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_lupdf(theta | alpha, beta)
.
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_lupdf
(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
dropping constant additive terms
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.