Inferences

An inference is an extrapolation from a state in one state-space to a state in another, where both state-spaces describe the same set of real objects. The space extrapolated from is the observed state-space; the space extrapolated to is the unobserved one.

The standing example

Take the observed state-space {cloudy, not cloudy} and the unobserved state-space {rain in the next 24 hours, no rain in the next 24 hours}. Both describe the same region of sky over the same interval. Moving from cloudy to rain in the next 24 hours — with or without a probability attached — is an inference.

Notice what the definition rules out. An extrapolation between spaces describing different objects is not an inference in this sense; it is an analogy, and it needs an extra argument that the objects are relevantly alike. Much of what goes wrong when a model is transported to a new population, a new plant or a new hospital is exactly this: the inference was licensed for one set of objects and was quietly applied to another.

An inference is a relation, not a number

It is tempting to identify an inference with the probability attached to it. The definition here keeps them apart, and the separation earns its keep. The inference is the structural fact — this observed state, that unobserved space — and the probability distribution is what is asserted about it. Two modellers can agree entirely about which inferences their model makes and disagree about every number.

This matters for the measurement that follows later. What gets measured is the inference: how far it still is from a deductive conclusion. That question makes sense whatever the numbers turn out to be, which is why the measure can be used to choose the numbers rather than presupposing them.

Conditional inference

In practice a model does not make one inference; it makes a family of them, one for each state of the observed space. Each member has the form given this observed state, the distribution over the unobserved space is such-and-such. The precipitation case study shows the pattern plainly: three conditions on prior observations, each carrying its own conditional probability of a wet year.

Writing a model out this way makes two things visible that a fitted equation tends to hide. First, the observed space is finite and enumerable, so every case the model can encounter is on the page. Second, some conditional inferences are supported by many historical cases and others by very few, and the difference is legible rather than buried in a standard error.

Deduction as the limiting case

If the observed state settles the unobserved one — if cloudy guaranteed rain — the inference would be a deduction. Nothing would be left to determine, and the missing information would be zero. Deduction is not a different kind of thing from inference on this account; it is the endpoint of a scale, reached when the residual uncertainty vanishes.

The other endpoint is equally instructive. If the observed state tells us nothing at all about the unobserved space, the inference is worthless, and the residual uncertainty equals the whole uncertainty we started with. Every real inference sits somewhere between, and locating it on that scale is the point of the measures introduced in the next group of pages.

What the model owes the world

Eventually the unobserved space is observed: it rains or it does not. The state that then obtains is the outcome, and comparing outcomes with the inferences that anticipated them is the only test the model faces that it cannot game. Statistical practice packages that comparison in many ways — scoring rules, calibration curves, likelihood on held-out data — and the assumptions underneath the packaging are worth understanding; the Stanford Encyclopedia's entry on the philosophy of statistics is a careful guide.