A wider prior gives the moves back by spending the band

An older observer’s colour matches are read by fitting six numbers that describe the eye: the rod weight, the lens’s age, the macula’s density, the amplitude and centre of a yellowing band in the lens, and the shift of the macular pigment’s peak. A single session of matches ties those numbers together loosely. The band is the newest of them: a band in the lens reads as age or macula found the yellow pigments an ageing lens grows adding a shoulder of absorption just past 400 nm that a fit without a band assigns to the lens’s age or to the macula, and a lost lens band goes to age or the macula found the same assignment whenever a fit misses a band it should have found. A second session keeps a band and the macula apart found a lens band at 445 nm and a macula peaking a few nanometres short trading almost completely when one session is all there is, and one session has a second valley found the fit, searched from enough starts, often settling where the macula’s peak sits ten to twenty nanometres from any real eye’s.

Two ways of holding the macula near the standard have been tried on those one-session experiments. A bound on the macula trades false moves for hidden ones fenced the shift within ±6 nm and closed the far valley, at the cost of real moves that, with one session’s noise on them, reach the fence and stop. A soft prior on the macula frees the lens band replaced the fence with a Gaussian prior four nanometres wide — one more residual in the fit, the shift over the prior’s width in units of the matches’ noise — so that a fit can go further when the matches insist. It kept the false moves away, found the real moves as seldom as the bound or more seldom, pulled every real three-nanometre move in to 1.2–1.9 nm, and in return found the lens band beside the macula in ten to twelve experiments of twenty where the unconstrained fit found three to seven.

A prior four nanometres wide pulls a three-nanometre move about halfway home. A wider prior pulls less. The prediction was that a prior six nanometres wide keeps the false moves at two or fewer of twenty, finds the real moves within two of the hard bound’s counts and keeps two thirds of the band’s gain over the unconstrained fit at 430 and 445 nm; and that at eight nanometres the far valley reopens at 415 nm. The alternative named was that the band’s gain falls as fast as the moves recover — that the band and the macula’s shift share one session’s information in a fixed proportion, and no width buys both.

One trade, read at three widths #

Summed over the six observers whose macula truly moved, the real moves found beyond twice their standard error are 14 of 120 under the four-nanometre prior, 22 under the six and 25 under the eight, against 27 for the unconstrained fit and 19 for the hard bound. Summed over the four observers whose band the four-nanometre prior freed — the band at 430 or 445 nm, the macula unmoved or moved short — the bands found are 44 of 80 at four nanometres, 34 at six and 25 at eight, against 22 unconstrained and 20 under the bound.

• The false moves stay away at every width, as predicted: at most two of twenty at every band.
• Six nanometres finds the real moves within two of the bound’s count for every observer, as predicted, and beside 415 nm finds more than the bound.
• Six nanometres keeps 55 per cent of the band’s gain, not two thirds. The prediction fails, narrowly.
• At eight nanometres the far valley reopens at 415 nm, as predicted: four fits of twenty fall short of −8 nm there.

On a plane of moves found against bands found the prior’s three widths and the unconstrained fit, which is the prior at infinite width, lie on one descending line. Each widening buys moves and spends bands, and nothing on the line buys both. That is the alternative the prediction named, and it holds in total: from four nanometres to eight the moves recover 11 of the 13 they had lost against the unconstrained fit, 85 per cent, and the band gives back 19 of its 22 gained, 86 per cent. The two shares are equal to a per cent.

The path between is not straight. From four nanometres to six the prior buys eight moves for ten bands; from six to eight it buys three moves for nine bands, three times the price a move. Six nanometres sits at the knee of the trade, where the cheap moves have been bought and the expensive ones have not. And the hard bound, at 19 moves and 20 bands, lies below and to the left of six nanometres’ 22 and 34: a prior six nanometres wide finds more real moves than the bound and half again as many bands. No weighting of the two goals prefers the fence.

The moves come back at six nanometres #

Under the six-nanometre prior the real moves are found in 7, 6, 3, 2, 3 and 1 experiments of twenty for the observers with the band at 415, 430 and 445 nm and the macula moved short and long, against the bound’s 5, 4, 4, 3, 3 and 0. Every count lies within two of the bound’s. Beside 415 nm, where the far valley is deepest and the bound’s fence stopped most real moves, the prior finds two more in each direction.

