Stan: A Probabilistic Programming Language

Abstract

Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm.

Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible.

Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.

1. Introduction

The goal of the Stan project is to provide a flexible probabilistic programming language for statistical modeling along with a suite of inference tools for fitting models that are robust, scalable, and efficient.

Stan differs from BUGS (Lunn, Thomas, Best, and Spiegelhalter 2000; Lunn, Spiegelhalter, Thomas, and Best 2009; Lunn, Jackson, Best, Thomas, and Spiegelhalter 2012) and JAGS (Plummer 2003) in two primary ways. First, Stan is based on a new imperative probabilistic programming language that is more flexible and expressive than the declarative graphical modeling languages underlying BUGS or JAGS, in ways such as declaring variables with types and supporting local variables and conditional statements. Second, Stan’s Markov chain Monte Carlo (MCMC) techniques are based on Hamiltonian Monte Carlo (HMC), a more efficient and robust sampler than Gibbs sampling or Metropolis-Hastings for models with complex posteriors.¹

Stan has the interfaces cmdstan for the command line shell, pystan for Python (Van Rossum et al. 2016), and rstan for R (R Core Team 2016). Stan also provides packages wrapping cmdstan, including MatlabStan for MATLAB, Stan.jl for Julia, StataStan for Stata, and MathematicaStan for Mathematica. These interfaces run on Windows, Mac OS X, and Linux, and are open-source licensed.

The next section provides an overview of how Stan works by way of an extended example, after which the details of Stan’s programming language and inference mechanisms are provided.

2. Core functionality

This section describes the use of Stan from the command line for estimating a Bayesian model using both MCMC sampling for full Bayesian inference and optimization to provide a point estimate at the posterior mode.

2.1. Program for estimating a Bernoulli parameter

Consider estimating the chance of success parameter for a Bernoulli distribution based on a sequence of observed binary outcomes. Figure 1 provides an implementation of such a model in Stan. The model treats the observed binary data, y[1],...,y[N], as independent and identically distributed, with success probability theta. The vectorized likelihood statement can also be coded using a loop as in BUGS, although it will run more slowly than the vectorized form:

Figure 1:

Stan program to estimate chance of success in the independent Bernoulli model.

for (n in 1:N)
  y[n] ~ bernoulli(theta);

A beta(1, 1) (i.e., uniform) prior is placed on theta, although there is no special behavior for conjugate priors in Stan. The prior could be dropped from the model altogether because parameters start with uniform distributions on their support, here constrained to be between 0 and 1 in the parameter declaration for theta.

2.2. Data format

Data for running Stan from the command line can be included in the R dump format. All of the variables declared in the data block of the Stan program must be defined in the data file. For example, 10 observations for the model in Figure 1 can be encoded as:

N <- 10
y <- c(0, 1, 0, 0, 0, 0, 0, 0, 0, 1)

This defines the contents of two variables, an integer N and a 10-element integer array y. The variable N is declared in the data block of the program as being an integer greater than or equal to zero; the variable y is declared as an integer array of size N with entries between 0 and 1 inclusive.

In rstan and pystan, data can also be passed directly through memory without the need to read or write to a file.

2.3. Compiling the model

After a C++ compiler and make are installed,² the Bernoulli model in Figure 1 can be translated to C++ and compiled with a single command. First, the directory must be changed to $stan, which we use as a shorthand for the directory in which Stan was unpacked.³

> cd $stan
> make replication/bernoulli

This produces an executable file bernoulli (bernoulli.exe on Windows) on the same path as the model. Forward slashes can be used with make on Windows.

2.4. Running the sampler

Command to sample from the model

The executable can be run with default options by specifying a path to the data file. The first command in the following example changes the current directory to that containing the model, which is where the data resides and where the executable is built. From there, the path to the data is just the file name bernoulli.data.R.

> cd $stan/replication/bernoulli
> ./bernoulli sample data file=bernoulli.data.R

For Windows, the ./ before the command should be removed. This call specifies that sampling should be performed with the model instantiated using the data in the specified file.

