= Prior probability
{title2=$\Pi(A)=\mathbb P(\Theta\in A)$}
{wiki}
= Prior
{synonym}
= Prior distribution
{synonym}
<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>:
$$
\mathbb P(\Theta\in A)=\int_A\pi(\theta)\,d\theta,\qquad \pi(\theta)\geq0,\qquad \int\pi(\theta)\,d\theta=1.
$$
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> $\operatorname{Beta}(2,2)$ 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,
$$
\mathbb P(\Theta>3/4)=\int_{3/4}^1 6\theta(1-\theta)\,d\theta=\frac5{32}=0.15625.
$$
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> $\operatorname{Beta}(1,1)$ would give <posterior mean> $9/12=3/4$, compared with $5/7$ under $\operatorname{Beta}(2,2)$: 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.
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