A third session finds the edge band and counts bands misplaced

An older eye’s colour matches carry two things a fit can confuse. The lens yellows with age and may carry a narrow absorption band of its own at short wavelengths; the macular pigment, a filter over the central retina, may peak a few nanometres from where the standard observer puts it. A macula three nanometres short is a lens band found the two indistinguishable to a fit that holds only one of them, and a second session keeps a band and the macula apart fitted both — six numbers: rod weight, lens age, macular density, the band’s height and centre, and the macula’s shift — on a census of nine simulated observers at sixty, twenty noisy experiments each.

The census has since been refined in one direction at a time. One session has a second valley found fits falling into a far valley where the macula’s shift runs past seven nanometres; a soft prior on the macula frees the lens band and a wider prior gives the moves back by spending the band closed it with a Gaussian prior on the shift, six nanometres wide at the knee of a trade. A second session leaves the prior one job found that with two sessions of matches the prior had only one task left — closing the far valley — and that one band resisted every setting. Beside 415 nm with the macula unmoved, two sessions find the lens band in twelve experiments of twenty with the prior or without, where every other observer’s band is found in sixteen to twenty.

A band at 415 nm sits against the lens’s own rising absorption at the short end of the spectrum, so its trade is with the lens’s age rather than with the macula. The census’s third design adds one more pair of lights: a blue-pumped phosphor LED against a lamp with a narrow cyan line at 487 nm. A line on the macula’s flank buys less than a repeated session built it to read the macula’s long flank, where no lens band reaches; but its phosphor LED also carries short-wave light. The prediction was that three sessions find the 415 nm band, with the macula unmoved, in at least seventeen experiments of twenty, and that the experiments losing it under two sessions are those whose fitted lens age runs three or more years old. If three sessions still found it in under fifteen, the band and the lens’s age would share the matches’ information at every design the census holds.

Three sessions find the band beside the lens’s edge #

With three sessions, the band at 415 nm with the macula unmoved is found within twenty nanometres of its place in eighteen experiments of twenty — at the best of five starts and under the six-nanometre prior alike — against twelve under two sessions.

The first half of the prediction holds. The figure’s other rows say what the third session costs and gives elsewhere: it finds the band in place in nineteen or twenty experiments at 445 nm whatever the macula did, and at 430 nm with the macula long; it finds it less often at 415 nm with the macula long, fifteen against sixteen, and at 430 nm with the macula short, sixteen against twenty. The third pair of lights reads short wavelengths for the band and long ones for the macula, and when the macula is moved short its flank and a band at 430 nm compete for the same new information.

The count in the figure is not the one the census has reported, and the difference is the second finding of this essay. It counts a band only where the fit places it within twenty nanometres of where it was put.

The band is lost to an older lens, but only half a year older #

Why two sessions lose the 415 nm band is a question about the fits that lose it, and the census holds them.

The eight two-session experiments that lose the band fit the lens at 60.7 to 65.3 years, averaging 62.8; the twelve that find it at 57.9 to 63.5, averaging 59.7. Every loss lies above 60.5 years, and every find but one at or below it. Only three of the eight losses run three or more years old.

So the mechanism the prediction named is the right one and its size was wrong. An older lens absorbs more at short wavelengths across the whole region the band occupies, and a fit that makes the lens half a year to five years older can account for the band’s absorption with the lens alone. The threshold that separates the two groups is not three years but half a year: a fit that puts the lens even slightly older than the truth has spent the band’s evidence on the lens. That is a sharper statement about identifiability than the prediction made. The lens and the band at 415 nm are so nearly the same direction in the matches that half a year of age is worth the band.

The third session breaks the tie, and the natural reading of how is through its short-wave light. Its phosphor LED, set against the cyan lamp, differs from it in the short-wave region where the lens’s broad slope and the band’s narrow shape part company, so it reads the lens’s age from a pair of lights in which the band has a different leverage, and the band is left to account for what the age does not. The census does not isolate that path; it shows only that the tie is broken.