Terminal output from sampler

The output is as follows, starting with a summary of the command line options used, including defaults; these are also written into the sample file as comments.

method = sample (Default)
  sample
    num_samples = 1000 (Default) 
    num_warmup = 1000 (Default) 
    save_warmup = 0 (Default)
    thin = 1 (Default)
    adapt
      engaged = 1 (Default) 
      gamma = 0.050000000000000003 (Default) 
      delta = 0.80000000000000004 (Default) 
      kappa = 0.75 (Default) 
      t0 = 10 (Default) 
      init_buffer = 75 (Default) 
      term_buffer = 50 (Default) 
      window = 25 (Default)
    algorithm = hmc (Default) 
      hmc
        engine = nuts (Default)
          nuts
            max_depth = 10 (Default)
        metric = diag_e (Default) 
        stepsize = 1 (Default) 
        stepsize_jitter = 0 (Default)
id = 0 (Default) 
data
  file = bernoulli.data.R
init = 2 (Default)
random
  seed = 2261934443
output
  file = output.csv (Default) 
  diagnostic_file = (Default) 
  refresh = 100 (Default)
Gradient evaluation took 8e-06 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.08 seconds. Adjust your expectations accordingly!
Iteration: 1 / 2000 [ 0%] (Warmup) 
Iteration: 100 / 2000 [ 5%] (Warmup)
...
Iteration: 1900 / 2000 [ 95%] (Sampling)
Iteration: 2000 / 2000 [100%] (Sampling)
 Elapsed Time: 0.009569 seconds (Warm-up)
               0.019888 seconds (Sampling)
               0.029457 seconds (Total)

The sampler configuration parameters are echoed; here they are all default values other than the data file.

The command line parameters marked Default may be explicitly set on the command line. Each value is preceded by the full path to it in the hierarchy; for instance, to set the maximum depth for the No-U-Turn sampler (NUTS), the command would be the following, where backslash indicates a continued line.

> ./bernoulli sample \
         algorithm=hmc engine=nuts max_depth=5 \
         data file=bernoulli.data.R

Help

A description of all configuration parameters including default values and constraints is available by executing

> ./bernoulli help-all

The sampler and its configuration are described at greater length in the manual (Stan Development Team 2016).

Sample file output

The output CSV file (comma-separated values), written by default to output.csv, starts with a summary of the configuration parameters for the run.

# stan_version_major = 2
# stan_version_minor = 14
# stan_version_patch = 0
# model = bernoulli_model
# method = sample (Default)
#   sample
#     num_samples = 1000 (Default)
#     num_warmup = 1000 (Default)
#     save_warmup = 0 (Default)
#     thin = 1 (Default)
#     adapt
#       engaged = 1 (Default)
#       gamma = 0.050000000000000003 (Default)
#       delta = 0.80000000000000004 (Default)
#       kappa = 0.75 (Default)
#       t0 = 10 (Default)
#       init_buffer = 75 (Default)
#       term_buffer = 50 (Default)
#       window = 25 (Default)
#     algorithm = hmc (Default)
#       hmc
#         engine = nuts (Default)
#           nuts
#             max_depth = 10 (Default)
#           metric = diag_e (Default)
#           stepsize = 1 (Default)
#           stepsize_jitter = 0 (Default)
# id = 0 (Default)
# data
#   file = bernoulli.data.R
# init = 2 (Default)
# random
#   seed = 2261934443
# output
#   file = output.csv (Default)
#   diagnostic_file = (Default)
#   refresh = 100 (Default)
lp__,accept_stat__,stepsize__,treedepth__,n_leapfrog__,\ divergent__,energy__,theta 
# Adaptation terminated
# Step size = 1.40687
# Diagonal elements of inverse mass matrix:
# 0.520458
−6.86458,0.987384,1.40687,1,1,0,6.91901,0.313323
−6.86458,0.348003,1.40687,2,3,0,8.2162,0.313323
−8.39704,0.533623,1.40687,2,3,0,9.37779,0.0775484 
...
−7.05295,0.933164,1.40687,1,1,0,7.14303,0.354908
−7.93385,0.686343,1.40687,2,3,0,8.05215,0.465703
#
#   Elapsed Time: 0.009569 seconds (Warm-up)
#                 0.019888 seconds (Sampling)
#                 0.029457 seconds (Total)

