Model selection — where it appears
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
The classes held-out pairs cannot rule out
A substitution matrix can choose how many distinct costs it has as part of the fit: try all 203 ways of grouping its six pairs into classes and keep the one that best predicts pairs it was not fitted to. On corpora from a generator with two costs, that choice finds the two on about a third of resamples, and no more often at 3,200 pairs than at 50. A spurious split gains the same bit or two of fit at every size, and held-out scoring charges it a fixed price. A penalty that grows with the corpus finds the two every time, and finds a third class half a bit away from 400 pairs.
A cost built from two properties
A substitution matrix fitted as a partition gives each class of pairs one cost, and the best number of classes can be chosen from the data. When the true costs are a sum of two terms — a transversion penalty and a graded difference between the letters — every pair costs something different, and the chosen partition's number of classes becomes a statement about the corpus: two classes at 25 sequence pairs, four at 200, five at 1,600. A model with the two terms and three parameters predicts a fresh corpus better than any partition at every size, and gets 191 of 200 alignments right from 50 pairs where the chosen partition needs 3,200. Told the wrong property it is worse than the plain two-class matrix, and nothing in its own fit says so.
A line the corpus finds
Substitution costs built as a transversion penalty plus the distance between the two letters' values on a graded property beat every partition of the pairs, when the model is told the property, and fall below the plainest matrix when told the wrong one. Fitting each letter's position on a line instead was predicted to find the property's order from about four times the told model's data and then behave like it. It finds the order on every corpus of 100 sequence pairs or more but one, and every miss starts with the two letters whose values differ by a fifth of a bit. But knowing the order is not knowing the costs. Its alignments are no better than six free classes' until 800 pairs, sixteen times the data the told model needed for the same score, and on a generator with no graded property at all it finds a line anyway, a quarter as long as a real one at 200 pairs, and pays for it against the plain partition.
A line with a prior toward the classes
Substitution costs fitted as the letters' positions on a free line learn the letters' order quickly, and on a generator with no graded property they invent a line anyway. Charging the line for its length, at a rate growing with the corpus as the partitions' charge does, was predicted to collapse the invented line from 50 pairs and to cost the real one little. It does neither. On the generator with no property the charged line still has a median length of 0.12 bits at 3,200 pairs. On the generator with a property it takes the letters' order away at every size up to 400 pairs where the free line had it on sixteen seeds of sixteen. Charging the line for the number of gaps it keeps, the partitions' own rule, collapses the invented line on every seed at every size and matches the chosen partition to the bit — and then needs 1,600 pairs before it believes a real line is there.
Named alongside it
The objects these essays reach for when they reach for this one.
Cost modelEstimatorFittingOverfittingSubstitution matrixAlignmentHonest limitCorpusCross validationParameter choicePrediction