19.2 Beta Proportion Distribution

19.2.1 Probability Density Function

If \(\mu \in (0, 1)\) and \(\kappa \in \mathbb{R}^+\), then for \(\theta \in (0,1)\), \[ \mathrm{Beta\_Proportion}(\theta|\mu,\kappa) = \frac{1}{\mathrm{B}(\mu \kappa, (1 - \mu) \kappa)} \, \theta^{\mu\kappa - 1} \, (1 - \theta)^{(1 - \mu)\kappa- 1} , \] 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 \(\mu \in (0, 1)\) and strictly positive parameter, \(\kappa > 0\).

19.2.2 Sampling Statement

theta ~ beta_proportion(mu, kappa)

Increment target log probability density with beta_proportion_lpdf( theta | mu, kappa) dropping constant additive terms.

19.2.3 Stan Functions

real beta_proportion_lpdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion density of theta in \((0,1)\) given mean mu and precision kappa

real beta_proportion_lcdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion cumulative distribution function of theta in \((0,1)\) given mean mu and precision kappa

real beta_proportion_lccdf(reals theta | reals mu, reals kappa)
The log of the beta_proportion complementary cumulative distribution function of theta in \((0,1)\) given mean mu and precision kappa

R beta_proportion_rng(reals mu, reals kappa)
Generate a beta_proportion variate with mean mu and precision kappa; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.