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19.1 Beta Distribution

19.1.1 Probability Density Function

If $\alpha \in \mathbb{R}^+$ and $\beta \in \mathbb{R}^+$, then for $\theta \in (0,1)$, $$\text{Beta}(\theta|\alpha,\beta) = \frac{1}{\mathrm{B}(\alpha,\beta)} \, \theta^{\alpha - 1} \, (1 - \theta)^{\beta - 1} ,$$ where the beta function $\mathrm{B}()$ is as defined in section combinatorial functions.

Warning: If $\theta = 0$ or $\theta = 1$, then the probability is 0 and the log probability is $-\infty$. Similarly, the distribution requires strictly positive parameters, $\alpha, \beta > 0$.

19.1.2 Sampling Statement

theta ~ beta(alpha, beta)

Increment target log probability density with beta_lpdf(theta | alpha, beta) dropping constant additive terms.

19.1.3 Stan Functions

real beta_lpdf(reals theta | reals alpha, reals beta)
The log of the beta density of theta in $[0,1]$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_cdf(reals theta, reals alpha, reals beta)
The beta cumulative distribution function of theta in $[0,1]$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_lcdf(reals theta | reals alpha, reals beta)
The log of the beta cumulative distribution function of theta in $[0,1]$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

real beta_lccdf(reals theta | reals alpha, reals beta)
The log of the beta complementary cumulative distribution function of theta in $[0,1]$ given positive prior successes (plus one) alpha and prior failures (plus one) beta

R beta_rng(reals alpha, reals beta)
Generate a beta variate with positive prior successes (plus one) alpha and prior failures (plus one) beta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.