A band held near the lens closes the valley the prior could not

An older eye’s colour matches hold two things a fit can trade against each other: a narrow absorption band the yellowing lens may carry at short wavelengths, and a macular pigment that may peak a few nanometres from the standard observer’s. A census of nine simulated observers at sixty — a band at 415, 430 or 445 nm and a macula moved three nanometres short, three long or not at all, twenty noisy experiments each — has been fitted, essay by essay, with six numbers: rod weight, lens age, macular density, the band’s height and centre, and the macula’s shift. One session has a second valley found a far valley where the shift runs past seven nanometres short; a bound on the macula trades false moves for hidden ones tried a hard limit on the shift and found it hiding real moves; a soft prior on the macula frees the lens band replaced it with a Gaussian prior six nanometres wide.

A third session finds the edge band and counts bands misplaced added the census’s third design — a phosphor LED set against a lamp with a cyan line at 487 nm — and found what it was built to find: the band beside 415 nm with the macula unmoved found in place in eighteen experiments of twenty, against twelve under two sessions. It also found what the census’s counting rule had hidden. A band counted as found wherever a fit put it, and twelve of 180 three-session fits put it more than twenty nanometres from its place, eleven between 480 and 500 nm on the macula’s long flank, every one with the macula fitted at least 2.7 nm short; under the six-nanometre prior two fits beside 415 nm still lay beyond 7.5 nm short, a valley the same prior had closed for two sessions.

The misplaced band and the misfitted macula were one event: a macula moved short takes absorption off its long flank, and a band at 495 nm puts it back. A lens band cannot sit at 495 nm — the lens’s absorption lies at the short end of the spectrum — so the fit is free to describe the matches with a parameter in a place its name does not allow. The prediction was that holding the band’s centre to 400–460 nm, from the census’s 390–500, empties the far valley — no fit beyond 7.5 nm short — finds the 415 nm band with the macula short in place in at least eighteen of twenty, and raises the moves found beside 430 nm with the macula short 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 remained with the band held, the far valley belonged to the macula alone.

Held near the lens, the band is found where it was put #

With the band’s centre held to 400–460 nm, the 415 nm band is found in place in nineteen experiments of twenty with the macula short, twenty with it unmoved and nineteen with it long, against sixteen, eighteen and fifteen with the band free. No fit, at any observer, places the band more than twenty nanometres from where it was put.

The second part of the prediction holds, and so does the stronger form behind it. Every misplaced fit becomes a fit in place, and the counts beside 415 nm rise by two to four. At 430 and 445 nm the counts move by one or not at all, which is what a bound that changes only the misplaced fits should do: those observers had few to change.

The far valley is empty #

Beside 415 nm with the macula three short, four free fits put the shift beyond 7.5 nm short; held, none does, and the furthest lies at −5.1 nm. Over all 180 three-session fits, with the band held and with or without the prior, none lies beyond 7.5 nm short.

The first part of the prediction holds, and it settles the question the alternative raised. The far valley was never the macula’s alone. A far-shifted macula needs something to put back the absorption its long flank loses, and with the band held near the lens there is nothing in the six numbers that can; the fit has no use for the far valley once its partner is out of reach. A second session leaves the prior one job found the prior’s one remaining job under two sessions to be closing this valley. Under three sessions the job belongs to the band’s limits instead, and the limits do it without a prior at all.

Every misplaced fit’s macula comes home #

Free, the twelve misplaced fits put the macula’s shift between −12.2 and +0.3 nm; held, every one lies within 2.3 nm of its observer’s true shift — the fit at −12.2 returns to −4.3, the one at +0.3 beside a macula truly three long returns to +3.7. Each of them had also left the real lens band undescribed; held, each places it within fourteen nanometres of where it was put, nine of the twelve within four.

So the bound repairs two parameters at once by removing one degree of freedom. That is the signature of a trade rather than of noise: when two parameters move together to describe the same feature of the data, constraining either one returns both.

The misplaced valley was barely deeper than the true one #

The bound changes the fits it was aimed at, and the question is what it costs them.

Holding the band near the lens raises the misplaced fits’ $\chi^2$ by 0.16 to 10.4 — under four for ten of the twelve. The 168 fits already in place move by at most 1.2. A rise of under four units, with six parameters fitted to several hundred residuals, is what noise alone gives between two fits of nearly equal merit.

That is the most useful number in this essay, because it says what a scientist reading these fits could and could not have known. The misplaced fits were the best fits by $\chi^2$, so a fit that reports only its best solution reported them. But in ten of twelve cases the true placement lay within a few units of $\chi^2$ — a difference the data can barely tell from chance. The matches did not prefer the misplaced band; they were nearly indifferent between it and the right one, and the fit’s search fell into whichever it reached first. A bound on the macula trades false moves for hidden ones bounded the parameter that was misbehaving, the shift, and paid in real moves hidden; bounding the partner, the band, costs nothing measurable, because the band’s limit is set by where a lens can absorb and the shift’s had to be chosen.

The moves beside 415 come back; the cost at 430 stays #

Beside 415 nm the held fits find the macula’s real move in fifteen experiments with the macula short and in all twenty with it long, against twelve and sixteen with the band free. Beside 430 nm with the macula short they find it in ten, against nine; with the prior, nine. The third part of the prediction fails.

The essay before had attributed the third session’s cost at 430 nm, with the macula short, to a band drawn toward the cyan line. The bound removes that possibility — no held band can approach 487 nm — and the cost remains. So the cost is the macula’s own: with the macula moved short and the band at 430 nm, the macula’s short flank and the band overlap in the region the third session’s phosphor LED reads, and the six numbers trade them there regardless of where the band is allowed to go. The two essays together locate the third session’s two costs: the misplaced bands beside 415 nm were a freedom the fit had and should not have had, and the cost at 430 nm is an overlap the matches cannot resolve.

