Bounded Discrete Distributions

Bounded discrete probability functions have support on $\{0,\dots ,N\}$ for some upper bound $N$.

Binomial distribution

Probability mass function

Suppose $N\in \mathbb{N}$ and $\theta \in [0,1]$, and $n\in \{0,\dots ,N\}$. $$\text{Binomial}(n|N,\theta )=(\genfrac{}{}{0ex}{}{N}{n}){\theta }^n(1-\theta {)}^{N-n}.$$

Log probability mass function

$$\begin{array}{rcl}\log \text{Binomial}(n|N,\theta )& =& \log \mathrm{\Gamma }(N+1)-\log \mathrm{\Gamma }(n+1)-\log \mathrm{\Gamma }(N-n+1)\\ & & +n\log \theta +(N-n)\log (1-\theta ),\end{array}$$

Gradient of log probability mass function

$$\frac{\partial }{\partial \theta }\log \text{Binomial}(n|N,\theta )=\frac{n}{\theta }-\frac{N-n}{1-\theta }$$

Distribution statement

n ~ binomial(N, theta)

Increment target log probability density with binomial_lupmf(n | N, theta).

Available since 2.0

Stan functions

real binomial_lpmf(ints n | ints N, reals theta)
The log binomial probability mass of n successes in N trials given chance of success theta

Available since 2.12

real binomial_lupmf(ints n | ints N, reals theta)
The log binomial probability mass of n successes in N trials given chance of success theta dropping constant additive terms

Available since 2.25

real binomial_cdf(ints n | ints N, reals theta)
The binomial cumulative distribution function of n successes in N trials given chance of success theta

Available since 2.0

real binomial_lcdf(ints n | ints N, reals theta)
The log of the binomial cumulative distribution function of n successes in N trials given chance of success theta

Available since 2.12

real binomial_lccdf(ints n | ints N, reals theta)
The log of the binomial complementary cumulative distribution function of n successes in N trials given chance of success theta

Available since 2.12

R binomial_rng(ints N, reals theta)
Generate a binomial variate with N trials and chance of success theta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.

Available since 2.18

Binomial distribution, logit parameterization

Stan also provides a version of the binomial probability mass function distribution with the chance of success parameterized on the unconstrained logistic scale.

Probability mass function

Suppose $N\in \mathbb{N}$, $\alpha \in \mathbb{R}$, and $n\in \{0,\dots ,N\}$. Then $$\begin{array}{rcl}\text{BinomialLogit}(n|N,\alpha )& =& \text{Binomial}(n|N,{\text{logit}}^{-1}(\alpha ))\\ & =& (\genfrac{}{}{0ex}{}{N}{n}){({\text{logit}}^{-1}(\alpha ))}^n{(1-{\text{logit}}^{-1}(\alpha ))}^{N-n}.\end{array}$$

Log probability mass function

$$\begin{array}{rcl}\log \text{BinomialLogit}(n|N,\alpha )& =& \log \mathrm{\Gamma }(N+1)-\log \mathrm{\Gamma }(n+1)-\log \mathrm{\Gamma }(N-n+1)\\ & & +n\log {\text{logit}}^{-1}(\alpha )+(N-n)\log (1-{\text{logit}}^{-1}(\alpha )),\end{array}$$

Gradient of log probability mass function

$$\frac{\partial }{\partial \alpha }\log \text{BinomialLogit}(n|N,\alpha )=\frac{n}{{\text{logit}}^{-1}(-\alpha )}-\frac{N-n}{{\text{logit}}^{-1}(\alpha )}$$

Distribution statement

n ~ binomial_logit(N, alpha)

Increment target log probability density with binomial_logit_lupmf(n | N, alpha).

Available since 2.0

Stan functions

real binomial_logit_lpmf(ints n | ints N, reals alpha)
The log binomial probability mass of n successes in N trials given logit-scaled chance of success alpha

Available since 2.12

real binomial_logit_lupmf(ints n | ints N, reals alpha)
The log binomial probability mass of n successes in N trials given logit-scaled chance of success alpha dropping constant additive terms

Available since 2.25

Binomial-logit generalized linear model (Logistic Regression)

Stan also supplies a single function for a generalized linear model with binomial distribution and logit link function, i.e., a function for logistic regression with aggregated outcomes. This provides a more efficient implementation of logistic regression than a manually written regression in terms of a binomial distribution and matrix multiplication.

