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12.7 Ordered Probit Distribution

12.7.1 Probability Mass Function

If KN with K>2, cRK1 such that ck<ck+1 for k{1,,K2}, and ηR, then for k{1,,K}, OrderedProbit(k | η,c)={1Φ(ηc1)if k=1,Φ(ηck1)Φ(ηck)if 1<k<K,andΦ(ηcK1)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 transformed data and generated quantities blocks