A second session leaves the prior one job

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. With one session of matches those numbers are loosely tied, and two kinds of trouble follow. One session has a second valley found the fit, searched thoroughly, often settling where the macula’s peak sits ten to twenty nanometres short of the standard, where no real macula is; and a second session keeps a band and the macula apart found a lens band near 430 or 445 nm and a slightly shifted macula trading almost completely within one session’s information.

A prior on the macula’s shift addresses both, at a price. A soft prior on the macula frees the lens band gave the fit a Gaussian prior four nanometres wide and found the far valley closed and the band freed, every real move shrunk. A wider prior gives the moves back by spending the band widened it and found one trade along the widths: 14 real moves and 44 bands at four nanometres, 22 and 34 at six, 25 and 25 at eight, against the unconstrained fit’s 27 and 22 — six nanometres at the knee, where the cheap moves have been bought back and the expensive ones have not.

That trade is a property of one session. A second session adds matches under different lights, and the joint census’s two-session fits already do most of what the prior was bought for. Taken at their best of five starts, two sessions find the lens band in twelve to twenty experiments of twenty and real moves in twelve to eighteen; they report an unmoved macula moved in two experiments at 415 nm, three at 430 and none at 445. The prediction was that the six-nanometre prior, applied to two sessions, brings those false moves to one or none at every band, finds every real move within two of the two-session fits’ counts, and finds the band in at least seventeen of twenty everywhere. If the real moves had fallen by more than three anywhere, the prior would cost a well-measured eye what it costs a poorly measured one, and should be dropped once the second session is in hand.

Three parts of the prediction fail, and none for the reason feared #

Under the six-nanometre prior, two sessions report an unmoved macula moved in two, two and no experiments of twenty for the bands at 415, 430 and 445 nm. They find the real moves in 13, 18, 13, 17, 14 and 12 experiments for the six moved observers, against 17, 18, 15, 17, 14 and 12 without the prior. They find the lens band within two experiments of their best fits’ count for every observer, and beside 415 nm with the macula unmoved in twelve, as without it.

• The false moves do not fall. Four of sixty remain, where the prediction wanted three or fewer and the best fits had five.
• One observer loses four real moves, beside 415 nm with the macula moved short — past the threshold the prediction named for dropping the prior.
• One observer’s band stays at twelve, under the seventeen predicted, with or without the prior.
• Every other count is unchanged or within one. For five observers of six the prior costs the real moves nothing at all.

Set beside one session under the same prior, the second session’s effect dwarfs the prior’s. One session under six nanometres finds seven, six, three, two, three and one of twenty; two sessions find twelve to eighteen with the prior or without it. The prior’s whole influence on two sessions is the four moves beside 415 nm, against a gain of ten to fifteen moves an observer from the second session.

The headline the threshold would give — drop the prior — is wrong for the observer that triggers it. The four moves lost beside 415 nm are not moves the prior hid. Most of them were never measurements.

The far valley in two sessions #

Of the twenty two-session experiments beside 415 nm with the macula three nanometres short, three fit the shift beyond 7.5 nm short at their best of five starts: −11.84, −7.73 and −11.00 nm, with standard errors of 1.31, 1.53 and 1.40, so all three count as moves. Under the prior they fit at −1.90, −3.05 and −0.94. The other seventeen experiments move by a tenth or two, and the mean fitted shift goes from −3.89 nm to −2.57.

The second valley one session has a second valley found ten to twenty nanometres short is not confined to one session. Two sessions narrow it and make it rarer, but beside a band at 415 nm, where the band sits on the lens’s own absorption edge and the macula’s short flank can stand in for both, a two-session fit searched from five starts can still settle there. When it does, it settles with a small standard error — the far valley is a genuine minimum of the fit, not a flat ridge — and the macula is reported moved by four times its true amount with the right sign.

Across all nine observers four best fits lie beyond 7.5 nm, every one of them with the macula truly moved short: the three beside 415 nm and one beside 430 nm, at −9.53. Under the prior none of 180 two-session fits passes six nanometres; the largest is 5.93, beside 445 nm, where the true move is three and the fit’s error 1.1.

