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12.7 Ordered Probit Distribution

12.7.1 Probability Mass Function

If \(K \in \mathbb{N}\) with \(K > 2\), \(c \in \mathbb{R}^{K-1}\) such that \(c_k < c_{k+1}\) for \(k \in \{1,\ldots,K-2\}\), and \(\eta \in \mathbb{R}\), then for \(k \in \{1,\ldots,K\}\), \[ \text{OrderedProbit}(k~|~\eta,c) = \left\{ \begin{array}{ll} 1 - \Phi(\eta - c_1) & \text{if } k = 1, \\[4pt] \Phi(\eta - c_{k-1}) - \Phi(\eta - c_{k}) & \text{if } 1 < k < K, \text{and} \\[4pt] \Phi(\eta - c_{K-1}) - 0 & \text{if } k = K. \end{array} \right. \] The \(k=K\) case is written with the redundant subtraction of zero to illustrate the parallelism of the cases; the \(k=1\) and \(k=K\) edge cases can be subsumed into the general definition by setting \(c_0 = -\infty\) and \(c_K = +\infty\) with \(\Phi(-\infty) = 0\) and \(\Phi(\infty) = 1\).

12.7.2 Sampling Statement

k ~ ordered_probit(eta, c)

Increment target log probability density with ordered_probit_lpmf( k | eta, c) dropping constant additive terms.

12.7.3 Stan Functions

real ordered_probit_lpmf(ints k | vector eta, vectors c)
The log ordered probit probability mass of k given linear predictors eta, and cutpoints c.

int ordered_probit_rng(real eta, vector c)
Generate an ordered probit variate with linear predictor eta and cutpoints c; may only be used in generated quantities block