Abstraction

The word model in "scientific model" points at abstraction. A model is not the thing; it is a description of the thing with most of the thing left out. Which parts are left out is the whole craft.

What an abstraction is

An abstraction is a description of a set of real objects that is removed from the differences among those objects when they are described in detail. The word chair is an abstraction: real chairs differ in height, colour, material and wear, and the word discards all of it while keeping whatever makes them chairs.

Put formally, an abstraction places several more detailed descriptions in a disjunction — a logical OR. The abstraction adult covers man OR woman. The abstraction above median rainfall covers every specific rainfall total above the median. Moving up a level of abstraction is always a matter of taking a union.

Conjunction moves the other way

A less abstract description is formed by placing descriptions in a conjunction — a logical AND. Brown AND chair is less abstract than chair; it applies to fewer objects and says more about each. Adding conjuncts refines; adding disjuncts coarsens. Those two operations generate the entire lattice of descriptions available to a modeller.

Because a state is a description, all of this applies directly to states. A state may be an abstraction from other states, and a state-space may be coarsened by merging its states or refined by splitting them. A state that abstracts from nothing further is called a way, and ways are where the refinement stops.

Why the grain is a quantitative decision

Abstraction is usually discussed as a matter of style — how much detail an explanation should include. Inside a predictive model it is nothing of the kind, because the level of abstraction fixes two quantities that pull in opposite directions.

• What the model can say. A fine unobserved space carries more entropy, so a successful prediction over it is worth more. Forecasting which of eight rainfall bands will occur is a far stronger claim than forecasting wet or dry.
• What the data can support. Every refinement of the observed space divides the historical record into smaller cells. Past some point each cell holds so few cases that its estimated probabilities are noise, and the apparent gain in precision is an illusion.

The optimum sits between, and it depends on how much data there is. This is the same trade-off that appears in statistics as the bias–variance decomposition, in histogram construction as bin-width selection, and in machine learning as model capacity. Stating it in terms of abstraction has one advantage: it makes clear that the choice is about descriptions, and therefore that it can be argued about in ordinary language before any arithmetic is done.

Abstraction hides disagreement

A practical warning. Coarse descriptions let people who disagree believe they agree. Two engineers may both accept the state fuel rod failed and mean different things by it — a breach of any size, or a breach large enough to matter — and the difference will not surface until the historical record is being classified, or worse, until the model is being used. Whenever a state-space is proposed, it is worth asking which finer descriptions have been disjoined into each state, and whether everyone would classify the same borderline case the same way.

Description and information

There is an old and productive connection between how much a description leaves out and how much information it carries, and it is the connection this primer exploits. The Stanford Encyclopedia's survey of conceptions of information is a good place to see how many distinct notions live under that one word, and how carefully they have to be kept apart.