• 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
      • 3.1.2 Binary 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
    • 3.8 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 Matrix Power
      • 5.13.5 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 Non-Stiff Solver
      • 9.2.2 Stiff Solver
      • 9.2.3 ODE System Function
      • 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
  • 10 Deprecated Functions
    • 10.1 integrate_ode_rk45, integrate_ode_adams, integrate_ode_bdf ODE Integrators
      • 10.1.1 Specifying an Ordinary Differential Equation as a Function
      • 10.1.2 Non-Stiff Solver
      • 10.1.3 Stiff Solver
      • 10.1.4 Arguments to the ODE solvers
  • 11 Conventions for Probability Functions
    • 11.1 Suffix Marks Type of Function
    • 11.2 Argument Order and the Vertical Bar
    • 11.3 Sampling Notation
    • 11.4 Finite Inputs
    • 11.5 Boundary Conditions
    • 11.6 Pseudorandom Number Generators
      • 11.6.1 Restricted to Transformed Data and Generated Quantities
      • 11.6.2 Limited Vectorization
    • 11.7 Cumulative Distribution Functions
    • 11.8 Vectorization
      • 11.8.1 Vectorized Function Signatures
      • 11.8.2 Evaluating Vectorized Log Probability Functions
      • 11.8.3 Evaluating Vectorized PRNG Functions
  • Discrete Distributions
  • 12 Binary Distributions
    • 12.1 Bernoulli Distribution
      • 12.1.1 Probability Mass Function
      • 12.1.2 Sampling Statement
      • 12.1.3 Stan Functions
    • 12.2 Bernoulli Distribution, Logit Parameterization
      • 12.2.1 Probability Mass Function
      • 12.2.2 Sampling Statement
      • 12.2.3 Stan Functions
    • 12.3 Bernoulli-Logit Generalized Linear Model (Logistic Regression)
      • 12.3.1 Probability Mass Function
      • 12.3.2 Sampling Statement
      • 12.3.3 Stan Functions
  • 13 Bounded Discrete Distributions
    • 13.1 Binomial Distribution
      • 13.1.1 Probability Mass Function
      • 13.1.2 Log Probability Mass Function
      • 13.1.3 Gradient of Log Probability Mass Function
      • 13.1.4 Sampling Statement
      • 13.1.5 Stan Functions
    • 13.2 Binomial Distribution, Logit Parameterization
      • 13.2.1 Probability Mass Function
      • 13.2.2 Log Probability Mass Function
      • 13.2.3 Gradient of Log Probability Mass Function
      • 13.2.4 Sampling Statement
      • 13.2.5 Stan Functions
    • 13.3 Beta-Binomial Distribution
      • 13.3.1 Probability Mass Function
      • 13.3.2 Sampling Statement
      • 13.3.3 Stan Functions
    • 13.4 Hypergeometric Distribution
      • 13.4.1 Probability Mass Function
      • 13.4.2 Sampling Statement
      • 13.4.3 Stan Functions
    • 13.5 Categorical Distribution
      • 13.5.1 Probability Mass Functions
      • 13.5.2 Sampling Statement
      • 13.5.3 Sampling Statement
      • 13.5.4 Stan Functions
    • 13.6 Categorical Logit Generalized Linear Model (Softmax Regression)
      • 13.6.1 Probability Mass Functions
      • 13.6.2 Sampling Statement
      • 13.6.3 Stan Functions
    • 13.7 Ordered Logistic Distribution
      • 13.7.1 Probability Mass Function
      • 13.7.2 Sampling Statement
      • 13.7.3 Stan Functions
    • 13.8 Ordered Logistic Generalized Linear Model (Ordinal Regression)
      • 13.8.1 Probability Mass Function
      • 13.8.2 Sampling Statement
      • 13.8.3 Stan Functions
    • 13.9 Ordered Probit Distribution
      • 13.9.1 Probability Mass Function
      • 13.9.2 Sampling Statement
      • 13.9.3 Stan Functions
  • 14 Unbounded Discrete Distributions
    • 14.1 Negative Binomial Distribution
      • 14.1.1 Probability Mass Function
      • 14.1.2 Sampling Statement
      • 14.1.3 Stan Functions
    • 14.2 Negative Binomial Distribution (alternative parameterization)
      • 14.2.1 Probability Mass Function
      • 14.2.2 Sampling Statement
      • 14.2.3 Stan Functions
    • 14.3 Negative Binomial Distribution (log alternative parameterization)
      • 14.3.1 Sampling Statement
      • 14.3.2 Stan Functions