Probability mass function

Suppose $N\in \mathbb{N}$, $x\in {\mathbb{R}}^{n\cdot m},\alpha \in {\mathbb{R}}^n,\beta \in {\mathbb{R}}^m$, and $n\in \{0,\dots ,N\}$. Then $$\begin{array}{rl}& \text{BinomialLogitGLM}(n|N,x,\alpha ,\beta )=\text{Binomial}(n|N,{\text{logit}}^{-1}({\alpha }_i+x_i\cdot \beta ))\\ & =(\genfrac{}{}{0ex}{}{N}{n}){({\text{logit}}^{-1}({\alpha }_i+\sum _{1\le j\le m}{x}_{ij}\cdot {\beta }_j))}^n{(1-{\text{logit}}^{-1}({\alpha }_i+\sum _{1\le j\le m}{x}_{ij}\cdot {\beta }_j))}^{N-n}.\end{array}$$

Distribution statement

n ~ binomial_logit_glm(N, x, alpha, beta)

Increment target log probability density with binomial_logit_glm_lupmf(n | N, x, alpha, beta).

Available since 2.34

Stan Functions

real binomial_logit_glm_lpmf(int n | int N, matrix x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(int n | int N, matrix x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

real binomial_logit_glm_lpmf(int n | int N, matrix x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(int n | int N, matrix x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

real binomial_logit_glm_lpmf(array[] int n | array[] int N, row_vector x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(array[] int n | array[] int N, row_vector x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

real binomial_logit_glm_lpmf(array[] int n | array[] int N, row_vector x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(array[] int n | array[] int N, row_vector x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

real binomial_logit_glm_lpmf(array[] int n | array[] int N, matrix x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(array[] int n | array[] int N, matrix x, real alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

real binomial_logit_glm_lpmf(array[] int n | array[] int N, matrix x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta).

Available since 2.34

real binomial_logit_glm_lupmf(array[] int n | array[] int N, matrix x, vector alpha, vector beta)
The log binomial probability mass of n given N trials and chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.34

Beta-binomial distribution

Probability mass function

If $N\in \mathbb{N}$, $\alpha \in {\mathbb{R}}^{+}$, and $\beta \in {\mathbb{R}}^{+}$, then for $n\in 0,\dots ,N$, $$\text{BetaBinomial}(n|N,\alpha ,\beta )=(\genfrac{}{}{0ex}{}{N}{n})\frac{\mathrm{B}(n+\alpha ,N-n+\beta )}{\mathrm{B}(\alpha ,\beta )},$$ where the beta function $\mathrm{B}(u,v)$ is defined for $u\in {\mathbb{R}}^{+}$ and $v\in {\mathbb{R}}^{+}$ by $$\mathrm{B}(u,v)=\frac{\mathrm{\Gamma }(u)\mathrm{\Gamma }(v)}{\mathrm{\Gamma }(u+v)}.$$

Distribution statement

n ~ beta_binomial(N, alpha, beta)

Increment target log probability density with beta_binomial_lupmf(n | N, alpha, beta).

Available since 2.0

Stan functions

real beta_binomial_lpmf(ints n | ints N, reals alpha, reals beta)
The log beta-binomial probability mass of n successes in N trials given prior success count (plus one) of alpha and prior failure count (plus one) of beta

Available since 2.12

real beta_binomial_lupmf(ints n | ints N, reals alpha, reals beta)
The log beta-binomial probability mass of n successes in N trials given prior success count (plus one) of alpha and prior failure count (plus one) of beta dropping constant additive terms

Available since 2.25

real beta_binomial_cdf(ints n | ints N, reals alpha, reals beta)
The beta-binomial cumulative distribution function of n successes in N trials given prior success count (plus one) of alpha and prior failure count (plus one) of beta

Available since 2.0

real beta_binomial_lcdf(ints n | ints N, reals alpha, reals beta)
The log of the beta-binomial cumulative distribution function of n successes in N trials given prior success count (plus one) of alpha and prior failure count (plus one) of beta

Available since 2.12

real beta_binomial_lccdf(ints n | ints N, reals alpha, reals beta)
The log of the beta-binomial complementary cumulative distribution function of n successes in N trials given prior success count (plus one) of alpha and prior failure count (plus one) of beta

Available since 2.12

R beta_binomial_rng(ints N, reals alpha, reals beta)
Generate a beta-binomial variate with N trials, prior success count (plus one) of alpha, and prior failure count (plus one) of 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.

