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  • 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
    • 2.4 Size 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
      • 3.7.2 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
    • 3.15 Special functions
  • 4 Complex-Valued Basic Functions
    • 4.1 Complex assignment and promotion
      • 4.1.1 Complex function arguments
    • 4.2 Complex constructors and accessors
      • 4.2.1 Complex constructors
      • 4.2.2 Complex accessors
    • 4.3 Complex arithmetic operators
      • 4.3.1 Unary operators
      • 4.3.2 Binary operators
    • 4.4 Complex comparison operators
    • 4.5 Complex (compound) assignment operators
    • 4.6 Complex special functions
    • 4.7 Complex exponential and power functions
    • 4.8 Complex trigonometric functions
    • 4.9 Complex hyperbolic trigonometric functions
  • 5 Array Operations
    • 5.1 Reductions
      • 5.1.1 Minimum and maximum
      • 5.1.2 Sum, product, and log sum of exp
      • 5.1.3 Sample mean, variance, and standard deviation
      • 5.1.4 Euclidean distance and squared distance
      • 5.1.5 Quantile
    • 5.2 Array size and dimension function
    • 5.3 Array broadcasting
    • 5.4 Array concatenation
    • 5.5 Sorting functions
    • 5.6 Reversing functions
  • 6 Matrix Operations
    • 6.1 Integer-valued matrix size functions
    • 6.2 Matrix arithmetic operators
      • 6.2.1 Negation prefix operators
      • 6.2.2 Infix matrix operators
      • 6.2.3 Broadcast infix operators
    • 6.3 Transposition operator
    • 6.4 Elementwise functions
    • 6.5 Dot products and specialized products
      • 6.5.1 Specialized products
    • 6.6 Reductions
      • 6.6.1 Log sum of exponents
      • 6.6.2 Minimum and maximum
      • 6.6.3 Sums and products
      • 6.6.4 Sample moments
      • 6.6.5 Quantile
    • 6.7 Broadcast functions
      • 6.7.1 Symmetrization
    • 6.8 Diagonal matrix functions
    • 6.9 Container construction functions
    • 6.10 Slicing and blocking functions
      • 6.10.1 Columns and rows
      • 6.10.2 Block operations
    • 6.11 Matrix concatenation
    • 6.12 Special matrix functions
      • 6.12.1 Softmax
      • 6.12.2 Cumulative sums
    • 6.13 Covariance functions
      • 6.13.1 Exponentiated quadratic covariance function
    • 6.14 Linear algebra functions and solvers
      • 6.14.1 Matrix division operators and functions
      • 6.14.2 Symmetric positive-definite matrix division functions
      • 6.14.3 Matrix exponential
      • 6.14.4 Matrix power
      • 6.14.5 Linear algebra functions
    • 6.15 Sort functions
    • 6.16 Reverse functions
  • 7 Sparse Matrix Operations
    • 7.1 Compressed row storage
    • 7.2 Conversion functions
      • 7.2.1 Dense to sparse conversion
      • 7.2.2 Sparse to dense conversion
    • 7.3 Sparse matrix arithmetic
      • 7.3.1 Sparse matrix multiplication
  • 8 Mixed Operations
  • 9 Compound Arithmetic and Assignment
    • 9.1 Compound addition and assignment
    • 9.2 Compound subtraction and assignment
    • 9.3 Compound multiplication and assignment
    • 9.4 Compound division and assignment
    • 9.5 Compound elementwise multiplication and assignment
    • 9.6 Compound elementwise division and assignment
  • 10 Higher-Order Functions
    • 10.1 Algebraic equation solver
      • 10.1.1 Specifying an algebraic equation as a function
      • 10.1.2 Call to the algebraic solver
    • 10.2 Ordinary differential equation (ODE) solvers
      • 10.2.1 Non-stiff solver
      • 10.2.2 Stiff solver
      • 10.2.3 Adjoint solver
      • 10.2.4 ODE system function
      • 10.2.5 Arguments to the ODE solvers
      • 10.2.6 Arguments to the adjoint ODE solver
    • 10.3 1D integrator
      • 10.3.1 Specifying an integrand as a function
      • 10.3.2 Call to the 1D integrator
    • 10.4 Reduce-sum function
      • 10.4.1 Specifying the reduce-sum function
      • 10.4.2 The partial sum function
    • 10.5 Map-rect function
      • 10.5.1 Specifying the mapped function
