A soft prior on the macula frees the lens band

An older observer’s colour matches are read by fitting six numbers that describe the eye — rod weight, lens age, macular density, a lens band’s amplitude and centre, and the macula’s peak shift — and a single session of matches leaves those numbers loosely tied. One session has a second valley found the fit, searched thoroughly, often settling where the macula’s peak sits ten to twenty nanometres from the standard, where no real macula is; a bound on the macula trades false moves for hidden ones held the shift within ±6 nm and closed that valley.

The bound worked and cost something. An unmoved macula, reported moved in eight experiments of twenty without the bound, was reported moved in one or two with it; but beside a lens band at 445 nm, a macula truly three nanometres short was found in three experiments where the census found seven, because a real move with one session’s noise on it reaches the wall of a ±6 nm bound and stops there. A hard bound is a wall: a fit cannot pass it however much the matches pay.

A softer constraint lets the matches pay. A Gaussian prior on the shift adds to the fit’s $\chi^2$ the square of the shift over the prior’s width, so a shift of eight nanometres costs four units against a width of four, and a fit that the matches push hard enough can still go there. The prediction was that a prior four nanometres wide keeps the false moves at two or fewer at every band, as the bound did, and restores the 445 nm observers’ real moves to within two of the census’s, because their far fits would cost the prior more than a unit only past about −8 nm and one session’s noise rarely carries them so far. The alternative was that the prior lets the false moves back, which would mean the far valley’s pull is stronger than a real move’s and only a hard bound or a second session holds it.

The band comes back, the moves do not #

Under a Gaussian prior four nanometres wide on the macula’s shift, one session’s best fits from five starts report an unmoved macula moved in none, one and one experiments of twenty at the bands at 415, 430 and 445 nm; they find a real three-nanometre move as seldom as the hard bound did or more seldom — two of twenty beside 445 nm, against the census’s seven — and fit it, on average, at between 1.2 and 1.9 nanometres. One fit of 180 passes six nanometres. With the band at 430 or 445 nm and the macula unmoved or moved short, the prior’s fits find the lens band in ten to twelve experiments of twenty, where the census’s found three to seven; with the macula moved long, no more often than the census, and beside 445 nm in two where it found five.

• The prediction holds for the false moves and fails for the real ones.
• The alternative fails: the far valley stays closed.
• What the prediction did not ask is the larger result: holding the macula near the standard returns the lens band to the fit.

Drawn as a trade, the nine observers fall into two groups. Those whose macula is unmoved or moved short gain up to seven experiments’ worth of found bands and lose one to five moves; the moves they lose are false where the macula never moved and real where it did. Those whose macula moved long gain no band and lose one or two moves, and the observer with its band at 445 nm and its macula moved long loses three bands as well.

The pattern has one reading. A macula and a lens band both absorb short-wavelength light, and one session’s matches cannot always say which of them did. Held loose, the fit gives the absorption to whichever fits the noise best, and the macula, with its shift free, often takes it; held near the standard, the macula cannot, and the band — which every simulated observer has — is found. Where the macula truly moved long, its real move and the band compete the other way, and holding the macula in costs the band too.

False moves #

Unbounded, the best of five starts reported an unmoved macula moved in eight, six and two experiments of twenty; the bound in one, two and one; the prior in none, one and one. The prior closes the far valley at least as well as the wall did. The valley lay ten to twenty nanometres out; at a prior width of four, a fit there pays six to twenty-five units of $\chi^2$ for the privilege, and the earlier essay found the far valley lower than the near one by under a unit in half the cases. The matches never pay that much.

Of all 180 one-session fits, one ends beyond six nanometres under the prior, and none beyond eight. The prior does not need a wall to keep fits in the plausible range; its cost rises fast enough that the matches’ small preference for the far valley never overcomes it.

Real moves #

Observer by observer, the prior’s counts sit at or below the bound’s. Of the six observers whose macula truly moved, the prior finds the move as often as the bound for two and less often for four: at 415 nm five and three, at 430 nm two and two, at 445 nm two and none. The census found six and five, four and three, seven and two.

The prediction reasoned about the far fits: a real move’s fits reaching past −6 nm would be stopped by the wall and let through by the prior. That part is right — the prior lets them through. What the prediction left out is that the prior also acts on every fit inside the wall. A fit at −3 nm pays a little under the prior, a fit at −4 a little more, and every fit is drawn some way towards nought, so that a real move of three is fitted smaller and its fitted size is less often twice its error.

How far the prior pulls #

Every real move of three nanometres is fitted under the prior at a mean of between 1.2 and 1.9 nanometres, with the right sign; the unmoved maculas at between −0.05 and +0.7. That is the ordinary shrinkage of a prior: a single session’s matches pin the macula’s shift with a standard error of a few nanometres, comparable with the prior’s width of four, and the fit’s answer is pulled about halfway from what the matches alone say towards what the prior says.

Shrinkage is not an error a prior makes by accident; it is what a prior is. A fit that reports 1.5 nanometres for a macula truly moved three is saying that one session’s matches, weighed against what maculas generally do, support about half the move. For a population study averaging many observers the shrinkage biases every move towards nought; for an individual it gives a smaller, more cautious answer, and less often a significant one.

One observer’s fits #

For the observer the bound served worst, the two constraints spread the fits differently. Under the bound the twenty fits spread from the wall at −6 nm to past nought; under the prior they gather nearer nought, at a mean of −1.51 nm, and only two report the move beyond twice their error. Neither recovers the census’s seven — which, the earlier essay showed, owed something to the census’s starts missing valleys that happened to sit on the right side.

For this observer one session simply does not carry enough to place a three-nanometre move beside a band at 445 nm, and every way of fitting it trades one error for another: free, the fit sometimes finds the far valley; walled, it stops at the wall; softly held, it shrinks. A macula three nanometres short is a lens band named the confusion at this band in its title, and the prior does not undo it.

