Stan Functions Reference
Overview
Built-In Functions
1
Void Functions
1.1
Print Statement
1.2
Reject Statement
2
Integer-Valued Basic Functions
2.1
Integer-Valued Arithmetic Operators
2.1.1
Binary Infix Operators
2.1.2
Unary Prefix Operators
2.2
Absolute Functions
2.3
Bound Functions
3
Real-Valued Basic Functions
3.1
Vectorization of Real-Valued Functions
3.1.1
Unary Function Vectorization
3.2
Mathematical Constants
3.3
Special Values
3.4
Log Probability Function
3.5
Logical Functions
3.5.1
Comparison Operators
3.5.2
Boolean Operators
3.5.3
Logical Functions
3.6
Real-Valued Arithmetic Operators
3.6.1
Binary Infix Operators
3.6.2
Unary Prefix Operators
3.7
Step-like Functions
3.7.1
Absolute Value Functions
3.7.2
Bounds Functions
3.7.3
Arithmetic Functions
3.7.4
Rounding Functions
3.8
Power and Logarithm Functions
3.9
Trigonometric Functions
3.10
Hyperbolic Trigonometric Functions
3.11
Link Functions
3.12
Probability-Related Functions
3.12.1
Normal Cumulative Distribution Functions
3.12.2
Other Probability-Related Functions
3.13
Combinatorial Functions
3.14
Composed Functions
4
Array Operations
4.1
Reductions
4.1.1
Minimum and Maximum
4.1.2
Sum, Product, and Log Sum of Exp
4.1.3
Sample Mean, Variance, and Standard Deviation
4.1.4
Euclidean Distance and Squared Distance
4.2
Array size and dimension function
4.3
Array broadcasting
4.4
Array concatenation
4.5
Sorting functions
4.6
Reversing functions
5
Matrix Operations
5.1
Integer-Valued Matrix Size Functions
5.2
Matrix Arithmetic Operators
5.2.1
Negation Prefix Operators
5.2.2
Infix Matrix Operators
5.2.3
Broadcast Infix Operators
5.2.4
Elementwise Arithmetic Operations
5.3
Transposition Operator
5.4
Elementwise Functions
5.5
Dot Products and Specialized Products
5.5.1
Specialized Products
5.6
Reductions
5.6.1
Log Sum of Exponents
5.6.2
Minimum and Maximum
5.6.3
Sums and Products
5.6.4
Sample Moments
5.7
Broadcast Functions
5.8
Diagonal Matrix Functions
5.9
Slicing and Blocking Functions
5.9.1
Columns and Rows
5.9.2
Block Operations
5.10
Matrix Concatenation
5.11
Special Matrix Functions
5.11.1
Softmax
5.11.2
Cumulative Sums
5.12
Covariance Functions
5.12.1
Exponentiated quadratic covariance function
5.13
Linear Algebra Functions and Solvers
5.13.1
Matrix Division Operators and Functions
5.13.2
Symmetric positive-definite matrix division functions
5.13.3
Matrix Exponential
5.13.4
Linear Algebra Functions
5.14
Sort Functions
5.15
Reverse Functions
6
Sparse Matrix Operations
6.1
Compressed Row Storage
6.2
Conversion Functions
6.2.1
Dense to Sparse Conversion
6.2.2
Sparse to Dense Conversion
6.3
Sparse Matrix Arithmetic
6.3.1
Sparse Matrix Multiplication
7
Mixed Operations
8
Compound Arithmetic and Assignment
8.1
Compound Addition and Assignment
8.2
Compound Subtraction and Assignment
8.3
Compound Multiplication and Assignment
8.4
Compound Division and Assignment
8.5
Compound Elementwise Multiplication and Assignment
8.6
Compound Elementwise Division and Assignment
9
Higher-Order Functions
9.1
Algebraic Equation Solver
9.1.1
Specifying an Algebraic Equation as a Function
9.1.2
Call to the Algebraic Solver
9.2
Ordinary Differential Equation (ODE) Solvers
9.2.1
Specifying an Ordinary Differential Equation as a Function
9.2.2
Non-Stiff Solver
9.2.3
Stiff Solver
9.2.4
Arguments to the ODE solvers
9.3
1D Integrator
9.3.1
Specifying an Integrand as a Function
9.3.2
Call to the 1D Integrator
9.4
Reduce-Sum Function
9.4.1
Specifying the Reduce-sum Function
9.4.2
The Partial sum Function
9.5
Map-Rect Function
9.5.1
Specifying the Mapped Function
9.5.2
Rectangular Map
Discrete Distributions
10
Conventions for Probability Functions
10.1
Suffix Marks Type of Function
10.2
Argument Order and the Vertical Bar
10.3
Sampling Notation
10.4
Finite Inputs
10.5
Boundary Conditions
10.6
Pseudorandom Number Generators
10.6.1
Restricted to Transformed Data and Generated Quantities
10.6.2
Limited Vectorization
10.7
Cumulative Distribution Functions
10.8
Vectorization
10.8.1
Vectorized Function Signatures
10.8.2
Evaluating Vectorized Log Probability Functions
10.8.3
Evaluating Vectorized PRNG Functions
