12.7 Ordered Probit Distribution
12.7.1 Probability Mass Function
If K∈N with K>2, c∈RK−1 such that ck<ck+1 for k∈{1,…,K−2}, and η∈R, then for k∈{1,…,K}, OrderedProbit(k | η,c)={1−Φ(η−c1)if k=1,Φ(η−ck−1)−Φ(η−ck)if 1<k<K,andΦ(η−cK−1)−0if k=K. 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 c0=−∞ and cK=+∞ with Φ(−∞)=0 and Φ(∞)=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