Normal Distribution
nothing is called 95% until it has been counted
Concept

Overfitting — where it appears

Fitting parameters that describe the noise of one sample rather than the structure behind it, improving the fit and worsening the prediction. Its size is an arithmetic function of the parameter count and the sample size, which is exactly what an information criterion's penalty estimates.

Named by 26 essays across 15 fields — each of them below, with the objects they name alongside it.

What each rule gives up against an oracle that is arithmetic. Expected squared error of the candidate each rule selects, minus the expected squared error of the best candidate in the table, over 500 draws of 120 rows. Both quantities are closed forms — σ_S²(1 + q/(n − q − 1)) — so the only Monte Carlo here is over which candidate got picked. The hold-out spends half its rows measuring what the criterion computes, and pays 1.8 times as much for it. Schwarz's criterion is worst because it is answering a different question: which candidate contains the truth, rather than which one forecasts best.

A criterion is a prediction of the hold-out

A rolling hold-out spends half the sample measuring what a criterion computes from all of it. Against an oracle that is arithmetic rather than an estimate, the criterion gives up 0.01701 and the hold-out 0.03200 — and the number the hold-out reports for its own winner is optimistic by more than either.

proxy · Order-selection The correction is not a property of the sample. tr(HΩ)/q for each of fifteen candidates, at ρ = 0.7. Two candidates that fit the same number of coefficients need corrections that differ by as much as 1.49, because one of them is fitting the persistent predictors and the other is not — so no single number can be right for both, and the scalar n/n_eff = 5.537 is above every one of them. The four predictors carry persistences 0.9, 0.6, 0.3, 0; at one persistence for every column the whole spread collapses and a scalar looks exactly as good as the trace.

A penalty is a trace

Akaike's 2q is not a count of coefficients. It is the answer a trace collapses to when the rows are independent — and once they are not, the trace is still the right object and is no longer the count.

effective · Order-selection The eighth was not a constant. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations between the candidates, over 800 draws apiece. The earlier field reports this flat at about an eighth across list length, on a table and a world it never varies. Vary how much the omitted coefficients are worth — one multiplier, with the table, the list, the law and the sample size all held — and it runs from 17.6% to 1.5%, a factor of 11.75. The world in which every candidate is true is the world in which the tuning list decides most; the world in which one candidate dominates is the world in which it decides nothing.

The eighth that was not a constant

How often a per-candidate tuning list changes which candidate wins is reported flat at about an eighth across list length. Vary how far apart the candidates are instead and it runs from 17.6% to 1.5%.

turnover · Order-selection The curvature is in the denominator. The optimism a Bartlett band of each width actually costs, divided by that width, on two ways of measuring the width, over 2000 draws at 120 rows. Measured in the weights the band spends — Σ w(k), which is what the earlier field levies its charges on — the reading falls from 0.9528 at two lags to 0.7486 at thirty, so a charge proportional to the summed weights is too dear at one end and too cheap at the other. Measured in the pairs the band uses — Σ w(k)(1 − k/n), because a lag of k is an average over n − k products — the same readings are flat from 4 lags up, at 0.0084 of χ² per width against 0.2359. The correction has no fitted parameter in it: it is a function of the window, the width and the sample size.

The width a band is measured in

A tapered covariance band spends 84% of its own weights at two lags and 74% at thirty. Every charge in the collection is a straight line through the origin in those weights, so it is too dear at one end and too cheap at the other.

curve · Criterion What each rung is made of. Each pair of searches, over 300 draws, split into the two effects its excess is the difference of. The overlap is what the second search loses by having the first already run at its own answer; the interaction is what the joint search finds by moving the first off it. They subtract to the excess exactly, on every draw, because the pinned supremum cancels. Two disjoint dictionaries of independent columns read an excess of 0.000011 and are made of 0.000514 and 0.000503. A break paired with a dictionary of step columns has an interaction of exactly 0 and is all overlap. And a break paired with an independent column has an overlap of -0.004395 against an interaction of 0.002364, which is what puts its excess below zero.