      • 10.5.2 Rectangular map
  • 11 Deprecated Functions
    • 11.1 integrate_ode_rk45, integrate_ode_adams, integrate_ode_bdf ODE integrators
      • 11.1.1 Specifying an ordinary differential equation as a function
      • 11.1.2 Non-stiff solver
      • 11.1.3 Stiff solver
      • 11.1.4 Arguments to the ODE solvers
  • 12 Conventions for Probability Functions
    • 12.1 Suffix marks type of function
    • 12.2 Argument order and the vertical bar
    • 12.3 Sampling notation
    • 12.4 Finite inputs
    • 12.5 Boundary conditions
    • 12.6 Pseudorandom number generators
      • 12.6.1 Restricted to transformed data and generated quantities
      • 12.6.2 Limited vectorization
    • 12.7 Cumulative distribution functions
    • 12.8 Vectorization
      • 12.8.1 Vectorized function signatures
      • 12.8.2 Evaluating vectorized log probability functions
      • 12.8.3 Evaluating vectorized PRNG functions
  • Discrete Distributions
  • 13 Binary Distributions
    • 13.1 Bernoulli distribution
      • 13.1.1 Probability mass function
      • 13.1.2 Sampling statement
      • 13.1.3 Stan Functions
    • 13.2 Bernoulli distribution, logit parameterization
      • 13.2.1 Probability mass function
      • 13.2.2 Sampling statement
      • 13.2.3 Stan Functions
    • 13.3 Bernoulli-logit generalized linear model (Logistic Regression)
      • 13.3.1 Probability mass function
      • 13.3.2 Sampling statement
      • 13.3.3 Stan Functions
  • 14 Bounded Discrete Distributions
    • 14.1 Binomial distribution
      • 14.1.1 Probability mass function
      • 14.1.2 Log probability mass function
      • 14.1.3 Gradient of log probability mass function
      • 14.1.4 Sampling statement
      • 14.1.5 Stan functions
    • 14.2 Binomial distribution, logit parameterization
      • 14.2.1 Probability mass function
      • 14.2.2 Log probability mass function
      • 14.2.3 Gradient of log probability mass function
      • 14.2.4 Sampling statement
      • 14.2.5 Stan functions
    • 14.3 Beta-binomial distribution
      • 14.3.1 Probability mass function
      • 14.3.2 Sampling statement
      • 14.3.3 Stan functions
    • 14.4 Hypergeometric distribution
      • 14.4.1 Probability mass function
      • 14.4.2 Sampling statement
      • 14.4.3 Stan functions
    • 14.5 Categorical distribution
      • 14.5.1 Probability mass functions
      • 14.5.2 Sampling statement
      • 14.5.3 Sampling statement
      • 14.5.4 Stan functions
    • 14.6 Categorical logit generalized linear model (softmax regression)
      • 14.6.1 Probability mass functions
      • 14.6.2 Sampling statement
      • 14.6.3 Stan functions
    • 14.7 Discrete range distribution
      • 14.7.1 Probability mass functions
      • 14.7.2 Sampling statement
      • 14.7.3 Stan functions
    • 14.8 Ordered logistic distribution
      • 14.8.1 Probability mass function
      • 14.8.2 Sampling statement
      • 14.8.3 Stan functions
    • 14.9 Ordered logistic generalized linear model (ordinal regression)
      • 14.9.1 Probability mass function
      • 14.9.2 Sampling statement
      • 14.9.3 Stan functions
    • 14.10 Ordered probit distribution
      • 14.10.1 Probability mass function
      • 14.10.2 Sampling statement
      • 14.10.3 Stan functions
  • 15 Unbounded Discrete Distributions
    • 15.1 Negative binomial distribution
      • 15.1.1 Probability mass function
      • 15.1.2 Sampling statement
      • 15.1.3 Stan functions
    • 15.2 Negative binomial distribution (alternative parameterization)
      • 15.2.1 Probability mass function
      • 15.2.2 Sampling statement
      • 15.2.3 Stan functions
    • 15.3 Negative binomial distribution (log alternative parameterization)
      • 15.3.1 Sampling statement
      • 15.3.2 Stan functions
    • 15.4 Negative-binomial-2-log generalized linear model (negative binomial regression)
      • 15.4.1 Probability mass function
      • 15.4.2 Sampling statement
      • 15.4.3 Stan functions
    • 15.5 Poisson distribution
