Prior probability ($\Pi (A)=\mathbb{P}(\Theta \in A)$)

Prior probability describes uncertainty before the current observations are incorporated. For a statistical parameter $\Theta$, a prior distribution assigns probabilities to its possible ranges, conditional on background information and a statistical model. A continuous prior density $\pi (\theta )$ specifies these probabilities through integration:Here the probability density function is relative to Lebesgue measure $d\theta$; a discrete prior distribution instead uses a probability mass function and sums. Density at one point is not the probability of that point. A nonnegative kernel with infinite total mass is an improper prior, whose use requires a separate check of posterior propriety.

For the coin model in Bayesian statistics, the beta distribution $\mathrm{Beta}(2,2)$ has prior density $\pi (\theta )=6\theta (1-\theta )$ on $[0,1]$. Its mean is $1/2$ and its variance is $1/20$. It favors moderate heads probabilities without ruling out values near zero or one. For example,The prior curve in Bayesian posterior makes this preference visible; moment matching for a beta prior explains how to choose a beta distribution from a desired mean and variance.

A prior distribution lets earlier evidence or elicited knowledge contribute to statistical inference when the current sample size is limited. It also exposes assumptions that can be checked by prior elicitation and a prior predictive check. Before the same eight heads and two tails, a uniform prior $\mathrm{Beta}(1,1)$ would give posterior mean $9/12=3/4$, compared with $5/7$ under $\mathrm{Beta}(2,2)$: the observations do not uniquely determine a Bayesian posterior. Even a uniform prior makes a choice of parameter coordinates; its child article explains why changing coordinates can destroy uniformity.

A class prevalence applicable before considering the current case-specific evidence. Under random sampling from the relevant reference population, it sets the prior class probability used in Bayes' theorem.

The prior mean is the expected value of a statistical parameter under its prior distribution, when this expected value exists. The posterior mean instead uses the posterior distribution after conditioning on observations.

Translate external information into a prior distribution by asking for representative values, uncertainty intervals and tail probabilities. Fit parameters of a suitable distribution to those judgments and inspect the resulting prior predictive distribution. For a probability believed near $0.1$, a beta distribution with parameters $0.1\kappa ,0.9\kappa$ has the desired mean; its concentration $\kappa$ can be chosen to match a stated probability of exceeding $0.15$. The words “near” and “unlikely” alone do not specify one unique prior.

For two exhaustive mutually exclusive models of positive prior probability, prior odds compare their probabilities before observing the data. They must be distinguished from the parameter prior distributions within those models. A Bayes factor multiplies these prior odds to produce posterior odds.

A prior density is the probability density function of a continuous prior distribution with respect to a specified reference measure. A proper prior density is nonnegative and integrates to one. Prior-density factors enter Bayesian posterior and RJ-MCMC acceptance formulas along with the likelihood and model prior probabilities.

A continuous spike-and-slab prior mixes a narrow continuous probability distribution around zero with a much wider component. It permits practical effect selection while retaining a continuous posterior density. Its narrow component is not the exact-zero mass of a point-null mixture prior. A latent component indicator aids computation; a wide-component draw can still have small magnitude, so posterior effect thresholds and component membership answer different questions.

A uniform prior on a finite-volume parameter region assigns constant probability density there and zero outside. On $[0,1]$ it is $\mathrm{Beta}(1,1)$, giving a beta distribution posterior after binomial distribution data. Uniformity depends on the coordinate: a nonlinear reparametrization introduces a Jacobian and generally does not preserve a uniform density.