A bound on the macula trades false moves for hidden ones

An older observer’s colour matches carry the shape of the observer’s eye, and a run of essays has read that shape by fitting six numbers to the matches: rod weight, lens age, macular density, the amplitude and centre of a band in the lens, and how far the macula’s absorption peak sits from the standard one. A second session keeps a band and the macula apart built the census every later essay has read — nine simulated observers, their bands at 415, 430 or 445 nm and their maculas moved three nanometres short, left alone or moved three long, matched in one, two or three sessions, twenty experiments each.

One session has a second valley fitted every one of those experiments again from three more starting points, and found the census’s own two starts had stopped in a worse valley in sixty-six of 540 — forty-five of them under a single session. The better valley almost always moved the macula further short, ten to twenty nanometres from the model’s, and one session, fitted thoroughly, reported a macula moved where none was in up to eight experiments of twenty. The counts the earlier essays quoted for one session had been flattered by the census’s starts missing that valley.

A macula ten to twenty nanometres from the standard is not a macula anyone has measured. Its peak moves with the mixture of lutein and zeaxanthin in it, and the mixtures real eyes carry move it by a few nanometres either way. The prediction was that a fit constrained to shifts within ±6 nm restores each observer’s one-session counts to within two of the census’s, and that about one bounded fit in four ends pressed against its bound — roughly the experiments that had a second valley — so that a wrong answer is replaced by a visibly constrained one. The alternative was that the counts stay inflated, which would mean the matches prefer a far-moved macula even within the bound, and the second valley has a near neighbour that a bound does not exclude.

The false moves go, and some true ones with them #

With every one-session experiment fitted from five starts and the macula’s shift held within ±6 nm, an unmoved macula is reported moved beyond twice its standard error in one, two and one experiments of twenty for the bands at 415, 430 and 445 nm — where the unbounded best fits reported eight, six and two and the census’s two starts one, two and four. Every observer’s count of found bands lies within two of the census’s, and seven of nine keep their count of moved maculas within two; the two that do not have their band at 445 nm, where a macula truly moved three nanometres is found in three and none of twenty, against the census’s seven and two. Thirty-one of 180 bounded fits end on the bound, every one an experiment whose unbounded best fit lay beyond it, at a median cost of 0.29 in $\chi^2$. Under two sessions no count moves by more than two and one fit ends on the bound.

• The prediction’s rescue holds for the false moves the second valley made.
• Its “every observer” does not: beside 445 nm the bound costs real moves.
• Its quarter is a sixth: thirty-one fits on the bound, not forty-five.

The false moves were the second valley’s work, and the bound removes them where the valley made them. At 415 nm the unbounded fits reported an unmoved macula moved in eight experiments of twenty and the bounded fits in one; at 430 nm, six and two. The census’s own two-start fits had reported one and two — right for the wrong reason, since their starts had simply not reached the far valley — and the bounded fits, from five starts, now report the same counts for a reason that can be stated.

At 445 nm there was less to remove: the unbounded fits reported two false moves and the census four, and the bound brings it to one. A band at 445 nm sits under the macula’s own absorption, where a band and a moved macula are hardest to tell apart, and the census’s count there was high for that reason rather than because of the far valley.

Counts observer by observer #

Observer by observer, the bounded counts track the census’s. Seven of the nine lie within two: at 415 nm five, one and four against the census’s six, one and five; at 430 nm four, two and three, exactly the census’s. The found-band counts lie within two for all nine. The bound has not just suppressed moves — the true moves at 415 and 430 nm, three nanometres either way, are still found as often as the census found them.

The two exceptions are both at 445 nm with the macula truly moved: three found against seven where it moved short, none against two where it moved long. These are the observers whose band and macula overlap most, and the bound, which keeps a fit’s macula near the standard, there leaves a real move unreported without the band taking it up: the band is found no more often under the bound than without it.