The count has always included misplaced bands #

The census counts a band as found when its fitted height exceeds twice its standard error. Nothing in that rule asks where the band is.

Under two sessions at their best, nine of 180 fits place the band more than twenty nanometres from where it was put, six of them between 485 and 500 nm; under the six-nanometre prior, five. Under three sessions twelve do, eleven of them between 480 and 500 nm; under the prior, ten. Every such fit is still counted in the census’s band count if its fitted height clears twice its error, and most do.

These are not lens bands. A band at 495 nm sits on the macula’s long flank, sixty to eighty nanometres from the lens bands the census placed, and a fit that puts it there is using it to describe the macula — usually a macula moved far short, the far valley one session has a second valley found. The census’s counts were made before any fit was asked where its band was, and so every count from the second session onward has carried a few of these. The two-session counts quoted in the opening paragraph are the counts restricted to bands in place; the counts at the best of five starts that the earlier essays reported were up to three higher beside 415 nm.

That correction does not overturn what those essays found, since the misplaced fits are a few in each twenty and every comparison was made with the same rule on both sides. It does change what the counts mean. A band found in nineteen of twenty includes, at some observers, a fit that found a different thing.

Every misplaced band comes with a macula fitted short #

The misplaced fits are few enough to read one by one, and they share one property. Of the 21 fits that place the band more than twenty nanometres out — nine under two sessions, twelve under three — every one fits the macula’s shift at least 2.7 nm shorter than the truth. Sixteen of them belong to the observers whose band was put at 415 nm, four to 430 nm and one to 445 nm. Beside 415 nm with the macula truly three long, four three-session fits put the band at 484 to 499 nm and the shift at −7.5 to +0.3, where it should be +3.

So the misplaced band and the misfitted macula are one event, not two. Moving the macula short lowers the absorption the standard observer expects on its long flank, and a band at 485 to 500 nm puts that absorption back; the pair describes nearly the same matches as a macula in its place and no band there. The lens band the observer really has is then left undescribed, which is why most of these fits come from the observer whose band is hardest to see, at 415 nm, where the lens’s own age can take its place. The fit has traded two parameters it could not separate for two others it can, and every one of the trades runs in the same direction.

With three sessions the prior leaves the far valley open #

The third session places more bands on the macula’s flank than the second, not fewer, and it is natural to ask whether the six-nanometre prior, which closed the far valley for two sessions, does so for three.

Beside 415 nm with the macula truly three nanometres short, four three-session fits at their best put the macula’s shift beyond 7.5 nm short — at −8.0, −9.7, −11.8 and −12.2 — and under the six-nanometre prior two still do, at −9.3 and −11.3. Under two sessions the same prior kept every one of 180 fits within six nanometres.

The prior is one pseudo-observation, the shift over six nanometres in units of the matches’ noise, and its pull is fixed. A third session adds a whole session of residuals to the matches, and with them half as much evidence again for whatever the matches support. Where the far valley was a shallow second minimum, as under two sessions, a fixed pull was enough to lift a fit out of it. The third session’s lights evidently deepen that valley — the natural reading is that the cyan line reads the long flank that a far-shifted macula and a misplaced band together describe — and the same pull is no longer enough. The prior’s single remaining job under two sessions is a job it cannot do under three.

More moves beside the long bands, fewer at 430 #

Counting only fits with the band in place, three sessions find the macula’s real move in eighteen of twenty beside 430 and 445 nm with the macula long, against seventeen and twelve under two sessions with the prior; beside 445 nm with the macula short in fourteen, against thirteen. Beside 430 nm with the macula short they find it in nine, against thirteen; and beside 415 nm in twelve and sixteen, against thirteen and seventeen.

So the third session is not a uniform improvement. It is decisive where the macula moved long, because a long-moved macula’s flank sits on the cyan line, and it costs where the macula moved short and the band sits at 430 nm, where the short-moved flank and the band compete for the third pair’s short-wave light. The essay that introduced the third design measured its value as a reduction in the correlation between the band and the shift; the census of moves shows that the correlation it reduces depends on which way the macula moved.

