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Prior, Likelihood, Posterior

P(θ∣D)⏟posterior  ∝  P(D∣θ)⏟likelihood×P(θ)⏟prior\underbrace{P(\theta \mid D)}_{\text{posterior}} \;\propto\; \underbrace{P(D \mid \theta)}_{\text{likelihood}} \times \underbrace{P(\theta)}_{\text{prior}}

Three objects, each with a distinct role.

The prior

Your belief about θ\theta before seeing the data, expressed as a distribution rather than a point. Its width matters as much as its centre: a tight prior resists evidence, a diffuse one yields to it quickly.

Priors come in three flavours in practice:

Informative, encoding real knowledge. A market-making strategy's win rate is very unlikely to exceed 60%, and a prior can say so.

Weakly informative, ruling out the absurd while staying open. This is usually the sensible default.

Uninformative, expressing near-ignorance, such as a flat prior. Note these are not as neutral as they look: flat in one parameterisation is not flat in another, so "no information" is harder to specify than it sounds.

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