Fits on the bound #

Thirty-one of the 180 one-session fits end on the bound, about one in six, where the prediction expected one in four. They are not spread evenly: six, eight and seven for the three observers whose macula truly moved three nanometres long, five and three at 445 nm short and unmoved, two at 415 nm unmoved, and none for the observers at 415 and 430 nm whose macula moved short.

The prediction counted the experiments that had a second valley; the census found forty-five one-session misses, and a miss’s better valley was not always beyond six nanometres. Every fit that ends on the bound is one whose unbounded best fit lay beyond it, but not every second valley lay that far. The pattern of where they fall says what pulls a fit to the bound: a macula truly moved long gives one session’s noise a start in that direction, and twenty-one of the thirty-one pressed fits belong to the three long-moved observers, while the observers moved short at 415 and 430 nm contribute none — a pattern the census reports and does not explain, since the earlier essay found the far valley lying short. A fit pressed on its bound is the visibly constrained answer the prediction hoped for: in an analysis of real matches, a reported macula sitting exactly at ±6 nm is a flag that the matches wanted more.

What the bound costs the fit #

The bound raises a pressed fit’s $\chi^2$ by a median of 0.29 and at most by 3.0. Half of the fits it turns back lose less than three tenths of a unit — the two valleys nearly level, as the earlier essay found many of them — and none loses as much as the 7.6 that first revealed a missed valley. The matches do not insist on the far valley; they find it marginally better, by amounts their own noise could produce.

That is what makes the bound defensible as a prior rather than a fudge. A constraint that cost the fit many units of $\chi^2$ would be overriding the data; one that costs a fraction of a unit is choosing between two answers the data cannot separate, on the strength of what maculas are known to do.

A real move behind the bound #

Drawn experiment by experiment for the observer the bound treats worst, the cost is visible. Unbounded, four of twenty fits put the macula past −6 nm, as far as −8.7; bounded, those fits stop at −6, and only three experiments of twenty now report the move beyond twice their error, where the census reported seven. The far fits here were not the second valley of the earlier essay — they are a few nanometres past the bound, not ten or twenty — but a real three-nanometre move that one session’s noise carries a little too far.

A bound placed at ±6 nm sits only three nanometres past a real move of three, and noise of a few nanometres reaches it. A macula three nanometres short is a lens band found exactly this confusion at the 445 nm band — a short macula and a band beside it trading places in the fit — and the bound, by forbidding the macula to go further, leaves the fit at the edge of its range, where its reported move is no longer twice its error.

Two sessions #

Under two sessions no observer’s count moves by more than two, and one fit of 180 ends on the bound. The second session’s matches already pin the macula near its true place — a second session keeps a band and the macula apart was the census’s founding result — and the far valley, which one session’s matches cannot rule out, is ruled out by the second session’s.

So the bound is a one-session remedy. It does for one session part of what a second session does — keeps the macula out of a valley real eyes do not occupy — and it does it without new data, at the price of trusting a prior about where maculas sit. Where that prior is tight against a real move, as at 445 nm, it costs what a second session would not.

Other remedies for the same session #

The census has tried several ways of rescuing a single session. A line on the macula’s flank buys less than a repeated session added a narrow-band match on the macula’s flank, and knowing the macula’s density frees two sessions, not the line on its peak gave the fit the macula’s density from an independent measurement; a density a tenth wrong costs the band more than fitting it found that such outside knowledge, slightly wrong, can do more harm than good.

A bound is the mildest form of outside knowledge — not a value, only a range — and it shares that last essay’s risk in a weaker form: where the truth sits near the range’s edge, the bound biases the fit towards the inside. A bound of ±6 nm is a fair range for real maculas; for a real macula moved by three, it leaves only three nanometres of room for noise, and one session’s noise uses it.

Where the bound leaves the band #

The worry with any constraint on one parameter is that the fit moves the excluded effect into another. A lost lens band goes to age or the macula found exactly that traffic when a band was missed — its absorption reappearing as a little extra lens age or a moved macula — and a bound on the macula invites the reverse: a real move the macula is not allowed to make reappearing as a band.

