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.
Scoring a search without spending dataslider: how many rows there are to divide, 3 positionswide5 views
What else it draws
The same object, drawn to answer the other questions the essays put to it.
The hold-out's regret as the sample is divided, against the criterion's, which has no division to make. Too few origins and the average of 15 squared errors is mostly noise; too many and the window the candidates are ranked at has nothing to do with the 120 rows the winner is refitted on. The minimum of the two is at P = 60 and it is still 1.8 times the flat line. The sweep was built to find the split at which a hold-out becomes worth its cost, and on independent rows there is not one.
The 95% point of the statistic's own distribution in each world, against the mean 95% point of five reference distributions built from one sample. Where the error variance is a function of the design, the two resamplings that detach a residual from its row fall short and the two multipliers that keep it there do not; where the rows repeat each other it is the other way round. With both defects at once the blocked multiplier — drawn once per run of 5 rows, so the residual never moves and its neighbours share a sign — is the closest of the five, at 2.999 against a truth of 3.803. It is still short by 0.804, and that shortfall is the next figure.
Regret of five selection rules over fifteen candidates as the design and the errors are made persistent at the same coefficient, 300 draws at each of 7 settings, scored on fresh rows. Akaike's criterion gives up 0.0193, 0.0196, 0.0294, 0.0428, 0.0666, 0.1075, 0.1281; charging each coefficient its own variance inflation, with the persistences read off the sample, gives up 0.0188, 0.0231, 0.0339, 0.0375, 0.0496, 0.0656, 0.0652, and with them given 0.0193, 0.0236, 0.0348, 0.0374, 0.0426, 0.0469, 0.0318. The hold-out gives up 0.0358, 0.0365, 0.0393, 0.0453, 0.0594, 0.0913, 0.1087 and Schwarz's criterion 0.0355, 0.0295, 0.0331, 0.0397, 0.0575, 0.0849, 0.1047.
The optimism of an in-sample fit of the widest candidate on 120 rows, in units of its own error variance, over 1500 draws, by persistence. Counted on the 40 rows straight after the sample: 11.1, 21.5, 35.6, 51.5, 65.5, 87.8, 119.1. Counted on rows 600 further on: 9.3, 18.3, 33.0, 50.6, 67.0, 96.1, 159.0. Charged by the long-series inflation: 10.0, 19.3, 34.7, 54.4, 74.3, 114.2, 234.1. Charged by the same inflation summed over the sample's rows: 10.0, 19.2, 34.2, 53.1, 71.8, 108.2, 208.8. Charged by twice the trace of the hat matrix against the errors' correlation on the sample's own design: 10.0, 18.7, 32.0, 47.5, 61.5, 85.2, 132.2.
Where it is used
6 essays draw this figure, each at the numbers its own argument is about, so the same picture answers 6 different questions.
- A criterion is a prediction of the hold-out Scoring a search without spending data
- Where the two searches cross Scoring a search without spending data
- Two defects and one resampling Scoring a search without spending data
- How long a block a multiplier shares Scoring a search without spending data
- A penalty that counts independent rows Scoring a search without spending data
- A charge the sample can afford Scoring a search without spending data