      • 15.5.1 Probability mass function
      • 15.5.2 Sampling statement
      • 15.5.3 Stan functions
    • 15.6 Poisson distribution, log parameterization
      • 15.6.1 Probability mass function
      • 15.6.2 Sampling statement
      • 15.6.3 Stan functions
    • 15.7 Poisson-log generalized linear model (Poisson regression)
      • 15.7.1 Probability mass function
      • 15.7.2 Sampling statement
      • 15.7.3 Stan functions
  • 16 Multivariate Discrete Distributions
    • 16.1 Multinomial distribution
      • 16.1.1 Probability mass function
      • 16.1.2 Sampling statement
      • 16.1.3 Stan functions
    • 16.2 Multinomial distribution, logit parameterization
      • 16.2.1 Probability mass function
      • 16.2.2 Sampling statement
      • 16.2.3 Stan functions
  • Continuous Distributions
  • 17 Unbounded Continuous Distributions
    • 17.1 Normal distribution
      • 17.1.1 Probability density function
      • 17.1.2 Sampling statement
      • 17.1.3 Stan functions
      • 17.1.4 Standard normal distribution
      • 17.1.5 Sampling statement
      • 17.1.6 Stan functions
    • 17.2 Normal-id generalized linear model (linear regression)
      • 17.2.1 Probability distribution function
      • 17.2.2 Sampling statement
      • 17.2.3 Stan functions
    • 17.3 Exponentially modified normal distribution
      • 17.3.1 Probability density function
      • 17.3.2 Sampling statement
      • 17.3.3 Stan functions
    • 17.4 Skew normal distribution
      • 17.4.1 Probability density function
      • 17.4.2 Sampling statement
      • 17.4.3 Stan functions
    • 17.5 Student-t distribution
      • 17.5.1 Probability density function
      • 17.5.2 Sampling statement
      • 17.5.3 Stan functions
    • 17.6 Cauchy distribution
      • 17.6.1 Probability density function
      • 17.6.2 Sampling statement
      • 17.6.3 Stan functions
    • 17.7 Double exponential (Laplace) distribution
      • 17.7.1 Probability density function
      • 17.7.2 Sampling statement
      • 17.7.3 Stan functions
    • 17.8 Logistic distribution
      • 17.8.1 Probability density function
      • 17.8.2 Sampling statement
      • 17.8.3 Stan functions
    • 17.9 Gumbel distribution
      • 17.9.1 Probability density function
      • 17.9.2 Sampling statement
      • 17.9.3 Stan functions
    • 17.10 Skew double exponential distribution
      • 17.10.1 Probability density function
      • 17.10.2 Sampling statement
      • 17.10.3 Stan functions
  • 18 Positive Continuous Distributions
    • 18.1 Lognormal distribution
      • 18.1.1 Probability density function
      • 18.1.2 Sampling statement
      • 18.1.3 Stan functions
    • 18.2 Chi-square distribution
      • 18.2.1 Probability density function
      • 18.2.2 Sampling statement
      • 18.2.3 Stan functions
    • 18.3 Inverse chi-square distribution
      • 18.3.1 Probability density function
      • 18.3.2 Sampling statement
      • 18.3.3 Stan functions
    • 18.4 Scaled inverse chi-square distribution
      • 18.4.1 Probability density function
      • 18.4.2 Sampling statement
      • 18.4.3 Stan functions
    • 18.5 Exponential distribution
      • 18.5.1 Probability density function
      • 18.5.2 Sampling statement
      • 18.5.3 Stan functions
    • 18.6 Gamma distribution
      • 18.6.1 Probability density function
      • 18.6.2 Sampling statement
      • 18.6.3 Stan functions
    • 18.7 Inverse gamma Distribution
      • 18.7.1 Probability density function
      • 18.7.2 Sampling statement
      • 18.7.3 Stan functions
    • 18.8 Weibull distribution
      • 18.8.1 Probability density function
      • 18.8.2 Sampling statement
      • 18.8.3 Stan functions
    • 18.9 Frechet distribution
      • 18.9.1 Probability density function
      • 18.9.2 Sampling statement
      • 18.9.3 Stan functions
    • 18.10 Rayleigh distribution
      • 18.10.1 Probability density function
      • 18.10.2 Sampling statement
      • 18.10.3 Stan functions
  • 19 Positive Lower-Bounded Distributions
    • 19.1 Pareto distribution
