Priors and Posteriors

The prior encodes belief before data, the posterior after; the choice of prior is a modeling decision to make explicit.

The prior

A prior distribution p(θ) states what is believed about a parameter before seeing the current data. It can encode genuine prior knowledge, physical constraints, or deliberate ignorance. The prior is where domain expertise enters a Bayesian analysis.

Kinds of priors

Kronos motion — data assimilation

Informative: concentrated, reflecting strong prior knowledge.
Weakly informative: broad but ruling out absurd values.
Non-informative or reference: designed to influence the posterior as little as possible.

The posterior

After data, the posterior p(θ | data) ∝ p(data | θ) p(θ) blends prior and evidence. With little data the prior dominates; with abundant data the likelihood dominates and the posterior concentrates near the maximum likelihood estimate regardless of a reasonable prior.

Priors are not free parameters to hide

The honest practice is to state the prior explicitly and test how sensitive conclusions are to it. If a modest change in prior swings the conclusion, the data are too weak to settle the question, and that fact should be reported rather than obscured.

Improper priors

Some non-informative priors do not integrate to one (for example, a flat prior over the whole real line). These improper priors can still yield a proper posterior, but they require care: not every improper prior gives a valid posterior, and the result must be checked.

A prior chosen for mathematical convenience should still be defensible on its own terms, not just because it makes the algebra close.