Available since 2.18

Hypergeometric distribution

Probability mass function

If $a\in \mathbb{N}$, $b\in \mathbb{N}$, and $N\in \{0,\dots ,a+b\}$, then for $n\in \{max(0,N-b),\dots ,min(a,N)\}$, $$\text{Hypergeometric}(n|N,a,b)=\frac{(\genfrac{}{}{0ex}{}{a}{n})(\genfrac{}{}{0ex}{}{b}{N-n})}{(\genfrac{}{}{0ex}{}{a+b}{N})}.$$

Distribution statement

n ~ hypergeometric(N, a, b)

Increment target log probability density with hypergeometric_lupmf(n | N, a, b).

Available since 2.0

Stan functions

real hypergeometric_lpmf(int n | int N, int a, int b)
The log hypergeometric probability mass of n successes in N trials given total success count of a and total failure count of b

Available since 2.12

real hypergeometric_lupmf(int n | int N, int a, int b)
The log hypergeometric probability mass of n successes in N trials given total success count of a and total failure count of b dropping constant additive terms

Available since 2.25

int hypergeometric_rng(int N, int a, int b)
Generate a hypergeometric variate with N trials, total success count of a, and total failure count of b; may only be used in transformed data and generated quantities blocks

Available since 2.18

Categorical distribution

Probability mass functions

If $N\in \mathbb{N}$, $N>0$, and if $\theta \in {\mathbb{R}}^N$ forms an $N$-simplex (i.e., has nonnegative entries summing to one), then for $y\in \{1,\dots ,N\}$, $$\text{Categorical}(y|\theta )={\theta }_y.$$ In addition, Stan provides a log-odds scaled categorical distribution, $$\text{CategoricalLogit}(y|\beta )=\text{Categorical}(y|\text{softmax}(\beta )).$$ See the definition of softmax for the definition of the softmax function.

Distribution statement

y ~ categorical(theta)

Increment target log probability density with categorical_lupmf(y | theta) dropping constant additive terms.

Available since 2.0

Distribution statement

y ~ categorical_logit(beta)

Increment target log probability density with categorical_logit_lupmf(y | beta).

Available since 2.4

Stan functions

All of the categorical distributions are vectorized so that the outcome y can be a single integer (type int) or an array of integers (type array[] int).

real categorical_lpmf(ints y | vector theta)
The log categorical probability mass function with outcome(s) y in $1:N$ given $N$-vector of outcome probabilities theta. The parameter theta must have non-negative entries that sum to one, but it need not be a variable declared as a simplex.

Available since 2.12

real categorical_lupmf(ints y | vector theta)
The log categorical probability mass function with outcome(s) y in $1:N$ given $N$-vector of outcome probabilities theta dropping constant additive terms. The parameter theta must have non-negative entries that sum to one, but it need not be a variable declared as a simplex.

Available since 2.25

real categorical_logit_lpmf(ints y | vector beta)
The log categorical probability mass function with outcome(s) y in $1:N$ given log-odds of outcomes beta.

Available since 2.12

real categorical_logit_lupmf(ints y | vector beta)
The log categorical probability mass function with outcome(s) y in $1:N$ given log-odds of outcomes beta dropping constant additive terms.

Available since 2.25

int categorical_rng(vector theta)
Generate a categorical variate with $N$-simplex distribution parameter theta; may only be used in transformed data and generated quantities blocks

Available since 2.0

int categorical_logit_rng(vector beta)
Generate a categorical variate with outcome in range $1:N$ from log-odds vector beta; may only be used in transformed data and generated quantities blocks

Available since 2.16

Categorical logit generalized linear model (softmax regression)

Stan also supplies a single function for a generalized linear model with categorical distribution and logit link function, i.e. a function for a softmax regression. This provides a more efficient implementation of softmax regression than a manually written regression in terms of a categorical distribution and matrix multiplication.

Note that the implementation does not put any restrictions on the coefficient matrix $\beta$. It is up to the user to use a reference category, a suitable prior or some other means of identifiability. See Multi-logit in the Stan User’s Guide.

Probability mass functions

If $N,M,K\in \mathbb{N}$, $N,M,K>0$, and if $x\in {\mathbb{R}}^{M\times K},\alpha \in {\mathbb{R}}^N,\beta \in {\mathbb{R}}^{K\cdot N}$, then for $y\in \{1,\dots ,N{\}}^M$, $$\begin{array}{rl}\text{CategoricalLogitGLM}(y|x,\alpha ,\beta )& =\prod _{1\le i\le M}\text{CategoricalLogit}(y_i|\alpha +x_i\cdot \beta )\\ & =\prod _{1\le i\le M}\text{Categorical}(y_i|softmax(\alpha +x_i\cdot \beta )).\end{array}$$ See the definition of softmax for the definition of the softmax function.