Stan’s behavior is fully specified by these configuration parameters, almost all of which have default values. By using the same version of Stan and these configuration parameters, exactly the same output file can be reproduced. The pseudorandom numbers generated by the sampler are fully determined by the seed (here randomly generated based on the time of the run, with value 2261934443) and the chain identifier (here 0). The identifier is used to advance the underlying pseudorandom number generator a sufficient number of values that using multiple chains with the same seed and different identifiers will draw from different subsequences of the pseudorandom number stream determined by the seed.

The output continues with a CSV header naming the columns of the output. For the default NUTS sampler in Stan 2.14.0, the output is as follows (on one line without the backslash).

lp__, accept_stat__, stepsize__, treedepth__, n_leapfrog__,\ divergent__, energy__, theta

The label lp__ is for log densities (up to an additive constant), accept_stat__ is for acceptance probabilities,⁴ stepsize__ is for the leapfrog integrator’s step size for simulating the Hamiltonian, treedepth__ is the depth of tree explored by the no-U-turn sampler (log base 2 of the number of log density and gradient evaluations), n_leapfrog__ is the number of density and gradient evaluations, divergent__ is a flag indicating a numerical instability during numerical integration resulting in the Hamiltonian not being conserved, and energy__ is the Hamiltonian value. The rest of the header will be the names of parameters; in this example, theta is the only parameter.

The results of step size and mass matrix adaptation are printed as comments.

# Adaptation terminated
# Step size = 1.40687
# Diagonal elements of inverse mass matrix:
# 0.520458

Unless adaptation is turned off, Stan uses the first half of the iterations to estimate a mass matrix and step size for numerical integration of the the Hamiltonian system. Stan uses a diagonal mass matrix by default, but may also be configured to use a dense mass matrix or unit mass matrix. The inverse mass matrix is estimated by regularizing the sample (co)variance of the latter half of the warmup iterations; see (Stan Development Team 2016) for full details. The rest of the file contains the sample, one draw per line, matching the header; here the parameter theta is the final value printed on each line, and each line corresponds to a draw from the posterior. The warmup iterations are excluded by default, but may be included with appropriate command configuration. The file ends with comments reporting the elapsed time.

−6.86458,0.987384,1.40687,1,1,0,6.91901,0.313323
−6.86458,0.348003,1.40687,2,3,0,8.2162,0.313323
−8.39704,0.533623,1.40687,2,3,0,9.37779,0.0775484 
...
−7.05295,0.933164,1.40687,1,1,0,7.14303,0.354908
−7.93385,0.686343,1.40687,2,3,0,8.05215,0.465703
#
# Elapsed Time: 0.009569 seconds (Warm-up)
#               0.019888 seconds (Sampling)
#               0.029457 seconds (Total)

It is evident from the values sampled for theta in the last column that there is a high degree of posterior uncertainty in the estimate of theta from the ten data points in the data file.

The log probabilities reported in the first column include not only the model log probabilities but also the Jacobian adjustment resulting from the transformation of the variables to unconstrained space. In this example, the Jacobian is the absolute derivative of the inverse logit function; see (Stan Development Team 2016) for the constrained parameter transforms and their Jacobians.

2.5. Sampler output analysis

Before performing output analysis, we recommend generating multiple independent chains in order to more effectively monitor convergence (Gelman and Rubin 1992; Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin 2013). Three more chains of draws can be created as follows.

./bernoulli sample data file=bernoulli.data.R random seed=2261934443 \
            id=1 output file=output1.csv
./bernoulli sample data file=bernoulli.data.R random seed=2261934443 \
            id=2 output file=output2.csv
./bernoulli sample data file=bernoulli.data.R random seed=2261934443 \
            id=3 output file=output3.csv

These calls illustrate how additional parameters are specified directly on the command line following the hierarchy given in the output. The backslash (\) at the end of a line indicates that the command continues on the next line; a caret (^) should be used in Windows.