The recovery is unequal across the bands. Beside 415 nm the moves come back strongly, from five and three under the narrow prior to eight and eight under the widest. Beside 430 and 445 nm they hardly come back at all: at every width they are found in two or three experiments of twenty, with the long move beside 445 nm found once at most. Those are the observers whose band lies closest to the macula’s own absorption, and where the band and the shift are hardest to tell apart one session gives up the move whichever way the prior leans.

That squares with what a second session keeps a band and the macula apart found: correlations between the band’s amplitude and the macula’s shift of 0.91 to 0.95 at 430 and 445 nm under one session. Where two numbers are that tightly tied, a prior on one of them decides the other, and whatever width it is given, the information to find both is not in the matches.

The band is spent where the move is short #

With the band at 430 nm and the macula moved short, the band is found in 12 experiments of twenty under the four-nanometre prior, 6 under the six and 4 under the eight, against the unconstrained fit’s 6. With the band at 445 nm and the macula unmoved it is found in 11 at every width, against the unconstrained fit’s 7. The other two freed observers fall between: at 430 nm unmoved from 11 to 9 to 6, and at 445 nm moved short from 10 to 8 to 4.

The pattern is the trade read observer by observer. Where the macula truly moved short, the move and the band compete for the same short-wave shortfall in the matches, and a prior that lets the shift go further lets it take the shortfall back from the band; at 430 nm the band’s gain is gone by six nanometres. Where the macula did not move, a wider prior has nothing to give back to the shift, the shift stays near nought, and the band keeps what the narrow prior bought — at 445 nm through eight nanometres. The band’s gain was always the moved macula’s loss. The four-nanometre prior freed the band by holding real short moves in, and the band survives a wider prior only where there was no move to release.

No width lets the false moves back #

With the macula unmoved, the six-nanometre prior reports it moved in 1, 2 and 1 experiments of twenty for the bands at 415, 430 and 445 nm, and the eight-nanometre prior in 1, 2 and 2 — against the bound’s 1, 2 and 1, the four-nanometre prior’s 0, 1 and 1, and eight, six and two from the best of five starts with no constraint at all. Fits putting an unmoved macula past six nanometres number none of sixty under the narrow prior, two under the six and four under the eight.

So the prior’s width can be chosen for the moves and the band without letting the false moves back in numbers. A false move needs a fit that travels past the standard and carries a standard error small enough to call the move real, and a wider prior allows the first more readily than the second, because its own width enters the shift’s error. Not always: of the four unmoved fits that pass six nanometres under the eight-nanometre prior, two carry errors of 2.6 and 4.6 nm small enough to be counted, and they are half of that width’s false moves at 415 and 445 nm. At six nanometres the two fits that pass six do so without being counted.

The far valley opens at 415 nm #

Beside 415 nm, with the macula truly three nanometres short, no fit falls short of −8 nm under the four-nanometre prior and the deepest lies at −4.4; under the six, two fall short of −8 and the deepest lies at −9.0; under the eight, four do, the deepest at −13.7. The far valley one session has a second valley found, ten to twenty nanometres short, is the place these fits are travelling to. With the band at 415 nm and the macula moved long the eight-nanometre prior sends two fits there, and with the band at 430 nm and the macula moved long, one, to −11.3.

The prediction placed the reopening at eight nanometres, and it is there; but it has begun at six. Two fits of twenty beyond −8 nm is a valley ajar rather than open, and those two fits are why the six-nanometre prior finds seven real short moves at 415 nm where the bound found five: a fit that goes to −9 nm with a standard error of 2.6 reports a move in the right direction and three times the right size.

That matters for what a found move means. Under the eight-nanometre prior, the mean fitted shift beside 415 nm with the macula moved short is −3.88 nm, past the true −3, its far fits pulling the mean beyond the truth. A count of moves found beyond twice their error does not distinguish a move found at −3 from one found at −13; the first is a measurement and the second is the far valley with the right sign.

Short moves grow back; two long ones shrink #

For the three observers whose macula moved short, the mean fitted shift grows in size at every widening: beside 415 nm from −1.85 to −2.75 to −3.88 nm, beside 430 nm from −1.23 to −1.53 to −1.67, beside 445 nm from −1.51 to −2.01 to −2.16. Moved long, beside 445 nm it grows from 1.63 to 2.27 to 3.01; beside 415 and 430 nm it falls, to 0.79 and 0.87 at eight nanometres.

