Why The Two Ends Are Not Mirror Images

On a list of six optional items, a threshold of one admits sixty-three profiles in sixty-four. A threshold of five admits seven in sixty-four.

Both rules are one step from an extreme. They are not reflections of each other, and the difference between them is a factor of nine.

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Six optional items, the whole staircase

Seven thresholds drawn. The tread is flat because the threshold is a whole number: there is nothing between one item and the next. A tread at the top admits everybody, which is a list that filters nothing at all.

The first two treads and the last two are both near an extreme and they are not mirror images: sixty-three in sixty-four against seven in sixty-four.

Computed from the threshold. Each item is counted as a separate coin: this is the unit of measurement, not an estimate of any population.

Where the asymmetry comes from

The number of profiles meeting exactly k items is a binomial coefficient, and those coefficients are symmetric: one way to meet none, six ways to meet one, fifteen to meet two, twenty to meet three, fifteen, six, one.

A threshold is not symmetric because it is a running total from the top. "At least one" sums everything except the single profile meeting none. "At least five" sums only the six profiles meeting five and the one meeting all six — seven in total.

So the two rules sit on opposite sides of a symmetric distribution and take wildly different slices of it: one takes sixty-three sixty-fourths, the other takes seven.

The practical reading

Loosening a strict rule is worth far more than loosening a loose one.

Going from six-of-six to five-of-six multiplies the intake by seven. Going from one-of-six to none-of-six multiplies it by about one and a sixtieth — which is to say, by nothing worth having.

The same move in the opposite direction: tightening from none to one costs essentially nothing, and tightening from five to six costs six sevenths of what was left.

The same single step is worth a factor of seven at one end of the staircase and nothing at all at the other. Where you are standing decides what a step is worth.

What this means for adjusting a list

A list that admits nobody is almost always at the strict end, and that is precisely where a one-step loosening pays the most. Before touching the items, move the threshold down one and recompute; the gain is usually larger than anything else available.

A list that admits everybody is at the loose end, and tightening by one step will not fix it. From a threshold of one on six items, moving to two changes the intake from sixty-three in sixty-four to fifty-seven in sixty-four — a few per cent. Getting to a meaningful filter requires moving several steps at once, into the middle of the staircase.

Why the middle moves fastest

The coefficients are largest in the middle, so that is where the running total changes most per step. On six items the fall from three to four is from twenty-one in thirty-two to eleven in thirty-two — almost exactly half, in one step.

This is the same acceleration described in the page on the steep part of the staircase, seen from the other direction: the ends are inert because almost no profiles live there, and the middle is violent because almost all of them do.

The compulsory part has no asymmetry at all

This is the contrast that makes the threshold's behaviour visible.

Every compulsory item halves the intake, wherever it sits in the list. The first one takes the intake from everything to a half; the eighth takes it from one in a hundred and twenty-eight to one in two hundred and fifty-six. The proportional effect is identical — exactly one half — at every position.

So one part of a list is perfectly uniform and the other is wildly uneven, and they look the same on the page. A reader has no way to tell from the layout which kind of parameter they are looking at.

Why both extremes are nearly flat

Because the bands at the ends of the distribution are tiny.

There is exactly one profile meeting none of eight items and exactly one meeting all eight. There are eight meeting exactly one and eight meeting exactly seven. Against two hundred and fifty-six, those are rounding errors, so the thresholds that add or remove them barely move the intake in absolute terms.

The middle bands hold twenty-eight, fifty-six and seventy profiles, which is where the movement is. The staircase is flat at the ends and steep in the middle for the same reason a bell is: the mass is in the middle, and a threshold is a cut through it.

The rule to carry

Ask where on the staircase the current rule sits before deciding what to change. Near the top, one step is a factor of several. Near the bottom, one step is a rounding error. In the middle, one step roughly halves or doubles.

Three different regimes, one parameter, and nothing in the wording of a rule tells you which regime you are in.

Near this one

• Choosing A Threshold On Purpose
  The threshold is the one parameter worth setting deliberately, and setting it takes under a minute. What the m
• The Staircase Of A Threshold
  Eight optional items and a rule saying how many are needed. The nine thresholds produce a staircase that is fl
• A Threshold Of None, And What It Is For
  Nine shapes in this record admit every profile. They look like lists and they filter nothing, and at least one
• A Threshold Of All, Which Is Not Optional At All
  Eight optional items requiring all eight admits one profile in two hundred and fifty-six — the same as eight c