The chains can be safely run in parallel under different processes; details of parallel execution depend on the operating system and the shell or terminal program. Although the same seed is used for each chain, the random numbers will in fact be independent as the chain identifier is used to skip the pseudorandom number generator ahead. Stan supplies a command line program bin/stansummary to summarize the output of one or more MCMC chains. Given a directory containing output from sampling,

> ls output*.csv
output.csv        output1.csv        output2.csv        output3.csv 

posterior summaries are printed using

> $stan/bin/stansummary output*.csv

The output is shown in Figure 2.⁵ Each row of the output summarizes a different value whose name is provided in the first column. These correspond to the columns in the output CSV files. The analysis includes estimates of the posterior mean (Mean) and standard deviation (StdDev). Quantiles for the median (50%) and the 90% posterior interval (5%, 95%) are also displayed; the quantiles printed can be configured.

Figure 2:

Output summary for the Bernoulli estimation model in Figure 1.

The remaining columns in the output provide an analysis of the sampling and its efficiency. The convergence diagnostic that is built into the bin/stansummary command is the estimated potential scale reduction statistic $\widehat{R}$ (Rhat); its value should be close to 1.0 when the chains have all converged to the same stationary distribution. Stan uses a more conservative version of $\widehat{R}$ than is usual in packages such as coda (Plummer, Best, Cowles, and Vines 2006), first splitting each chain in half to diagnose nonstationary chains; see (Gelman et al. 2013) and (Stan Development Team 2016) for definitions.

The column N_eff reports the effective sample size for a chain. Because MCMC methods produce a sample containing correlated draws in each chain, estimates such as posterior means are not expected to be as precise as they would be with truly independent draws. The effective sample size is an estimate of the number of independent draws that would lead to the same expected precision. The Monte Carlo standard error (MCSE) is an estimate of the error in estimating the posterior mean based on dividing the posterior standard deviation estimate by the square root of the effective sample size (sd / sqrt(n_eff)). Geyer (2011) provides a thorough introduction to effective sample size and MCSE estimation. Stan uses more conservative estimates based on both within-chain and cross-chain convergence; see (Gelman et al. 2013) and (Stan Development Team 2016) for motivation and definitions.

Because estimation accuracy is governed by the square root of the effective sample size, effective sample size per second (or its inverse) is the most relevant statistic for comparing the efficiency of sampler implementations. Compared to BUGS and JAGS, Stan is often relatively slow per iteration but relatively fast to generate a target effective sample size.

In this example, the estimated effective sample size (n_eff) is 1776, which is far greater than we typically need for inference. The posterior mean here is estimated to be 0.25 with an MCSE of 0.003. Because the model is conjugate, the exact posterior is known to be p(θ | y) = Beta(θ | 3, 9), which has a mean of 3/(3 + 9) = 0.25 and a mode of (3 − 1)/(3 + 9 − 2) = 0.2.

2.6. Estimators

Stan provides several ways to compute point estimates of parameters. The standard Bayesian approach is to use posterior means or medians, as computed by MCMC. The posterior mode, when it exists, is another popular estimator (providing what is sometimes called the maximum a posteriori (MAP) estimate). Stan programs can be interpreted as defining penalized log likelihood functions rather than posterior log densities, in which case the mode is the penalized maximum likelihood estimate (MLE).

Modes with optimization

The posterior mode (or penalized MLE) of the parameters conditioned on the data given the model can be found by using one of Stan’s built-in optimizers. The following command invokes optimization for the Bernoulli model using default configuration parameters.