The census reads the bands and does not find it. Under the bound, the 445 nm observers’ bands are found in three, six and three experiments of twenty, against the census’s three, seven and five; at 415 and 430 nm every observer’s band count lies within two of the census’s. The bound does not inflate the band. Where a real move is pressed against it, the fit stops at the edge and reports the move less often; it does not invent a band to carry the difference.

That is the better of the two failures a constraint can have. A fit that silently re-routes an effect into a neighbouring parameter produces confident wrong answers in a place nobody is looking; a fit that stops at its constraint produces a visible one — a shift sitting at ±6 nm — and a reported count that is too low. The one-session analyst sees the flag and can read the low count for what it is.

What a vision scientist should take from this #

Bound the macula’s shift when fitting one session. A ±6 nm bound removes the false moves a single session’s far valley makes, at a cost in $\chi^2$ too small for the matches to resist.

Report the fits that end on the bound. One in six one-session fits does; each is a fit the data pushed past the plausible range, and reading it as a measured shift of six nanometres would be wrong.

Expect the bound to hide moves beside a band at 445 nm. There the macula and the band trade places, and a real move pressed against the bound is reported less often.

How the experiments were refitted #

The census is the joint census’s: an observer of sixty with a lens band of fixed height centred at 415, 430 or 445 nm and a macula shifted by −3, 0 or +3 nm; one session of matches, or two; twenty experiments each, with the census’s own setting noise from its own seeds. Each experiment is fitted by the census’s damped Gauss–Newton descent from five starts — the census’s two, with the band at 410 and 450 nm, and the audit’s three, the truth and the census’s start with the band at 430 and 470 nm — with the shift held between −6 and +6 nm, and the best fit kept. A band or a moved macula is found when its fitted value exceeds twice its standard error from the fit’s covariance. A fit ends on the bound when its shift lies within a hundredth of a nanometre of ±6; its cost is its $\chi^2$ less the unbounded best fit’s over the census’s two starts and the audit’s three.

What this does not reach #

One bound. ±6 nm is one choice; a bound of ±8 would leave real three-nanometre moves more room and the far valley less, and the census has not read where the trade balances.

A hard bound. A soft prior — a penalty growing with the shift — would let a strong signal past and hold a weak one; the census reads only a wall.

Simulated observers. Every observer is built from the model the fit uses; a real eye’s macula has a shape as well as a position, which the six numbers do not carry.

Still open: whether a softer prior keeps the real moves #

The hard bound removes the false moves and hides some real ones, both because a fit cannot pass it at any price. A penalty that grows with the shift — the $\chi^2$ plus the square of the shift over a prior width — lets a fit go further when the matches insist and holds it back when they barely prefer the far valley.

The calculation is the one-session census refitted from five starts with a Gaussian prior on the shift of width four nanometres instead of the bound, the counts read as here. The prediction is that the prior keeps the false moves at two or fewer at every band, as the bound does, and restores the 445 nm observers’ real moves to within two of the census’s, because their far fits cost the prior more than a unit only past about −8 nm and the matches’ noise rarely carries them there. If the prior lets the false moves back, the far valley’s pull is stronger than a real move’s, and only a hard bound or a second session holds it. A soft prior on the macula frees the lens band makes that calculation.

A constraint is a claim about the population #

The habit here is about adding prior knowledge to a fit.

A bound on the macula’s shift is a statement that real eyes do not move it further, and it is a good statement — about maculas in general. Applied to one fit, it becomes a statement about this eye, and for an eye whose macula sits near the edge of the range, the statement is nearly false. The census’s observers moved three nanometres sat halfway to the bound, and one session’s noise carried them past it.

The failure mode is to judge a prior by whether it removes the errors it was introduced for, and not by what it does to cases near its edge. The test is to read the constrained fit on cases that are true and near the boundary — here, a real three-nanometre move beside the band that confuses it most — as well as on the cases the constraint was meant to correct.