Molecular Geometry & Bonding
what the shape decides
Concept

Model selection — where it appears

Deciding which of several candidate models a body of data supports. The honest version asks what the data can rule out rather than which model fits best, since a better fit is always available from more parameters.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

What it would cost to destroy each refutation. For every discordant pair in the collection, the measurement error that would be needed to reverse it, as a fraction of that predictor's own spread of measurements. Nothing is quoted: the question is not what the uncertainties are but how large they would have to be. The cheapest to destroy needs 3.4 per cent of the range and the dearest needs 51.6.

The error bar that would be needed

Two things a pair of measurements can say without any model both treat their numbers as exact. The quoted measurements carry no uncertainties, and inventing some would be worse than having none — so the question is asked the other way round. Not what the errors are, but how large they would have to be. The three refutations that seemed most worrying turn out to be the sturdiest of the lot.

bonding · Models The bracket has no upper end. Every price is multiplied by √2 over √(1 + r² − 2ρr), where ρ is the correlation between the two measurements' errors and r is the ratio of their sizes. For equal precision the factor is 1/√(1 − ρ), which is one at independence and unbounded at perfect correlation. The other extreme the question asked for is not a number.

The other end of the bracket is not a number

Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.

bonding · Models Five candidates, five failures, five different places. Every point of the square, with each cheap diagnostic's failing pair joined by a line. The five tested candidates fail on five different pairs involving 10 different points — no line shares an end with another. Had they all failed on one corner the lines would have converged on it, and the honest conclusion would have been that composites are safe away from that corner.

Five failures in five different places

Seven quantities a mean field produces for nothing have been tried as diagnostics, and none of them is usable. The question left is whether that is one finding or seven — whether the same awkward corner of the square breaks every candidate, or each is broken somewhere else. Each is broken somewhere else. Five candidates, five failing pairs, ten systems, and not one of them appearing twice.

wrong · Approximation One ranking, three chains. The collection's priced claims in order of fragility, with each claim placed in its own predictor's column. Two claims sharing a predictor share its error correlation, so their prices carry the same factor and their relative order cannot be changed by any correlation structure whatever — a column is rigid. Claims in different columns can be reordered at a price. So the ranking is not one ordering but three chains interleaved, and only the within-column statements need no assumption about anybody's errors.

Three chains and ninety orderings ruled out

Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

bonding · Models

Named alongside it

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

Error propagationMeasurement uncertaintyMonotonicityPredictorUnderdeterminationClosed formComposite methodCorrelationCorrelation energyError cancellationMean-field approximationModel limit

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