Two effects in one number

How much two searches over one sample share is measured as the net of two things — ground both of them find, and configurations only the joint search reaches. One extra supremum per draw separates them exactly.

separate · Break point How much of one search the other has already found. Five pairs of searches on one sample, on a scale whose zero and one are both fixed by construction. Zero is two searches over disjoint sets of independent columns: they remove shares of the residual sum that add, at 0.8 standard errors from exactly additive, and they read 0.004. One is a break search paired with a step column it contains, which reads exactly one on every draw because the step adds nothing at all. Between them: two dictionaries of step columns cut a few rows apart read 0.125, and the pair the earlier field measured — a break and a whitening window, both reading the same residual series — reads 0.762, three quarters of the way to one search containing the other. And below zero, a break paired with a search over independent columns reads -0.306: the joint search finds configurations neither half of it contains, so charging the two separately under-charges.

Two searches that share nothing

Two searches over independent columns remove shares of the residual sum that add exactly. On the scale a chi-square point is quoted on they look super-additive by a fifth of a unit, and none of it is overlap.

apart · Criterion Every candidate is behind by what its parameter count says. Each dot is one of the fifteen subsets of four predictors, fitted on a rolling window of 80 rows and scored against the benchmark out of sample over 60 origins, at a null where every one of them contains the truth. The line is σ²(q₀/(R − q₀ − 1) − q/(R − q − 1)), which is arithmetic on two integers and a window length. Most of these pairs are not nested — a subset of two predictors and a different subset of two share neither model — and the closed form does not care: the displacement is a statement about how many coefficients each side estimates. The candidates of the benchmark's own dimension sit at zero.

The displacement is a parameter count

A nested variant is behind its benchmark out of sample before anything is searched for. The closed form for how far turns out to have nothing about nesting in it — only two integers and a window length — and it prices a table where no candidate contains any other.

select · Multiplicity Most of the rise is the optimiser's, and under one law it is not. The rise in log-likelihood from the tapered plug-in to the maximum over the same eight-lag band, beside what the same optimiser produces on a sample generated from the plug-in's own covariance — where the family is correctly specified by construction and there is nothing to find. Under AR(1) at 0.8 the raw rise is 5.72 and the manufactured baseline is 4.79, leaving 0.93 at 1.8 standard errors; under long memory the excess is 0.14, at 0.2. Under the moving average it is 11.87 at 19.4 standard errors, on every draw. The taper is a shrinkage, and it costs nothing where the sequence decays smoothly and a great deal where it stops dead.

The plug-in and the maximum

A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.

family · Dependence What a disagreement costs, split on whether it decided anything. The regret from choosing the tuning parameter per candidate, on the draws where the candidates disagreed, split on whether the disagreement changed which candidate the table selects. Over 1200 draws at each list length: when the winner changes the regret is 0.02215, 0.03029, 0.03145; when it does not it is -0.00243, -0.00069, -0.00065 — negative, and small enough that it is inside two standard errors of nothing at every length. The whole of the cost lives in the first column, and the second column is not merely small but slightly the wrong sign: when the table's answer is unaffected, letting each candidate use its own window is a very slightly better rule than making them share one. So a disagreement about the tuning parameter is not a cost. A disagreement that changes the winner is.

The quarrel that changes the winner

A disagreement about the tuning parameter costs 0.031 when it changes which candidate the table selects and −0.0007 when it does not. The distance between the values disagreed about has nothing to do with it.

apiece · Order-selection Two factors, opposite directions. The two factors the cost of a per-candidate tuning parameter is a product of, as the candidates are pulled apart, over 800 draws at each of 5 separations. How often the candidates disagree about the tuning parameter rises from 31.8% to 88.8%; the share of those disagreements that change which candidate the table selects falls from 51.6% to 1.7%. So the setting where the candidates quarrel most about the tuning parameter is the setting where the quarrel matters least, and a sweep that reads the rate and stops has read the factor pointing the wrong way.