The two long moves that shrink do so for the same reason the short move at 415 nm overshoots. The far valley lies short whatever the true move, so a fit freed to travel can find it from a long macula as well as from a short one; beside 415 and 430 nm with the macula moved long, three fits at eight nanometres travel there, and their −8 to −11 nm drags the mean down. Read through its mean, a wider prior looks like it shrinks a long move; read fit by fit, the typical long move barely changes — its median stays at two nanometres beside 415 nm and grows from 2.4 to 2.5 beside 430 — and a few have fallen into the valley on the wrong side.

A macula three nanometres short is a lens band argued that a short macula and a lens band are, to one session, nearly the same observation. This is that argument at the level of a prior: the width decides how much of a short-wave shortfall is assigned to each, and a long macula, which no lens band imitates, is the one case where the assignment should be easy. It is easy beside 445 nm and fails beside 415 and 430 only because the valley short of the standard does not care which way the macula truly moved.

What a vision scientist should take from this #

A prior’s width is a choice of trade, not a choice of accuracy. Across four, six and eight nanometres one session’s experiments recover real moves and lose the lens band in nearly fixed proportion. Choose the width for which mistake is dearer in the study at hand, and report it.

Six nanometres is the defensible default for one session. It finds the real moves at least as often as a hard bound for four observers of six and within one of it for the other two, finds more lens bands than the bound everywhere the band is hard, keeps false moves at the bound’s level, and sits at the knee of the trade. The real span of lutein-to-zeaxanthin mixtures, a few nanometres either way, puts the true shifts well inside its reach.

No width substitutes for a second session. Beside 430 and 445 nm the band and the macula share one session’s information so tightly that every width finds two or three real moves of twenty. The cure there is more matches, not a better prior.

How the fits were made #

Each simulated observer is sixty years old with a lens band of fixed amplitude at 415, 430 or 445 nm and a macula whose peak is moved by −3, 0 or +3 nm; one session is the census’s single design of colour matches against a family of surfaces under two lights, with Gaussian noise of the census’s size and the census’s seeds. Every experiment is fitted by Levenberg–Marquardt from five starts — the census’s two, the truth and two more band centres — with the prior’s residual, the shift times the noise over the prior’s width, appended to the matches, and the census’s ±30 nm limits on the shift. The best fit of five by $\chi^2$ is kept; its covariance is the inverse of the normal matrix, the prior’s residual included, so that the prior’s width enters each standard error. A band or a move counts as found beyond twice its standard error. The four-nanometre counts and the bound’s are those of the earlier essays.

What this does not reach #

Two sessions. Every count here is one session’s, and the trade is a property of one session’s information; with a second, the band and the shift may stop competing for it.

The far valley’s depth. Whether fits that fall short of −8 nm would be caught by a reader looking at their $\chi^2$ against the near valley’s is not asked; they are counted as the fit reports them.

Other priors. A Gaussian centred on nought is one shape; a prior centred on a population’s mean shift, or one with heavier tails, would trade differently.

Still open: whether a prior still pays with two sessions #

Two sessions already find the lens band in eighteen to twenty experiments of twenty and real moves in twelve to nineteen, but they still report an unmoved macula moved in two experiments at 415 nm and three at 430. With the band and the shift no longer competing for one session’s information, a prior should cost the moves little.

The calculation is the joint census’s two-session experiments fitted from the same five starts under the six-nanometre prior, the false moves, real moves and bands counted as here. The prediction is that the false moves fall to one or none at every band, that every real move is found within two experiments of the two-session fits’ counts, and that the band is found in at least seventeen of twenty everywhere. If the real moves fall by more than three anywhere, a prior costs a well-measured eye what it costs a poorly measured one, and should be dropped once the second session is in hand.

A dial that was read as a fix #

The habit here is about what a tuning parameter is.

The narrow prior was found to buy the lens band, and the natural reading was that a prior is a fix for the band’s poor identification, to be widened until its costs disappear. Widening it showed that the costs and the gains move together along one line: the band was bought with the very moves the prior held in, and paying the moves back spends the band at nearly the same rate. Only the bound sat off the line, below it.

The failure mode is to tune a constraint until one cost vanishes and to read the result as an improvement, when the parameter is moving along a trade between two quantities that the data cannot both supply. The test is to read the constraint at several settings, plot what each buys against what it spends, and choose the setting knowing which point on the trade it is.