    • 14.4 Negative-Binomial-2-Log Generalized Linear Model (Negative Binomial Regression)
      • 14.4.1 Probability Mass Function
      • 14.4.2 Sampling Statement
      • 14.4.3 Stan Functions
    • 14.5 Poisson Distribution
      • 14.5.1 Probability Mass Function
      • 14.5.2 Sampling Statement
      • 14.5.3 Stan Functions
    • 14.6 Poisson Distribution, Log Parameterization
      • 14.6.1 Probability Mass Function
      • 14.6.2 Sampling Statement
      • 14.6.3 Stan Functions
    • 14.7 Poisson-Log Generalized Linear Model (Poisson Regression)
      • 14.7.1 Probability Mass Function
      • 14.7.2 Sampling Statement
      • 14.7.3 Stan Functions
  • 15 Multivariate Discrete Distributions
    • 15.1 Multinomial Distribution
      • 15.1.1 Probability Mass Function
      • 15.1.2 Sampling Statement
      • 15.1.3 Stan Functions
    • 15.2 Multinomial Distribution, Logit Parameterization
      • 15.2.1 Probability Mass Function
      • 15.2.2 Sampling Statement
      • 15.2.3 Stan Functions
  • Continuous Distributions
  • 16 Unbounded Continuous Distributions
    • 16.1 Normal Distribution
      • 16.1.1 Probability Density Function
      • 16.1.2 Sampling Statement
      • 16.1.3 Stan Functions
      • 16.1.4 Standard Normal Distribution
      • 16.1.5 Sampling Statement
      • 16.1.6 Stan Functions
    • 16.2 Normal-Id Generalized Linear Model (Linear Regression)
      • 16.2.1 Probability Distribution Function
      • 16.2.2 Sampling Statement
      • 16.2.3 Stan Functions
    • 16.3 Exponentially Modified Normal Distribution
      • 16.3.1 Probability Density Function
      • 16.3.2 Sampling Statement
      • 16.3.3 Stan Functions
    • 16.4 Skew Normal Distribution
      • 16.4.1 Probability Density Function
      • 16.4.2 Sampling Statement
      • 16.4.3 Stan Functions
    • 16.5 Student-T Distribution
      • 16.5.1 Probability Density Function
      • 16.5.2 Sampling Statement
      • 16.5.3 Stan Functions
    • 16.6 Cauchy Distribution
      • 16.6.1 Probability Density Function
      • 16.6.2 Sampling Statement
      • 16.6.3 Stan Functions
    • 16.7 Double Exponential (Laplace) Distribution
      • 16.7.1 Probability Density Function
      • 16.7.2 Sampling Statement
      • 16.7.3 Stan Functions
    • 16.8 Logistic Distribution
      • 16.8.1 Probability Density Function
      • 16.8.2 Sampling Statement
      • 16.8.3 Stan Functions
    • 16.9 Gumbel Distribution
      • 16.9.1 Probability Density Function
      • 16.9.2 Sampling Statement
      • 16.9.3 Stan Functions
  • 17 Positive Continuous Distributions
    • 17.1 Lognormal Distribution
      • 17.1.1 Probability Density Function
      • 17.1.2 Sampling Statement
      • 17.1.3 Stan Functions
    • 17.2 Chi-Square Distribution
      • 17.2.1 Probability Density Function
      • 17.2.2 Sampling Statement
      • 17.2.3 Stan Functions
    • 17.3 Inverse Chi-Square Distribution
      • 17.3.1 Probability Density Function
      • 17.3.2 Sampling Statement
      • 17.3.3 Stan Functions
    • 17.4 Scaled Inverse Chi-Square Distribution
      • 17.4.1 Probability Density Function
      • 17.4.2 Sampling Statement
      • 17.4.3 Stan Functions
    • 17.5 Exponential Distribution
      • 17.5.1 Probability Density Function
      • 17.5.2 Sampling Statement
      • 17.5.3 Stan Functions
    • 17.6 Gamma Distribution
      • 17.6.1 Probability Density Function
      • 17.6.2 Sampling Statement
      • 17.6.3 Stan Functions
    • 17.7 Inverse Gamma Distribution
      • 17.7.1 Probability Density Function
      • 17.7.2 Sampling Statement
      • 17.7.3 Stan Functions
    • 17.8 Weibull Distribution
      • 17.8.1 Probability Density Function
      • 17.8.2 Sampling Statement
      • 17.8.3 Stan Functions
    • 17.9 Frechet Distribution
      • 17.9.1 Probability Density Function
      • 17.9.2 Sampling Statement
      • 17.9.3 Stan Functions
    • 17.10 Rayleigh Distribution
      • 17.10.1 Probability Density Function
      • 17.10.2 Sampling Statement
      • 17.10.3 Stan Functions