So two of the four moves lost beside 415 nm are fits the prior drew back from −11 or −12 nm to within twice their error of nought, and one drawn back to −3.05 is still found. The other two lost are fits that sat just past twice their error at the best — −2.34 nm against a threshold of 2.34, −2.26 against 2.26 — and just under it with the prior, each pulled by less than a tenth of a nanometre by a prior whose variance is thirty times the fit’s own. Of the seventeen moves two sessions found at their best beside 415 nm, three were the far valley’s; under the prior thirteen are found, and every one of them is a measurement of a three-nanometre move.

The false moves that remain are honest #

With the macula unmoved, no two-session fit under the prior goes beyond 5.1 nm, and four of sixty report a move beyond twice the standard error. A test that calls a move whenever a fitted shift lies beyond twice its standard error, applied to fits whose errors are honest and whose shifts scatter about nought, calls one by chance in 4.55 per cent of them: 2.7 in sixty. Four is within the scatter of that count over three groups of twenty.

So the false moves the prediction expected the prior to remove are not the far valley’s. With one session, an unmoved macula was reported moved in eight, six and two experiments from the best of five starts, nearly all of them fits in the far valley, and every prior and bound removed most of those. With two sessions the false moves sit within five nanometres of nought and carry errors near 1.1 nm: they are the fluctuations a two-standard-error criterion is built to admit at its stated rate, and no prior that leaves real three-nanometre moves alone can remove them. A criterion at three standard errors would admit about one in four hundred, and would also lose some of the real moves.

The band is the second session’s, not the prior’s #

The two-session fits find the lens band in eighteen to twenty experiments for eight observers of nine, with the prior or without it, and the prior moves no count by more than two — beside 415 nm with the macula short, from seventeen to nineteen. Beside 415 nm with the macula unmoved, both find the band in twelve.

Under one session the prior bought bands by holding the macula in. With two sessions there is little left to buy: the second session’s lights separate the band’s absorption from the macula’s shift on their own, as a second session keeps a band and the macula apart found when it introduced the design. The gain beside 415 nm with the macula short is two fits whose band sat just under twice its error and crosses it. The far valley shows in the band too, but not in the count: the fits at −11.84 and −11.00 nm had found the band at amplitudes of 0.058 and 0.060 against a true 0.1, having given part of its absorption to the macula, and under the prior they find it at 0.168 and 0.135 — as far above the truth as they had been below it, so the prior trades one error in the band’s size for another of the same order.

The one observer the band stays hard for is the 415 nm band with an unmoved macula. That band sits on the steep edge of the lens’s own ageing absorption, where a band in the lens reads as age or macula found a band’s absorption readily assigned to the lens’s age instead, and a lost lens band goes to age or the macula found a band at 415 nm, when it is lost, going to three to five years of lens age. A prior on the macula cannot help with a trade between the band and the lens’s age, and the count is unchanged because the prior has nothing to act on there.

Why the prior is nearly inert #

Under the six-nanometre prior, one session’s mean standard error on the macula’s shift runs from ±1.80 to ±3.28 nm across the nine observers; two sessions’ from ±0.97 to ±1.27. A Gaussian prior acts on a fit in proportion to the ratio of the parameter’s own variance to the prior’s, so with one session a six-nanometre prior carries a tenth to a third of the weight the matches carry, and with two sessions about a thirtieth.

That is why the prior’s cost and its benefit both fall away together. With two sessions the prior moves a near-valley fit by a few per cent of its value — a tenth of a nanometre on a three-nanometre move — and the only fits it moves far are those far from nought, where the prior’s residual grows with the square of the shift while the matches’ preference between valleys is a few units of $\chi^2$. At −11 nm the prior’s residual costs more than ten times what it costs at −3, while the matches’ preference between the two valleys is a few units of $\chi^2$, and the far valley loses.

The means come back to the truth #

Under the prior, every two-session observer’s mean fitted shift lies within 0.43 nm of the truth: −2.57, −2.87 and −3.17 for the moves short, 2.93, 2.96 and 2.64 for the moves long, and −0.27, −0.14 and −0.10 for the unmoved maculas. At their best without the prior the same means are −3.89, −3.27 and −3.35 short. One session under the same prior shrinks every move by a third or more and pushes the unmoved 415 nm observer to +0.59.