> ./bernoulli optimize data file=bernoulli.data.R
method = optimize
optimize 
  algorithm = lbfgs (Default)
    lbfgs
      init_alpha = 0.001 (Default) 
      tol_obj = 9.9999999999999998e-13 (Default) 
      tol_rel_obj = 10000 (Default) 
      tol_grad = 1e-08 (Default) 
      tol_rel_grad = 10000000 (Default) 
      tol_param = 1e-08 (Default) 
      history_size = 5 (Default)
  iter = 2000 (Default) 
  save_iterations = 0 (Default)
id = 0 (Default) 
data
  file = bernoulli.data.R
init = 2 (Default) 
random 
  seed = 2265697650
output
  file = output.csv (Default) 
  diagnostic_file = (Default) 
  refresh = 100 (Default)
initial log joint probability = −6.19875
    Iter     log prob         ||dx||        ||grad|| 
       5     −5.00402     0.00118182     1.60451e-05
   alpha     alpha0    # evals   Notes
       1          1          8
Optimization terminated normally:
  Convergence detected: relative gradient magnitude is below tolerance

The final lines of the output indicate normal termination after seven iterations by convergence of the objective function (here the log density or penalized log likelihood) to within the default tolerance of 1e-08. The other values include final value of the log probability function (log prob), length of the difference between the current iteration’s value of the parameter vector and the previous value (||dx||), and the length of the gradient vector (||grad||).

The optimizer terminates when any of the log density, gradient, or parameter values are within their specified tolerance. The default optimizer uses the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, a quasi-Newton method which employs gradients and a memory and time efficient approximation to the Hessian (Nocedal and Wright 2006).

Optimizer output file

By default, optimization results are written into output.csv in the form of a valid CSV file.

# stan_version_major = 2
# stan_version_minor = 14
# stan_version_patch = 0
# model = bernoulli_model
# method = optimize
#   optimize
#     algorithm = lbfgs (Default)
#       lbfgs
#         init_alpha = 0.001 (Default)
#         tol_obj = 9.9999999999999998e-13 (Default)
#         tol_rel_obj = 10000 (Default)
#         tol_grad = 1e-08 (Default)
#         tol_rel_grad = 10000000 (Default)
#         tol_param = 1e-08 (Default)
#         history_size = 5 (Default)
#     iter = 2000 (Default)
#     save_iterations = 0 (Default)
# id = 0 (Default)
# data
#   file = bernoulli.data.R
# init = 2 (Default)
# random
#   seed = 2265697650
# output
#    file = output.csv (Default)
#    diagnostic_file = (Default)
#    refresh = 100 (Default)
lp__,theta 
−5.00402,0.200002

As with the sampler output, the configuration of the optimizer is dumped as CSV comments (lines beginning with #). Then there is a header, listing the log density, lp__, and the single parameter name, theta. The next line shows that the posterior mode for theta is 0.200002, matching the true posterior mode of 0.20 very closely.

Optimization is carried out on the unconstrained parameter space, but without the Jacobian adjustment to the log density. This ensures modes are defined with respect to the constrained parameter space as declared in the parameters block and used in the model specification.

2.7. Diagnostic mode

Stan provides a diagnostic mode that evaluates the log density and its gradient at the initial parameter values (either user supplied or generated randomly based on the specified or default seed).

> ./bernoulli diagnose data file=bernoulli.data.R
method = diagnose
  diagnose
    test = gradient (Default)
    gradient
      epsilon = 9.9999999999999995e-07 (Default) 
      error = 9.9999999999999995e-07 (Default)
id = 0 (Default) 
data
  file = bernoulli.data.R
init = 2 (Default) 
random
  seed = 2266249495
output
  file = output.csv (Default) 
  diagnostic_file = (Default) 
  refresh = 100 (Default)
TEST GRADIENT MODE
 Log probability=−6.77412
 param idx        value       model       finite diff         error
         0     −1.25293    0.333671          0.333671  −9.40188e-11

Here, a random initialization is used and the initial log density is −6.77412 and the single parameter theta, here represented by index 0, has a value of −1.25293 on the unconstrained scale (inverse-logit transformed to 0.222 on the constrained scale). The derivative supplied by the model and by a finite differences calculation are the same to within −9.40188e-11. Non-finite log densities or derivatives indicate a problem with the model in terms of constraints on parameter values, function input constraints being violated, boundary conditions arising in function evaluations, and sometimes overflow or underflow issues with floating-point calculations. Large relative discrepancies between the model’s gradient calculation and finite differences can indicate a bug in the model or even in Stan’s algorithmic differentiation for a function in the model.