Two factors pointing opposite ways

As the candidates on a table are pulled apart, they quarrel about the tuning parameter three times as often and the quarrel decides the winner thirty times less often. A sweep that reads the first factor has read the one pointing the wrong way.

turnover · Order-selection Both halves grow; the difference does not. The control pair's two components and their difference, against how much each of its two searches can find, over 1200 draws at each dictionary size. Two disjoint sets of independent columns are additive at every size — the excess stays inside a standard error or two of zero throughout — and it is not because there is nothing there. The overlap grows from 0.000112 at two columns to 0.000870 at ten, a factor of 7.76, and the interaction grows with it, staying within a factor of two of the overlap at every size. Two searches competing for one residual sum share ground and find configurations neither has alone, in almost equal measure, and their difference is what the earlier field's scale calls zero.

What a zero is made of

Two disjoint dictionaries of independent columns read an excess of 0.000116 and are made of an overlap of 0.000583 and an interaction of 0.000467. The control the whole scale is anchored on reads zero because two effects cancel.

separate · Break point A test between nested models, under a null that is true. 1000 comparisons: an AR(1) truth, forecast by a fitted AR(1) and by a fitted AR4 whose extra coefficients are zero. In population the two forecasts are identical, so every rejection is false. The larger model's mean squared error is 1.1663 against 1.0583 — worse, by exactly the noise in estimating coefficients that are not there — and the ordinary test therefore declares the smaller model significantly better 67.2% of the time. Read one-sided in the direction anybody asks about, it finds the larger model better 0.0% of the time. Adding the squared difference between the two forecasts back into the loss differential puts the level at 4.9%.

When one model contains the other

The comparison a forecaster most often wants is between a model and the same model with one more term. That is exactly the comparison the standard test cannot make — and it fails by declaring the smaller model significantly better, more confidently the more data it is given.

evaluation · Forecast The crossing is in the dependence, not in the split. Regret of each rule as the design and the errors are made persistent at the same coefficient, scored on fresh rows because the closed form assumes exactly what is being taken away. An optimism theorem counts rows; when the rows repeat each other there are fewer of them than there are rows, the penalty is too small for the fit it is correcting, and the criterion starts buying coefficients it should not — its average winner grows from 3.31 coefficients to 3.90. The hold-out never used the theorem and overtakes at ρ ≈ 0.81. Schwarz's criterion, worst of the three on independent rows, is best on repeating ones — its heavier penalty is right for the wrong reason.

Where the two searches cross

The obvious dial between a criterion and a hold-out is how much of the sample to hold out, and moving it never changes the answer. The dial that does is one nobody chooses — how much each row repeats the one before it — and the two rules change places at about 0.81.

proxy · Forecast A line in the right width beats two curves. How far each candidate charge sits from the measured optimism across the plateau, in units of each width's own standard error, over 2000 draws. The straight line through the origin in the band's summed weights — which is what the earlier field levies — misses by 0.2382 per width. The same straight line in the pairs the band actually uses, Σ w(k)(1 − k/n), misses by 0.0095. A fitted power law misses by 0.0293 and a fitted decaying rate by 0.0172, both on one fitted constant more. The deferral this field answers asked for a curve; the answer is a line, in a variable with nothing fitted in it.

A line that beats two curves

A deferral asked for a curve. Fitted against the same measurements, a straight line in a variable nobody had to fit describes the plateau better than either curve does with a constant more — and for three windows out of four it does not.

curve · Criterion The split depends on the order. How much two searches share, measured both ways round, over 300 draws. Pinning the first search at its own answer and searching the second gives one overlap; pinning the second and searching the first gives another. A break paired with an independent column reads -0.004395 one way and 0.002163 the other, at 8.70 paired standard errors and on opposite sides of zero. The excess the two components subtract to is the same in both orders by construction, so what changes is only how it is attributed. There is no order-free way to say which of two searches found ground both can reach, and the two orders bracket it.

A split that depends on the order

Run the second search first and pin that instead, and the same draw gives a different overlap and a different interaction — with the same difference. And one pair has no second order at all.

separate · Break point Flat along a row, apart between them. The probability that a per-candidate tuning list changes the winner, at three list lengths on three candidate tables, over 800 draws in each of the nine cells. Along a row — the reading the earlier field takes — it moves by a factor of at most 1.21, so that field's invariant survives on every table. Down a column it moves by up to 1.98. The list length is the dial that does not move this number and the table is one that does, and the earlier field varied only the first.