Distribution statement

y ~ categorical_logit_glm(x, alpha, beta)

Increment target log probability density with categorical_logit_glm_lupmf(y | x, alpha, beta).

Available since 2.23

Stan functions

real categorical_logit_glm_lpmf(int y | row_vector x, vector alpha, matrix beta)
The log categorical probability mass function with outcome y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta.

Available since 2.23

real categorical_logit_glm_lupmf(int y | row_vector x, vector alpha, matrix beta)
The log categorical probability mass function with outcome y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta dropping constant additive terms.

Available since 2.25

real categorical_logit_glm_lpmf(int y | matrix x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta.

Available since 2.23

real categorical_logit_glm_lupmf(int y | matrix x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta dropping constant additive terms.

Available since 2.25

real categorical_logit_glm_lpmf(array[] int y | row_vector x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta.

Available since 2.23

real categorical_logit_glm_lupmf(array[] int y | row_vector x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta dropping constant additive terms.

Available since 2.25

real categorical_logit_glm_lpmf(array[] int y | matrix x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta.

Available since 2.23

real categorical_logit_glm_lupmf(array[] int y | matrix x, vector alpha, matrix beta)
The log categorical probability mass function with outcomes y in $1:N$ given $N$-vector of log-odds of outcomes alpha + x * beta dropping constant additive terms.

Available since 2.25

Discrete range distribution

Probability mass functions

If $l,u\in \mathbb{Z}$ are lower and upper bounds ($l\le u$), then for any integer $y\in \{l,\dots ,u\}$, $$\text{DiscreteRange}(y|l,u)=\frac{1}{u-l+1}.$$

Distribution statement

y ~ discrete_range(l, u)

Increment the target log probability density with discrete_range_lupmf(y | l, u) dropping constant additive terms.

Available since 2.26

Stan functions

All of the discrete range distributions are vectorized so that the outcome y and the bounds l, u can be a single integer (type int) or an array of integers (type array[] int).

real discrete_range_lpmf(ints y | ints l, ints u)
The log probability mass function with outcome(s) y in $l:u$.

Available since 2.26

real discrete_range_lupmf(ints y | ints l, ints u)
The log probability mass function with outcome(s) y in $l:u$ dropping constant additive terms.

Available since 2.26

real discrete_range_cdf(ints y | ints l, ints u)
The discrete range cumulative distribution function for the given y, lower and upper bounds.

Available since 2.26

real discrete_range_lcdf(ints y | ints l, ints u)
The log of the discrete range cumulative distribution function for the given y, lower and upper bounds.

Available since 2.26

real discrete_range_lccdf(ints y | ints l, ints u)
The log of the discrete range complementary cumulative distribution function for the given y, lower and upper bounds.

Available since 2.26

ints discrete_range_rng(ints l, ints u)
Generate a discrete variate between the given lower and upper bounds; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.

Available since 2.26

Ordered logistic distribution

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 $k\in \{1,\dots ,K-2\}$, and $\eta \in \mathbb{R}$, then for $k\in \{1,\dots ,K\}$, $$\text{OrderedLogistic}(k|\eta ,c)=\{\begin{array}{ll}1-{\text{logit}}^{-1}(\eta -c_1)& \text{if }k=1,\\ {\text{logit}}^{-1}(\eta -c_{k-1})-{\text{logit}}^{-1}(\eta -c_k)& \text{if }1<k<K,\text{and}\\ {\text{logit}}^{-1}(\eta -c_{K-1})-0& \text{if }k=K.\end{array}$$ 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 $c_0=-\mathrm{\infty }$ and $c_K=+\mathrm{\infty }$ with ${\text{logit}}^{-1}(-\mathrm{\infty })=0$ and ${\text{logit}}^{-1}(\mathrm{\infty })=1$.

Distribution statement

k ~ ordered_logistic(eta, c)

Increment target log probability density with ordered_logistic_lupmf(k | eta, c).

Available since 2.0

Stan functions

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

Available since 2.18

real ordered_logistic_lupmf(ints k | vector eta, vectors c)
The log ordered logistic probability mass of k given linear predictors eta, and cutpoints c dropping constant additive terms.