      • 19.1.1 Probability density function
      • 19.1.2 Sampling statement
      • 19.1.3 Stan functions
    • 19.2 Pareto type 2 distribution
      • 19.2.1 Probability density function
      • 19.2.2 Sampling statement
      • 19.2.3 Stan functions
    • 19.3 Wiener First Passage Time Distribution
      • 19.3.1 Probability density function
      • 19.3.2 Sampling statement
      • 19.3.3 Stan functions
      • 19.3.4 boundaries
  • 20 Continuous Distributions on [0, 1]
    • 20.1 Beta distribution
      • 20.1.1 Probability density function
      • 20.1.2 Sampling statement
      • 20.1.3 Stan functions
    • 20.2 Beta proportion distribution
      • 20.2.1 Probability density function
      • 20.2.2 Sampling statement
      • 20.2.3 Stan functions
  • 21 Circular Distributions
    • 21.1 Von Mises distribution
      • 21.1.1 Probability density function
      • 21.1.2 Sampling statement
      • 21.1.3 Stan functions
      • 21.1.4 Numerical stability
  • 22 Bounded Continuous Distributions
    • 22.1 Uniform distribution
      • 22.1.1 Probability density function
      • 22.1.2 Sampling statement
      • 22.1.3 Stan functions
  • 23 Distributions over Unbounded Vectors
    • 23.1 Multivariate normal distribution
      • 23.1.1 Probability density function
      • 23.1.2 Sampling statement
      • 23.1.3 Stan functions
    • 23.2 Multivariate normal distribution, precision parameterization
      • 23.2.1 Probability density function
      • 23.2.2 Sampling statement
      • 23.2.3 Stan functions
    • 23.3 Multivariate normal distribution, Cholesky parameterization
      • 23.3.1 Probability density function
      • 23.3.2 Sampling statement
      • 23.3.3 Stan functions
    • 23.4 Multivariate Gaussian process distribution
      • 23.4.1 Probability density function
      • 23.4.2 Sampling statement
      • 23.4.3 Stan functions
    • 23.5 Multivariate Gaussian process distribution, Cholesky parameterization
      • 23.5.1 Probability density function
      • 23.5.2 Sampling statement
      • 23.5.3 Stan functions
    • 23.6 Multivariate Student-t distribution
      • 23.6.1 Probability density function
      • 23.6.2 Sampling statement
      • 23.6.3 Stan functions
    • 23.7 Gaussian dynamic linear models
      • 23.7.1 Sampling statement
      • 23.7.2 Stan functions
  • 24 Simplex Distributions
    • 24.1 Dirichlet distribution
      • 24.1.1 Probability density function
      • 24.1.2 Meaning of Dirichlet parameters
      • 24.1.3 Sampling statement
      • 24.1.4 Stan functions
  • 25 Correlation Matrix Distributions
    • 25.1 LKJ correlation distribution
      • 25.1.1 Probability density function
      • 25.1.2 Sampling statement
      • 25.1.3 Stan functions
    • 25.2 Cholesky LKJ correlation distribution
      • 25.2.1 Probability density function
      • 25.2.2 Sampling statement
      • 25.2.3 Stan functions
  • 26 Covariance Matrix Distributions
    • 26.1 Wishart distribution
      • 26.1.1 Probability density function
      • 26.1.2 Sampling statement
      • 26.1.3 Stan functions
    • 26.2 Inverse Wishart distribution
      • 26.2.1 Probability density function
      • 26.2.2 Sampling statement
      • 26.2.3 Stan functions
  • Additional Distributions
  • 27 Hidden Markov Models
    • 27.1 Stan functions
  • Appendix
  • 28 Mathematical Functions
    • 28.1 Beta
    • 28.2 Incomplete beta
    • 28.3 Gamma
    • 28.4 Digamma
  • References

Stan Functions Reference

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23 Distributions over Unbounded Vectors

The unbounded vector probability distributions have support on all of RK for some fixed K.

  • Multivariate Normal Distribution
  • Multivariate Normal Distribution, Precision Parameterization
  • Multivariate Normal Distribution, Cholesky Parameterization
  • Multivariate Gaussian Process Distribution
  • Multivariate Gaussian Process Distribution, Cholesky parameterization
  • Multivariate Student-T Distribution
  • Gaussian Dynamic Linear Models