2.8. Roadmap for the rest of the paper

Now that the key functionality of Stan has been demonstrated, the remaining sections cover specific aspects of Stan’s architecture. Section 3 covers variable data type declarations as well as expressions and type inference, Section 4 describes the top-level blocks and execution of a Stan program, Section 5 lays out the available statements, and Section 6 the built-in math, matrix, and probability function library. Section 7 lays out MCMC and optimization-based inference. There are two appendices, Appendix A outlining the development process and Appendix B detailing the library dependencies.

3. Data types

All expressions in Stan are statically typed, including variables. This means their type is declared at compile time as part of the model, and does not change throughout the execution of the program. This is the same behavior as is found in compiled programming languages such as C/C++, Fortran, and Java, but is unlike the behavior of interpreted languages such as BUGS, R, and Python. Statically typing the variables (as well as declaring them in appropriate blocks based on usage) makes Stan programs easier to read and easier to debug by making explicit the modeling decisions and expression types.

3.1. Primitive types

The primitive types of Stan are real and int, which are used to represent continuous and integer values. These values are represented directly in C++ as types double and int. Integer expressions can be used anywhere a real value is required, but not vice-versa.

3.2. Vector and matrix types

Stan supports vectors, row vectors, and matrices with the usual access operations. Vectors are declared with their sizes and matrices with their number of rows and columns. Vector, row vector, and matrix elements are accessed using bracket notation, as in y[3] for the third element of a vector or row vector and a[2, 3] for the element in the third column of the second row of a matrix. Indexing begins from 1. The notation a[2] accesses the second row of matrix a.

Multiple indexing may be applied, with syntax following that of R and MATLAB. For example, if a is a vector containing at least four elements, then a[2:4] has elements a[2], a[3], and a[4]. Either or both bounds may be omitted, in which case they default to the first and last element in the vector. For example, a[3:] has two fewer elements than a and is defined so that a[3:][i] evaluates to a[2 + i]. If n is an array of integers, then a[n] is a vector with the same size as n and is defined so that a[n][i] evaluates to a[n[i]].

All vector and matrix types contain real values and may not be declared to hold integers. Collections of integers are represented using arrays.

3.3. Array types

An array may have entries of any other type. For example, arrays of integers and reals are allowed, as are arrays of vectors or arrays of matrices.

Higher-dimensional arrays are intrinsically arrays of arrays. An entry in a two-dimensional array y may be accessed as y[1,2]. The expression y[1] by itself denotes the one-dimensional array whose values correspond to the first row of y. Thus y[1][2] has the same value as y[1,2].⁶ Unlike integers, which may be used where real values are required, arrays of integers may not be used where real arrays are required.⁷

The manual contains a chapter discussing the efficiency tradeoffs and motivations for separating arrays and matrices.

3.4. Constrained variable types

Variables may be declared with constraints. The constraints have different effects depending on the block in which the variable is declared.

Integer and real types may be provided with lower bounds, upper bounds, or both. This includes the types used in arrays, and the real types used in vectors and matrices.

Vector types may be constrained to be unit simplexes (all entries non-negative and summing to one), unit length vectors (sum of squares is one), ordered (entries are in ascending order), positive ordered (entries in ascending order, all non-negative), using the types simplex[K], unit_vector[K], ordered[K], and positive_ordered[K], where K is the size of the vector. Matrices may be constrained to be covariance or precision matrices (symmetric, positive definite) or correlation matrices (symmetric, positive definite, unit diagonal), using the types cov_matrix[K] and corr_matrix[K]. For efficient and stable arithmetic, matrices may also be defined to be Cholesky factors of covariance or correlation matrices, using the types cholesky_factor_cov[K] and cholesky_factor_corr[K].

3.5. Expressions

The syntax of Stan is defined in terms of expressions and statements. Expressions denote values of a particular type. Statements represent operations such as assignment and incrementing the log density as well as control structures such as for loops and conditionals.

