see #

The 95% points above rest on the sixty largest of twelve hundred draws, and the 99% points on the twelve largest. That is enough to see the overlap climb, which it does by tens of units, and not enough to say much about an interaction of a unit or two at the extreme.

The step dictionaries show the limit. At the 99% point their interaction is 1.10 with the first search pinned and 6.97 with the second — the same pair, the same draws, read through two sequential runs whose 99% points each rest on twelve values. A difference of six units between two such readings is within what twelve values in a skewed tail can produce, so the essay does not lean on either. What it leans on is the 95% point and below, where the interaction sits within two units of its average on every live pair in both orders, against an overlap that moves by between two and thirty-two units over the same levels.

The asymmetry between what is resolved and what is not runs in the essay’s favour. A claim that the overlap grows into the tail needs only that a large effect be visible through sampling noise, and it is, at every level and in both orders. A claim that the interaction does not grow would need the noise to be small relative to a unit, and at the 95% point it is; at the 99% point it is not, and the two orders’ disagreement there is the honest measure of how far the instrument reaches.

Still open: whether the law moves the interaction’s tail #

Everything here is under the first-order autoregression. The arithmetic of adding tails does not depend on the law, so the overlap’s climb should be the same under any of the field’s laws; the interaction is a property of what the joint search can reach, and that is exactly what the earlier field found moving by half between laws for the break and the window. If the interaction stayed near its average under the moving average and long memory too, the rule this essay supports — simulate the joint statistic, and expect the over-charge to be mostly the arithmetic of tails — would carry across the field’s laws; if under long memory the interaction grew into the tail, the break and the window would have a tail of their own that no reading of averages could predict.

The claim, and where it stops #

This essay takes the field’s split of an excess into overlap and interaction, read at quantiles of the three statistics instead of at their means. The claims are that the split at a quantile is exact and its interaction never negative; that on the two independent dictionaries the overlap goes from −0.02 on average to 2.44 at the 95% point while the interaction stays at 0.06; that on the step dictionaries the interaction is between 4.55 and 4.90 from the median to the 95% point while the overlap goes from −0.73 to 31.02; that on the break and a column the interaction at the 95% point is exactly zero; that at the critical value the split’s overlap therefore no longer measures shared ground; and that the per-draw interaction’s own tail is several times its mean while the gap between the two statistics’ 95% points is close to it; and that with the second search pinned instead the interaction again stays within two units of its average at the 95% point while the overlap climbs, on every live pair.

What stays out, named as a decision: other laws, deferred above; and the power of a test built on any of these components, which needs alternatives. The boundary against the overlap at the critical value is that it measured the over-charge at the tail and this essay says which term of the field’s split it lands in.