States and state-spaces
Everything else in this primer rests on two definitions. They look almost too plain to be worth writing down, and they are the reason the rest of the argument can be made at all.
A state is a description, and therefore a proposition
A state is a description of a set of real objects. Rain in the next twenty-four hours is a description of a region of the earth's surface. Cladding failure during the transient is a description of a fuel rod. Relapse within five years is a description of a patient cohort.
Because a description asserts something that may hold or fail to hold, a state is a proposition. That single observation is what ties the whole apparatus to logic. Propositions can be conjoined, disjoined and negated; they can stand in entailment relations; they can be assigned probabilities. A model built out of states is therefore an object that logic can operate on, rather than a black box that emits numbers.
A state-space is complete and exclusive
A state-space is a set of states of a real object that is complete, in the sense that the objects it describes are described by exactly one of its states at any given instant. The set {rain in the next 24 hours, no rain in the next 24 hours} is a state-space for a region of the earth. So is {dry, light, moderate, heavy}, at a finer grain.
Completeness is doing two jobs at once. Every case must land somewhere — no object may fall outside the space — and no case may land in two places. Mathematically, a state-space is a partition of the objects under description, and the two conditions are what let probabilities over the space sum to one. Drop either and the arithmetic in the later pages stops working.
The requirement is easy to state and easy to violate. A survey category set of {employed, unemployed, student, retired} is not a state-space: a working student belongs to two, and a person on unpaid leave to none. Repairs of that kind — adding an "other" state, or making the categories mutually exclusive by fiat — are not bookkeeping. They change what the model is about.
Observed and unobserved spaces
Models involve at least two state-spaces describing the same objects. The observed space holds what is measured — cloud cover, a laboratory value, last year's rainfall. The unobserved space holds what is to be predicted. Moving from a state in the first to a state in the second is exactly what an inference is.
Keeping the two apart matters more than it seems. A great many modelling failures amount to smuggling something from the unobserved space into the observed one — using a laboratory result that in practice is only ordered after the diagnosis, or a rainfall total that is not available until after the forecast is due. The model scores beautifully and is useless, because the observed space it was built on cannot be observed when the prediction has to be made.
Choosing the grain
Nothing forces a particular state-space on the modeller. {wet year, dry year}, {above median, below median} and a set of eight precipitation bands all describe the same weather. The choice among them is a modelling decision with consequences in both directions: a coarse space is easy to predict and says little; a fine space says a great deal and is hard to predict from limited data.
That trade-off is not a nuisance to be settled by convention. It is the central quantitative question of the method described in these pages, and it is what abstraction and ways are for: they give the vocabulary for saying how coarse a description is, so that the coarseness can be optimised rather than guessed.
Probability enters here
Once a state-space is fixed, a probability distribution over it is a well-defined object, and the question of what those numbers mean — long-run frequencies, degrees of belief, propensities — becomes live. This primer stays deliberately ecumenical about it, because the machinery of the later pages works under more than one reading. Readers who want the dispute laid out properly should see the Stanford Encyclopedia's account of interpretations of probability.