With the band held, three of 180 fits rest on the bound at 460 nm, all beside 445 nm; every other fit lies within fourteen nanometres of its place. The three at the bound are the bound’s only visible effect on observers whose bands were never misplaced, and they are fits that, free, would have gone a few nanometres further; their bands and moves are counted as before.

Four ways to close one valley #

The far valley has now been met four times in this run, and each time something different closed it, which says more about the valley than any one of them.

Under one session it was the deepest problem the census had. One session has a second valley found fits falling into it from the census’s own starts; a bound on the macula trades false moves for hidden ones closed it by limiting the shift, at the price of real moves; a soft prior on the macula frees the lens band closed it more gently with a prior, and a wider prior gives the moves back by spending the band found the prior’s width a trade between the band and the moves, with six nanometres at its knee. Under two sessions the same six-nanometre prior kept every one of 180 fits within six nanometres of the truth — the valley had become shallow enough for a fixed pull to lift every fit out of it. Under three it was deep enough again that two fits stayed in, and here the band’s limit closes it without any prior.

Read in that order, the valley is one thing: a macula moved far short paired with something that puts its long flank back. Under two sessions every fit beyond 7.5 nm short placed its band between 490 and 500 nm, and under three every one placed it between 475 and 498 — a band on the flank itself. Whether the one-session valley was the same pair is not checked here. The valley deepened with the third session because the third session’s cyan line reads exactly the flank the pair describes, and so made the pair a better description of the matches than it had been with two. A prior on one member of the pair weakens as the data grow; a limit on the other member does not, because it does not depend on the data at all.

That changes what the earlier priors were for. They were asked to do a job — keep the shift from a place the matches nearly supported — that a physical limit on the band does better, and the census’s counts under every prior since the first carried a few fits in which the band had wandered to the flank. The essay before recounted with only bands in place; with the band held, the recount and the count agree, and no recount is needed.

What a vision scientist should take from this #

Bound a parameter by what it is, not by what it is doing. The far valley was closed by limiting the band to where a lens band can physically sit, and the bound cost nothing measurable. Limiting the shift, which was the parameter visibly running away, had cost real moves.

A best fit can be a near tie. Ten of the twelve misplaced fits beat the true placement by under four units of $\chi^2$. A reported best solution says nothing about whether a physically different solution lies just behind it; the bound, or a second search from the other valley, is how to find out.

One design’s costs can have different causes. The third session’s cost beside 415 nm was a freedom in the fit and its cost beside 430 nm with the macula short is an overlap in the matches; the same test — remove the freedom — separates them.

How the fits were made #

The census and its noise are the joint census’s: nine observers at sixty, twenty experiments each, three sessions of light pairs over the census’s surface family with the matching task’s setting noise added from the census’s own seeds. Each experiment is fitted for six numbers by a damped Gauss–Newton descent from five starts — the truth, and the census’s start with the band centred at 410, 430, 450 and 470 nm, the last brought to the bound — with the shift limited to ±30 nm as before and the band’s centre to 400–460 nm, with no prior on the shift and with the six-nanometre one. A band counts as found when its height exceeds twice its standard error and as in place within twenty nanometres of where it was put; a move counts when the shift exceeds twice its error with the band in place. The $\chi^2$ cost of the bound is the best held fit’s $\chi^2$ less the best free fit’s, for the same experiment.

What this does not reach #

The bound’s edges are a judgement. 400–460 nm brackets the census’s bands with room; a real lens band could sit outside it, and an observer whose band lay near 460 nm would find it pinned, as three fits here are.

The overlap at 430 nm is located, not resolved. That the macula’s short flank and a 430 nm band share the third session’s information is the reading the bound leaves; no design that separates them is tried.

Three sessions and one age. Whether the bound matters at one or two sessions, where the prior already closed the valley, is not computed.

Still open: whether a fourth pair of lights frees the band at 430 #

The third session’s remaining cost is beside 430 nm with the macula three short, where the held fits find the real move in ten experiments of twenty against thirteen under two sessions. A pair of lights that differ where the band at 430 nm absorbs and the macula’s short flank does not — a narrow line near 425 nm against a broad source — would read the band without the macula.

The calculation is the census’s three-session design with a fourth pair added, a lamp with a narrow line at 425 nm against the phosphor LED, fitted with the band held to 400–460 nm, the moves and bands counted in place as here. The prediction is that beside 430 nm with the macula short the move is found in at least fourteen of twenty and the band in at least eighteen, while the 415 nm observers keep their counts within one. If the move stays near ten, the overlap lies in the macula’s shape rather than in the lights, and the band at 430 nm and a short macula cannot be separated by colour matching at all.

A freedom the fit should not have had #

The habit here is about the difference between a fit’s parameters and the things they are named for.

The census’s six numbers included a band centre free to range over the macula’s flank, and for three essays the fit used that freedom whenever the matches were nearly indifferent — which, in ten of twelve cases, they were. Nothing in the fit’s own output flagged it: the misplaced fits had the lowest $\chi^2$, passed every significance test, and reported a lens band that no lens could have. Asking where the band was found them; limiting it to where it could be removed them, along with the far valley a prior had been asked to close.

The failure mode is to let a parameter range over values its physical meaning excludes, and then to read its significance as a detection. The test is to bound each parameter by what it represents and see which solutions move — and to read the $\chi^2$ cost of the bound as the measure of how nearly the data could tell them apart.