11
Binary Distributions
11.1
Bernoulli Distribution
11.1.1
Probability Mass Function
11.1.2
Sampling Statement
11.1.3
Stan Functions
11.2
Bernoulli Distribution, Logit Parameterization
11.2.1
Probability Mass Function
11.2.2
Sampling Statement
11.2.3
Stan Functions
11.3
Bernoulli-Logit Generalized Linear Model (Logistic Regression)
11.3.1
Probability Mass Function
11.3.2
Sampling Statement
11.3.3
Stan Functions
12
Bounded Discrete Distributions
12.1
Binomial Distribution
12.1.1
Probability Mass Function
12.1.2
Log Probability Mass Function
12.1.3
Gradient of Log Probability Mass Function
12.1.4
Sampling Statement
12.1.5
Stan Functions
12.2
Binomial Distribution, Logit Parameterization
12.2.1
Probability Mass Function
12.2.2
Log Probability Mass Function
12.2.3
Gradient of Log Probability Mass Function
12.2.4
Sampling Statement
12.2.5
Stan Functions
12.3
Beta-Binomial Distribution
12.3.1
Probability Mass Function
12.3.2
Sampling Statement
12.3.3
Stan Functions
12.4
Hypergeometric Distribution
12.4.1
Probability Mass Function
12.4.2
Sampling Statement
12.4.3
Stan Functions
12.5
Categorical Distribution
12.5.1
Probability Mass Functions
12.5.2
Sampling Statement
12.5.3
Sampling Statement
12.5.4
Stan Functions
12.6
Categorical Logit Generalized Linear Model (Softmax Regression)
12.6.1
Probability Mass Functions
12.6.2
Sampling Statement
12.6.3
Stan Functions
12.7
Ordered Logistic Distribution
12.7.1
Probability Mass Function
12.7.2
Sampling Statement
12.7.3
Stan Functions
12.8
Ordered Logistic Generalized Linear Model (Ordinal Regression)
12.8.1
Probability Mass Function
12.8.2
Sampling Statement
12.8.3
Stan Functions
12.9
Ordered Probit Distribution
12.9.1
Probability Mass Function
12.9.2
Sampling Statement
12.9.3
Stan Functions
13
Unbounded Discrete Distributions
13.1
Negative Binomial Distribution
13.1.1
Probability Mass Function
13.1.2
Sampling Statement
13.1.3
Stan Functions
13.2
Negative Binomial Distribution (alternative parameterization)
13.2.1
Probability Mass Function
13.2.2
Sampling Statement
13.2.3
Stan Functions
13.3
Negative Binomial Distribution (log alternative parameterization)
13.3.1
Sampling Statement
13.3.2
Stan Functions
13.4
Negative-Binomial-2-Log Generalized Linear Model (Negative Binomial Regression)
13.4.1
Probability Mass Function
13.4.2
Sampling Statement
13.4.3
Stan Functions
13.5
Poisson Distribution
13.5.1
Probability Mass Function
13.5.2
Sampling Statement
13.5.3
Stan Functions
13.6
Poisson Distribution, Log Parameterization
13.6.1
Probability Mass Function
13.6.2
Sampling Statement
13.6.3
Stan Functions
13.7
Poisson-Log Generalized Linear Model (Poisson Regression)
13.7.1
Probability Mass Function
13.7.2
Sampling Statement
13.7.3
Stan Functions
14
Multivariate Discrete Distributions
14.1
Multinomial Distribution
14.1.1
Probability Mass Function
14.1.2
Sampling Statement
14.1.3
Stan Functions
Continuous Distributions
15
Unbounded Continuous Distributions
15.1
Normal Distribution
15.1.1
Probability Density Function
15.1.2
Sampling Statement
15.1.3
Stan Functions
15.1.4
Standard Normal Distribution
15.1.5
Sampling Statement
15.1.6
Stan Functions
15.2
Normal-Id Generalized Linear Model (Linear Regression)
15.2.1
Probability Distribution Function
15.2.2
Sampling Statement
15.2.3
Stan Functions
15.3
Exponentially Modified Normal Distribution
15.3.1
Probability Density Function
15.3.2
Sampling Statement
15.3.3
Stan Functions
15.4
Skew Normal Distribution
15.4.1
Probability Density Function
15.4.2
Sampling Statement
15.4.3
Stan Functions
15.5
Student-T Distribution
15.5.1
Probability Density Function
15.5.2
Sampling Statement
15.5.3
Stan Functions
15.6
Cauchy Distribution
15.6.1
Probability Density Function
15.6.2
Sampling Statement
15.6.3
Stan Functions
15.7
Double Exponential (Laplace) Distribution
15.7.1
Probability Density Function
15.7.2
Sampling Statement
15.7.3
Stan Functions
15.8
Logistic Distribution
15.8.1
Probability Density Function
15.8.2
Sampling Statement
15.8.3
Stan Functions
15.9
Gumbel Distribution
15.9.1
Probability Density Function
15.9.2