Available since 2.25

int ordered_logistic_rng(real eta, vector c)
Generate an ordered logistic variate with linear predictor eta and cutpoints c; may only be used in transformed data and generated quantities blocks

Available since 2.0

Ordered logistic generalized linear model (ordinal regression)

Probability mass function

If $N,M,K\in \mathbb{N}$ with $N,M>0$, $K>2$, $c\in {\mathbb{R}}^{K-1}$ such that $c_k<c_{k+1}$ for $k\in \{1,\dots ,K-2\}$, and $x\in {\mathbb{R}}^{N\times M},\beta \in {\mathbb{R}}^M$, then for $y\in \{1,\dots ,K{\}}^N$, $$\begin{array}{r}\\ & \text{OrderedLogisticGLM}(y|x,\beta ,c)\\ & =\prod _{1\le i\le N}\text{OrderedLogistic}(y_i|x_i\cdot \beta ,c)\\ & =\prod _{1\le i\le N}\{\begin{array}{ll}1-{\text{logit}}^{-1}(x_i\cdot \beta -c_1)& \text{if }y=1,\\ {\text{logit}}^{-1}(x_i\cdot \beta -c_{y-1})-{\text{logit}}^{-1}(x_i\cdot \beta -c_y)& \text{if }1<y<K,\text{and}\\ {\text{logit}}^{-1}(x_i\cdot \beta -c_{K-1})-0& \text{if }y=K.\end{array}\end{array}$$ The $k=K$ case is written with the redundant subtraction of zero to illustrate the parallelism of the cases; the $y=1$ and $y=K$ edge cases can be subsumed into the general definition by setting $c_0=-\mathrm{\infty }$ and $c_K=+\mathrm{\infty }$ with ${\text{logit}}^{-1}(-\mathrm{\infty })=0$ and ${\text{logit}}^{-1}(\mathrm{\infty })=1$.

Distribution statement

y ~ ordered_logistic_glm(x, beta, c)

Increment target log probability density with ordered_logistic_lupmf(y | x, beta, c).

Available since 2.23

Stan functions

real ordered_logistic_glm_lpmf(int y | row_vector x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c. The cutpoints c must be ordered.

Available since 2.23

real ordered_logistic_glm_lupmf(int y | row_vector x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c dropping constant additive terms. The cutpoints c must be ordered.

Available since 2.25

real ordered_logistic_glm_lpmf(int y | matrix x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c. The cutpoints c must be ordered.

Available since 2.23

real ordered_logistic_glm_lupmf(int y | matrix x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c dropping constant additive terms. The cutpoints c must be ordered.

Available since 2.25

real ordered_logistic_glm_lpmf(array[] int y | row_vector x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c. The cutpoints c must be ordered.

Available since 2.23

real ordered_logistic_glm_lupmf(array[] int y | row_vector x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c dropping constant additive terms. The cutpoints c must be ordered.

Available since 2.25

real ordered_logistic_glm_lpmf(array[] int y | matrix x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c. The cutpoints c must be ordered.

Available since 2.23

real ordered_logistic_glm_lupmf(array[] int y | matrix x, vector beta, vector c)
The log ordered logistic probability mass of y, given linear predictors x * beta, and cutpoints c dropping constant additive terms. The cutpoints c must be ordered.

Available since 2.25

Ordered probit distribution

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 $k\in \{1,\dots ,K-2\}$, and $\eta \in \mathbb{R}$, then for $k\in \{1,\dots ,K\}$, $$\text{OrderedProbit}(k|\eta ,c)=\{\begin{array}{ll}1-\mathrm{\Phi }(\eta -c_1)& \text{if }k=1,\\ \mathrm{\Phi }(\eta -c_{k-1})-\mathrm{\Phi }(\eta -c_k)& \text{if }1<k<K,\text{and}\\ \mathrm{\Phi }(\eta -c_{K-1})-0& \text{if }k=K.\end{array}$$ 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 $c_0=-\mathrm{\infty }$ and $c_K=+\mathrm{\infty }$ with $\mathrm{\Phi }(-\mathrm{\infty })=0$ and $\mathrm{\Phi }(\mathrm{\infty })=1$.

Distribution statement

k ~ ordered_probit(eta, c)

Increment target log probability density with ordered_probit_lupmf(k | eta, c).

Available since 2.19

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.

Available since 2.18

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

Available since 2.25

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

Available since 2.19

real ordered_probit_lupmf(ints k | real eta, vectors c)
The log ordered probit probability mass of k given linear predictor eta, and cutpoints c dropping constant additive terms.

Available since 2.19

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

Available since 2.18