The band, found again #

The band count is where the prior pays for itself. At 430 nm, with the macula moved short and unmoved, the prior finds the band in twelve and eleven experiments of twenty against the census’s six and six; at 445 nm in ten and eleven against three and seven. At 415 nm, where band and macula overlap least, the prior adds five bands where the macula moved short — twelve against seven — and none where it is unmoved or moved long. The hard bound, for comparison, left every band count within two of the census’s — its wall caught only the fits that reached it, and left the rest of the macula’s freedom to compete with the band.

A lost lens band goes to age or the macula found the band’s absorption, when the fit missed it, reappearing as a little extra lens age or a moved macula. The prior closes one of those two routes. With the macula’s shift costly, the fit’s cheapest way to explain the short-wavelength absorption is the band that is really there, and one session finds it far more often.

The cost where the macula moved long #

The trade is not free for every observer. With the macula moved three nanometres long, the prior finds the band no more often than the census at any band, and beside 445 nm in two experiments where the census found five. A macula moved long and a band at 445 nm push the observer’s matches in directions that, in one session, partly offset; holding the macula near the standard leaves the fit to explain a long move’s effect with something else, and the band, pulled in the wrong direction, is found less often.

Real maculas move both ways — the mixture of lutein and zeaxanthin sets the peak a few nanometres either side of the standard — so a population would hold observers of both kinds. The prior’s gain in bands for the short-moved and unmoved observers is larger than its loss for the long-moved ones in the census, twenty-seven experiments’ worth against four, but a population weighted differently would weigh it differently.

Outside knowledge, three ways #

The census has now given a single session outside knowledge three ways, and the three show a pattern. Knowing the macula’s density frees two sessions, not the line on its peak supplied the macula’s density from an independent measurement and found it freeing the fit when the measurement was right; a density a tenth wrong costs the band more than fitting it found the same knowledge, slightly wrong, costing the band more than leaving the density free. A value is strong knowledge and strongly wrong when it is wrong.

The bound and the prior supply knowledge about the macula’s shift as a range rather than a value. The bound says nothing at all inside its range and everything at its wall; the prior says a little everywhere. That is why the bound left the band where the census had it — inside the wall the macula competed as freely as before — and the prior returned it: a gentle pull on every fit is what moves the shared absorption from the macula to the band, not a wall that only the far fits ever meet.

A line on the macula’s flank buys less than a repeated session is the reminder of what outside knowledge is being compared with. A second session of matches gives the fit new evidence about both the band and the macula at once; every kind of prior knowledge gives evidence about one of them and buys the other only by subtraction. The prior’s gain in bands is real, and it is bought with the macula’s moves, where a second session would have bought both.

What a vision scientist should take from this #

For one session, prefer a soft prior on the macula’s shift to a hard bound when the lens band is the quantity of interest. It closes the far valley as well as the bound did and returns the band to the fit for most observers.

Do not read a prior’s shift as the macula’s shift. Under a four-nanometre prior a real three-nanometre move is fitted at about one and a half; the prior’s answer is a shrunk one, and significant less often than the move is real.

For the macula’s own shift, take a second session. A second session keeps a band and the macula apart found two sessions placing both, and neither the wall nor the prior makes one session do that.

How the experiments were refitted #

The census is the joint census’s, one session of matches for nine simulated observers of sixty — a lens band of fixed height at 415, 430 or 445 nm and a macula shifted −3, 0 or +3 nm — with the census’s 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 and the audit’s three — with the census’s limits of ±30 nm on the shift and one more residual: the shift over four nanometres, in units of the matches’ noise, set against nought. The best of the five fits is kept. A band or a moved macula is found when its fitted value exceeds twice its standard error, computed from the fit’s covariance with the prior’s residual included, so that the prior narrows the shift’s error as it shrinks the shift. The bound’s and the census’s counts are the earlier essay’s.

What this does not reach #

One width. Four nanometres is one choice; a wider prior would shrink less and close the far valley less firmly, and the census has not read the width at which the two balance.

A prior centred on the standard. Every prior here pulls towards no shift; an observer group known to have, say, more zeaxanthin would want a prior centred elsewhere.

One session only. The prior was not run on two sessions, where the bound changed almost nothing and the prior’s shrinkage would be smaller still.

Still open: whether a wider prior keeps both the band and the moves #

A prior four nanometres wide shrinks a three-nanometre move about halfway and frees the band. A wider prior shrinks less, and if it still costs the far valley enough to keep it closed, it might keep most of the band’s gain and give back some of the moves.

The calculation is the one-session census refitted from five starts under Gaussian priors six and eight nanometres wide, the false moves, real moves and bands counted as here. The prediction is that at six nanometres the false moves stay at two or fewer, the real moves rise to within two of the bound’s counts, and the band keeps two thirds of its gain over the census at 430 and 445 nm; at eight, the far valley reopens at 415 nm. If the band’s gain falls as fast as the moves recover, the band and the macula’s shift share one session’s information in a fixed proportion, and no prior width buys both.

Knowledge about one number frees another #

The habit here is about what a constraint on one parameter does to the others.

The prior was chosen for the macula, to stop false moves, and judged by the macula’s counts it does that and costs real moves too. The larger effect was on a different number, the lens band, which the prior never mentions: with the macula held near what maculas usually do, the absorption the two had been competing for went to the band, and one session found it twice as often.

The failure mode is to judge a constraint only by the parameter it constrains. In a fit where parameters compete for the same evidence, loosening one’s grip or tightening it moves the others, and the effect can be largest where nobody looked. The test is to read every parameter’s recovery before and after — here, the band beside the macula — and to report the trade rather than the target.