A table and a list

A nested ladder of candidates differing by one coefficient was predicted to turn over more often at every list length. It turns over less at every one, and its list changes the winner half as often.

turnover · Order-selection Cut the charge and the width follows it. The band width each charge picks, averaged over 400 draws of 120 rows under AR(1) at 0.8, with the standard deviation across draws beside it. Schwarz's charge — half a log n a lag, which is 2.39 here — picks 3.67. Akaike's picks 6.02. Charging the numbers the window actually leaves free, which is half the width, picks 10.12; charging what the optimism measures, 0.767 of that, picks 14.15. A charge and the width it buys are very nearly reciprocal, which is what a likelihood rising at a fixed rate a lag implies and is why the four answers span a factor of 3.86. The width that was actually best on the draw averages 13.90 and moves by 10.30 from draw to draw — three times as much as any rule's answer does.

A width that moves and an error that does not

Four charges give four widths a factor of four apart and four errors half a per cent apart. The derived charge wins, significantly, by a quarter of what was on offer — and none of the four is an estimate of anything.

dimension · Criterion What each criterion selects, at 50 observations. 700 series from an AR(2) with coefficients 0.6 and -0.3, every order from 0 to 8 fitted to the same 42 responses so the log-likelihoods are comparable. AIC finds the true order 55.1% of the time and lands above it 25.7%; BIC finds it 54.1% and lands above it 4.0%. The closed form for one extra lag is P(χ²₁ > 2) = 15.73% for AIC, which does not depend on n at all, and P(χ²₁ > ln n) = 4.79% for BIC at this size, which falls to zero. Under the true order is the other failure and it is BIC's: 41.9% against 19.1%.

Choosing the order

One criterion is consistent and one is not, which is the whole of what gets said about them. At two hundred observations the consistent one is right 95% of the time and the other 70%; at fifty they are both right 54% of the time and wrong in opposite directions, and consistency has not started to mean anything yet.

forecast · Order-selection A wider band is always a better fit. The likelihood maximised over the band, at five widths, averaged over 30 samples. A band at L lags is a band at L + 1 with the last entry held at zero, so the families are nested and the maximised likelihood cannot fall — it does not, on any draw. What it does is rise at 0.984 of log-likelihood a lag. A parameter that is doing nothing buys half a unit in expectation and Akaike's criterion charges one, so this is a criterion very nearly indifferent between every width on offer. The dashed line is what a charge of one unit a lag would exactly cancel. Nothing in the fit chooses a width, and what does choose one is a charge somebody has to pick.

Nothing in the fit picks the width

A wider band is always a better fit, and it is better by about one unit of log-likelihood a lag — which is the order of what a criterion charges for a parameter. Three defensible rules choose widths a factor of three apart.

family · Order-selection R² against the number of useless predictors, n = 30. The response is pure noise and so is every predictor, so the true relationship is nothing at all. R² rises from 0.000 to 0.648 anyway, following k/(n − 1) — which is what a criterion that rewards higher R² is actually rewarding.

R² is not a measure of fit

Adding a predictor with no relationship to anything cannot reduce R², and in expectation raises it by 1/(n − 1). Twenty useless predictors on thirty points give an R² of 0.69 from pure noise.

regression · Summary The ranking on the left, the weights on the right. Eight moving-average forecasts of an AR(1) at φ = 0.4895, the persistence at which the best of them exactly ties the 60-observation benchmark. On the left, each candidate's expected squared error in units of the series' own variance: the smallest belongs to L = 2, at 1.0156. On the right, the weight each carries in the variance-minimising combination of all eight — and the best of them carries 0.00000. The two ends of the family carry 1.0172 of the weight between them, and the combination they make is worth 0.7817, which is 23.0% below the best single forecast. Both columns are closed forms in φ. Which forecast to keep and which forecasts to use are different questions, and this is a set where the answers share nothing.