A third session measures every shift finer #

Under the prior, the third session narrows the shift’s mean standard error at every observer, to between 78 and 96 per cent of the two-session error; every three-session error lies between ±0.92 and ±1.19 nm. The prior changes no observer’s count of bands or moves in place by more than two.

That is the background for the far valley’s survival. A shift measured to a nanometre is measured to a sixth of the prior’s width, so the prior contributes a few per cent of the information about it, and less with every session added. The prior was always going to fade as the data grew; what was not obvious is that the far valley would not fade with it. More data made the true minimum sharper and left the false one in place.

What a vision scientist should take from this #

A count of detections should say where they were detected. The census counted a lens band wherever a fit put one, and a few fits in every observer put it on the macula’s flank, sixty to eighty nanometres away. A detection criterion that tests only a parameter’s size against its error will count a parameter that has moved to describe something else.

An identifiability failure has a size, and it can be smaller than expected. The band beside the lens’s edge is lost whenever the fit makes the lens half a year older than it is. The mechanism was named correctly in advance and its scale was not, and the scale is the more useful number: it says how nearly the two parameters share a direction in the matches.

A prior’s job changes as the data grows. The six-nanometre prior closed a far valley under two sessions and fails to under three, not because the prior changed but because the data deepened the valley faster than it sharpened the prior’s pull.

How the fits were made #

The census is nine observers at sixty: a lens band at 415, 430 or 445 nm with the macula moved three nanometres short, three long or not at all. Each experiment adds the matching task’s setting noise, from the census’s own seeds, to the matches the observer would make for one, two or three sessions of light pairs over the census’s surface family, and fits six numbers by a damped Gauss–Newton descent within the census’s ±30 nm limits on the shift. The best fit is the lowest $\chi^2$ of the census’s two starts and three more; the prior adds one residual, the shift over six nanometres times the noise. A band counts as found when its height exceeds twice its standard error and as in place when its centre lies within twenty nanometres of where it was put; a move counts when the shift exceeds twice its error, and in place when the band is.

What this does not reach #

One age, nine observers, twenty experiments each. The half-year threshold is this census’s; at another age the lens’s slope differs and so may the trade.

The misplaced fits are counted, not dissected. That a band at 495 nm describes a far-shifted macula’s flank is the reading of where they fall and which way their shifts go; each was not refitted with its band held in place.

The prior’s width is held at six. A prior narrower with each session would keep its pull; whether that closes the three-session valley without costing the real moves is not computed.

Still open: whether a band held near the lens closes the valley the prior cannot #

The far valley under three sessions is a band on the macula’s flank paired with a macula far short. A fit whose band centre is bounded to the region a lens band can occupy — 400 to 460 nm — cannot place it there, and the far valley’s fits would have to find another description or none.

The calculation is the three-session census fitted again from the same five starts with the band’s centre bounded to 400–460 nm, with and without the six-nanometre prior, the counts read in place as here. The prediction is that no fit then lies beyond 7.5 nm short, that the 415 nm band with the macula short is found in place in at least eighteen of twenty, and that the moves found beside 430 nm with the macula short rise from nine to at least twelve, because the third session’s cost there was a band drawn toward the cyan line. If the far fits remain with the band bounded, the far valley belongs to the macula alone, and only a bound on the shift itself can close it.

A band held near the lens closes the valley the prior could not made that calculation: no held fit is left in the far valley, the 415 nm band is found in nineteen, and the cost at 430 nm stays.

A count that never asked where #

The habit here is about what a detection count contains.

The census reported bands found by a rule that compared each fitted height with its error, and for five essays every comparison was made with that rule. Asking where each counted band sat — a question nobody had put to the census — found a handful of bands in every set of 180 sitting on the macula’s flank, and showed that the third session, which finds the lens-edge band as predicted, also puts more bands there and deepens a valley the prior had closed.

The failure mode is to read a significance test on a parameter as a detection of the thing the parameter was named for. The test is to check that the detected parameter is where that thing can be — and to recount, before comparing designs, with only the detections that are.