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 generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.