The weight that is a vector

Two forecasts have a best combination and one number describes it. Eight have a best combination too, and the vector describing it puts nothing at all on the forecast with the smallest mean squared error.

search · Rank The ladder is the same ladder under every law. Each pair's overlap under each of the four laws, over 400 draws apiece. The scale is fixed at both ends by construction: two searches over disjoint sets of independent columns read -0.001, 0.013, -0.002, 0.004, and a break search paired with a step column it already contains reads exactly one under every law. Between them the pair that reads one residual series twice runs 0.763, 0.805, 0.577, 0.752 — lowest under long memory, where a whitening has most to do and the break search has least left to find that the whitening has not taken. The rung that moves most is the pair of step dictionaries, from -0.148 under a moving average to 0.419 under a break; and the pair that is negative is negative under all four.

Three quarters of the way to one search

The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.

apart · Criterion One window for the table, or one each. The regret of the same fifteen-candidate table under AR(1) at 0.8 over 150 draws, with the window attached three ways. Chosen once from the fullest candidate's residuals it gives up 0.02518. Chosen from each candidate's own residuals, with the covariance estimate still shared, it gives up 0.02799 — a paired cost of 0.00281 at 2.0 standard errors for the tuning parameter alone. Estimating the covariance per candidate as well costs 0.01087, so the objection already on record is about 3.9 times the size of the one that was not.

A window for every candidate

The window and the order a whitening needs are chosen once, from the fullest candidate, on an argument that was made about an estimated covariance. A tuning parameter is not a covariance, and the two cost different amounts.

together · Order-selection What the interval covers once the order is chosen as well. 1200 series of 40 observations from an AR(1) with φ = 0.7, at each horizon, on one set of seeds. The upper line is the interval computed at the true parameters — it covers 95.5% on average, which is the check that σ²Σψ² is the right formula rather than a claim about anything a forecaster can do. The lower line is the same formula fed σ̂² and φ̂: 93.6% at one step and 90.8% at 6. The third line chooses the order by AIC from the same data before computing the interval, which costs a further 0.8 points at h = 6.

The interval after the choice

Estimating the coefficients of a known model costs a 95% forecast interval about two points of coverage. Choosing which coefficients to estimate, from the same forty observations, costs another four and a half — so the step nobody records in the output is the more expensive of the two.

forecast · Order-selection The scale moves the width; the curve does not. The band width each charge picks, averaged over 150 draws of 120 rows. The two conventions — a unit a lag and half a log n a lag — pick 6.08 and 3.65 lags. The four charges derived from the measured optimism pick 15.05, 15.60, 15.47 and 15.44, against a best width on the draw of 14.13. So the scale a charge is levied on moves the width by a factor of 4.27 and the shape of the charge moves it by 3.6%. None of the six is an estimate of the draw's own best width: the correlations are -0.006, -0.003, 0.017, -0.001, 0.016, -0.004.

What a better charge buys

Four charges derived from the same measurements pick band widths within six per cent of each other and deliver errors within two per cent of the gap any of them leaves. The scale a charge is levied on decides the width; the shape of the charge decides nothing.

curve · Criterion The same dial, on a list that steps by one. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations, for both tuning parameters at a matched list length of 8. The sieve order runs from 13.1% to 2.6%, a factor of 5.00; the whitening window, from 16.4% to 1.5%, a factor of 11.75. What is held is the number of options, the table, the law, the sample size and the seeds; what cannot be held is the size of a step, since an integer step and a geometric step are different amounts of change. The dial moves both, and it moves them by 2.35 times as much on one as on the other.

A step that is not a ratio

Run the separation sweep on a tuning list of integers rather than a geometric ladder and the two factors still point opposite ways. The invariant does not survive: along a row of integers the probability moves by 2.163 where along the geometric ladder it moves by 1.208.

turnover · Order-selection

Named alongside it

The objects these essays reach for when they reach for this one.

Model selectionMonte CarloInformation criterionDependenceSelection effectDegrees of freedomNested modelsBandwidth selectionBenchmark forecastRegretSpecification searchClosed form

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