  • 18 Positive Lower-Bounded Distributions
    • 18.1 Pareto Distribution
      • 18.1.1 Probability Density Function
      • 18.1.2 Sampling Statement
      • 18.1.3 Stan Functions
    • 18.2 Pareto Type 2 Distribution
      • 18.2.1 Probability Density Function
      • 18.2.2 Sampling Statement
      • 18.2.3 Stan Functions
    • 18.3 Wiener First Passage Time Distribution
      • 18.3.1 Probability Density Function
      • 18.3.2 Sampling Statement
      • 18.3.3 Stan Functions
      • 18.3.4 Boundaries
  • 19 Continuous Distributions on [0, 1]
    • 19.1 Beta Distribution
      • 19.1.1 Probability Density Function
      • 19.1.2 Sampling Statement
      • 19.1.3 Stan Functions
    • 19.2 Beta Proportion Distribution
      • 19.2.1 Probability Density Function
      • 19.2.2 Sampling Statement
      • 19.2.3 Stan Functions
  • 20 Circular Distributions
    • 20.1 Von Mises Distribution
      • 20.1.1 Probability Density Function
      • 20.1.2 Sampling Statement
      • 20.1.3 Stan Functions
      • 20.1.4 Numerical Stability
  • 21 Bounded Continuous Distributions
    • 21.1 Uniform Distribution
      • 21.1.1 Probability Density Function
      • 21.1.2 Sampling Statement
      • 21.1.3 Stan Functions
  • 22 Distributions over Unbounded Vectors
    • 22.1 Multivariate Normal Distribution
      • 22.1.1 Probability Density Function
      • 22.1.2 Sampling Statement
      • 22.1.3 Stan Functions
    • 22.2 Multivariate Normal Distribution, Precision Parameterization
      • 22.2.1 Probability Density Function
      • 22.2.2 Sampling Statement
      • 22.2.3 Stan Functions
    • 22.3 Multivariate Normal Distribution, Cholesky Parameterization
      • 22.3.1 Probability Density Function
      • 22.3.2 Sampling Statement
      • 22.3.3 Stan Functions
    • 22.4 Multivariate Gaussian Process Distribution
      • 22.4.1 Probability Density Function
      • 22.4.2 Sampling Statement
      • 22.4.3 Stan Functions
    • 22.5 Multivariate Gaussian Process Distribution, Cholesky parameterization
      • 22.5.1 Probability Density Function
      • 22.5.2 Sampling Statement
      • 22.5.3 Stan Functions
    • 22.6 Multivariate Student-T Distribution
      • 22.6.1 Probability Density Function
      • 22.6.2 Sampling Statement
      • 22.6.3 Stan Functions
    • 22.7 Gaussian Dynamic Linear Models
      • 22.7.1 Sampling Statement
      • 22.7.2 Stan Functions
  • 23 Simplex Distributions
    • 23.1 Dirichlet Distribution
      • 23.1.1 Probability Density Function
      • 23.1.2 Meaning of Dirichlet Parameters
      • 23.1.3 Sampling Statement
      • 23.1.4 Stan Functions
  • 24 Correlation Matrix Distributions
    • 24.1 LKJ Correlation Distribution
      • 24.1.1 Probability Density Function
      • 24.1.2 Sampling Statement
      • 24.1.3 Stan Functions
    • 24.2 Cholesky LKJ Correlation Distribution
      • 24.2.1 Probability Density Function
      • 24.2.2 Sampling Statement
      • 24.2.3 Stan Functions
  • 25 Covariance Matrix Distributions
    • 25.1 Wishart Distribution
      • 25.1.1 Probability Density Function
      • 25.1.2 Sampling Statement
      • 25.1.3 Stan Functions
    • 25.2 Inverse Wishart Distribution
      • 25.2.1 Probability Density Function
      • 25.2.2 Sampling Statement
      • 25.2.3 Stan Functions
  • Additional Distributions
  • 26 Hidden Markov Models
    • 26.1 Stan functions
  • Appendix
  • 27 Mathematical Functions
    • 27.1 Beta
    • 27.2 Incomplete Beta
    • 27.3 Gamma
    • 27.4 Digamma
  • References

Stan Functions Reference

This is an old version, view current version.

5 Matrix Operations

  • Integer-Valued Matrix Size Functions
  • Matrix Arithmetic Operators
  • Transposition Operator
  • Elementwise Functions
  • Dot Products and Specialized Products
  • Reductions
  • Broadcast Functions
  • Diagonal Matrix Functions
  • Slicing and Blocking Functions
  • Matrix Concatenation
  • Special Matrix Functions
  • Covariance Functions
  • Linear Algebra Functions and Solvers
  • Sort Functions
  • Reverse Functions