Sampling Statement
15.9.3
Stan Functions
16
Positive Continuous Distributions
16.1
Lognormal Distribution
16.1.1
Probability Density Function
16.1.2
Sampling Statement
16.1.3
Stan Functions
16.2
Chi-Square Distribution
16.2.1
Probability Density Function
16.2.2
Sampling Statement
16.2.3
Stan Functions
16.3
Inverse Chi-Square Distribution
16.3.1
Probability Density Function
16.3.2
Sampling Statement
16.3.3
Stan Functions
16.4
Scaled Inverse Chi-Square Distribution
16.4.1
Probability Density Function
16.4.2
Sampling Statement
16.4.3
Stan Functions
16.5
Exponential Distribution
16.5.1
Probability Density Function
16.5.2
Sampling Statement
16.5.3
Stan Functions
16.6
Gamma Distribution
16.6.1
Probability Density Function
16.6.2
Sampling Statement
16.6.3
Stan Functions
16.7
Inverse Gamma Distribution
16.7.1
Probability Density Function
16.7.2
Sampling Statement
16.7.3
Stan Functions
16.8
Weibull Distribution
16.8.1
Probability Density Function
16.8.2
Sampling Statement
16.8.3
Stan Functions
16.9
Frechet Distribution
16.9.1
Probability Density Function
16.9.2
Sampling Statement
16.9.3
Stan Functions
16.10
Rayleigh Distribution
16.10.1
Probability Density Function
16.10.2
Sampling Statement
16.10.3
Stan Functions
17
Positive Lower-Bounded Distributions
17.1
Pareto Distribution
17.1.1
Probability Density Function
17.1.2
Sampling Statement
17.1.3
Stan Functions
17.2
Pareto Type 2 Distribution
17.2.1
Probability Density Function
17.2.2
Sampling Statement
17.2.3
Stan Functions
17.3
Wiener First Passage Time Distribution
17.3.1
Probability Density Function
17.3.2
Sampling Statement
17.3.3
Stan Functions
17.3.4
Boundaries
18
Continuous Distributions on [0, 1]
18.1
Beta Distribution
18.1.1
Probability Density Function
18.1.2
Sampling Statement
18.1.3
Stan Functions
18.2
Beta Proportion Distribution
18.2.1
Probability Density Function
18.2.2
Sampling Statement
18.2.3
Stan Functions
19
Circular Distributions
19.1
Von Mises Distribution
19.1.1
Probability Density Function
19.1.2
Sampling Statement
19.1.3
Stan Functions
19.1.4
Numerical Stability
20
Bounded Continuous Distributions
20.1
Uniform Distribution
20.1.1
Probability Density Function
20.1.2
Sampling Statement
20.1.3
Stan Functions
21
Distributions over Unbounded Vectors
21.1
Multivariate Normal Distribution
21.1.1
Probability Density Function
21.1.2
Sampling Statement
21.1.3
Stan Functions
21.2
Multivariate Normal Distribution, Precision Parameterization
21.2.1
Probability Density Function
21.2.2
Sampling Statement
21.2.3
Stan Functions
21.3
Multivariate Normal Distribution, Cholesky Parameterization
21.3.1
Probability Density Function
21.3.2
Sampling Statement
21.3.3
Stan Functions
21.4
Multivariate Gaussian Process Distribution
21.4.1
Probability Density Function
21.4.2
Sampling Statement
21.4.3
Stan Functions
21.5
Multivariate Gaussian Process Distribution, Cholesky parameterization
21.5.1
Probability Density Function
21.5.2
Sampling Statement
21.5.3
Stan Functions
21.6
Multivariate Student-T Distribution
21.6.1
Probability Density Function
21.6.2
Sampling Statement
21.6.3
Stan Functions
21.7
Gaussian Dynamic Linear Models
21.7.1
Sampling Statement
21.7.2
Stan Functions
22
Simplex Distributions
22.1
Dirichlet Distribution
22.1.1
Probability Density Function
22.1.2
Meaning of Dirichlet Parameters
22.1.3
Sampling Statement
22.1.4
Stan Functions
23
Correlation Matrix Distributions
23.1
LKJ Correlation Distribution
23.1.1
Probability Density Function
23.1.2
Sampling Statement
23.1.3
Stan Functions
23.2
Cholesky LKJ Correlation Distribution
23.2.1
Probability Density Function
23.2.2
Sampling Statement
23.2.3
Stan Functions
24
Covariance Matrix Distributions
24.1
Wishart Distribution
24.1.1
Probability Density Function
24.1.2
Sampling Statement
24.1.3
Stan Functions
24.2
Inverse Wishart Distribution
24.2.1
Probability Density Function
24.2.2
Sampling Statement
24.2.3
Stan Functions
Appendix
25
Mathematical Functions
25.1
Beta
25.2
Incomplete Beta
25.3
Gamma
25.4
Digamma
References
Stan Functions Reference
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Built-In Functions