The best fits’ overshoot beside 415 nm is the far valley seen through a mean: three fits near −11 nm in twenty pull a mean of −2.6 to −3.9. Beside 430 nm one fit at −9.5 does the same on a smaller scale. A count of moves found beyond twice their error treats those fits as successes, and a mean treats them as a bias; the prior removes them from both readings at once.

What a vision scientist should take from this #

With two sessions, a six-nanometre prior on the macula’s shift is a guard rather than a constraint. It moves ordinary fits by a few per cent of their error and pulls far-valley fits back to the near valley; it is worth keeping for that alone.

A count of moves found is not a count of moves measured. Beside 415 nm three of seventeen two-session moves were fits four times the true size, and a criterion at twice the standard error cannot tell them from the rest. Read the fitted shifts, not only the count past a threshold.

The false moves left are the criterion’s. Two-session false moves sit at the rate a two-standard-error test admits by chance; lowering them is a choice of criterion, and it trades directly against the real moves.

The 415 nm band with an unmoved macula needs more than a prior on the macula. Its trade is with the lens’s age, and two sessions resolve it only twelve times in twenty.

How the fits were made #

The joint census’s two-session design pairs its colour matches under two lights, each match a difference in CIELAB between two lights’ renderings of a family of surfaces for an observer described by the six numbers; every experiment adds Gaussian noise of the census’s size to the matches with the census’s seeds, so that the two-session experiments here are the census’s own. Each is fitted by Levenberg–Marquardt from five starts — the census’s two, the truth and two more band centres — with one more residual, the shift times the noise over six nanometres, and the census’s ±30 nm limits; the best fit of five by $\chi^2$ is kept, and its covariance, the inverse of the normal matrix with the prior’s residual in it, gives each standard error. The best fits without the prior are the census’s two starts and the audit’s three, taken at whichever has the lower $\chi^2$ by more than a hundredth. A band or a move counts as found beyond twice its standard error.

What this does not reach #

Other criteria. Every count uses twice the standard error; the trade between false and real moves at other thresholds is not read.

The far valley’s frequency in real data. The census’s experiments are simulated with the model’s own observer; how often a real observer’s two-session fit would find the far valley depends on how well the model describes that eye.

Three sessions. The census also holds a three-session design, which adds a cyan lamp; it is not refitted here.

Still open: whether a third session finds the band beside the lens’s edge #

Two sessions find every lens band in eighteen to twenty experiments of twenty but one: a band at 415 nm with the macula unmoved, found in twelve whatever the prior, because its trade is with the lens’s age rather than the macula. The census’s third design adds a phosphor lamp against a cyan one, whose short-wave light reaches the lens’s edge.

The calculation is the joint census’s three-session experiments for the 415 nm observers, fitted from the same five starts with and without the six-nanometre prior, the bands, moves and the fitted lens age read as here. The prediction is that three sessions find the 415 nm band in at least seventeen of twenty with the macula unmoved, and that the experiments that lose it under two sessions are the ones whose fitted lens age runs three or more years old. If three sessions still find it in under fifteen, the band and the lens’s age share the matches’ information at every design the census holds, and the lens’s age must be measured separately before a band near its edge can be read.

A third session finds the edge band and counts bands misplaced made that calculation: three sessions find the band in place in eighteen, the losses under two fit the lens only half a year older, and the census’s counts turn out to have included bands placed on the macula’s flank.

A loss that was a correction #

The habit here is about reading a fall in a success count.

The prior lowered the count of real moves found beside 415 nm by four, past the threshold set in advance for calling the prior a cost. Reading the fits rather than the count showed three of the moves it lowered were fits at four times the true size, which the count had scored as successes because their sign was right. The prior’s change was the removal of three wrong answers that happened to point the right way, and the count, which could see only the sign, read it as a loss.

The failure mode is to read a fall in a count of successes as a loss of accuracy, when the count scores by a criterion that admits wrong answers alongside right ones. The test is to look at the values behind the count before and after the change, and ask whether what left the count was ever right.