Stan provides the usual kinds of expressions found in programming languages. This includes variables, literals denoting integers, real values or strings, binary and unary operators over expressions, and function application.

Type inference

The type of a numeric literal is determined by whether or not it contains a period or scientific notation; for example, 20 has type int whereas 20.0 and 2e+1 have type real.

The type of applying an operator or a function to one or more expressions is determined by the available signatures for the function. For example, the multiplication operator (*) has a signature that maps two int arguments to an int and two real arguments to a real result. Another signature for the same operator maps a row_vector and a vector to a real result.

Type promotion

If necessary, an integer type will be promoted to a real value. For example, multiplying an int by a real produces a real result by promoting the int argument to a real.

4. Top-level blocks and program execution

In the rest of this paper, we will concentrate on the modeling language and how compiled programs are executed. These details are the same whether a Stan program is being used by one of the built-in samplers or optimizers or being used externally by a user-defined sampler or optimizer.

We begin with an example that will be used throughout the rest of this section. (Gelman et al. 2013, Section 5.1) define a hierarchical model of the incidence of tumors in rats in control groups across trials; a very similar model is defined for mortality rates in pediatric surgeries across hospitals in (Lunn et al. 2000, 2009, Examples, Volume 1). A Stan implementation is provided in Figure 3. In the rest of this section, we will walk through what the meaning of the various blocks are for the execution of the program.

Figure 3:

Hierarchical binomial model with posterior predictive quantities, coded in Stan.

4.1. Data block

A Stan program starts with an (optional) data block, which declares the data required to fit the model. This is a very different approach to modeling and declarations than in BUGS and JAGS, which determine which variables are data and which are parameters at run time based on the shape of the data input to them. These declarations make it possible to compile Stan to much more efficient code.⁸ Missing data models may still be coded in Stan, but the missing values must be declared as parameters; see (Stan Development Team 2016) for examples of missing data, censored data, and truncated data models.

In the model in Figure 3, the data block declares an integer variable J for the number of groups in the hierarchical model. The arrays y and n have size J, with y[j] being the number of positive outcomes in n[j] trials.

All of these variables are declared with a lower-bound constraint restricting their values to be greater than or equal to zero. The constraint language for Stan is not strong enough to restrict each y[j] to be less than or equal to n[j].

The data for a Stan program is read in once as the C++ object representing the program is constructed. After the data is read in, the constraints are validated. If the data does not satisfy the declared constraints, the program will throw an exception with an informative error message, which is displayed to the user in the command line, R, and Python interfaces.

4.2. Transformed data block

The Stan program in Figure 3 does not have a transformed data block. A transformed data block may be used to define new variables that can be computed based on the data. For example, standardized versions of data can be defined in a transformed data block or Bernoulli trials can be summed to model as binomial. Any constants needed by the program should also be defined in the transformed data block.

The transformed data block starts with a sequence of variable declarations and continues with a sequence of statements defining the variables. For example, the following transformed data block declares a vector x_std, then defines it to be the standardization of x:

transformed data {
  vector[N] x_std = (x - mean(x)) / sd(x);
}

The transformed data block is executed during construction, after the data is read in. Any data variables declared in the data block may be used in the variable declarations or statements. Transformed data variables may be used after they are declared, although care must be taken to ensure they are defined before they are used. Any constraints declared on transformed data variables are validated after all of the statements are executed, with execution terminating with an informative error message at the first variable with an invalid value.

4.3. Parameter block

The parameter block in the program in Figure 3 defines three parameters. The parameter theta[j] represents the probability of success in group j. The prior on each theta[j] is parameterized by a mean chance of success lambda and count kappa. Both theta[j] and lambda are constrained to fall between zero and one. The Pareto distribution requires a strictly positive lower bound, so kappa is constrained to be greater than or equal to a conservative bound of 0.1 to match the support of the Pareto hyperprior it receives in the model block.

The parameter block is executed every time the log density is evaluated. This may be multiple times per iteration of a sampling or optimization algorithm.

Implicit change of variables to unconstrained space

The probability distribution defined by a Stan program is intended to have unconstrained support (i.e., no points of zero probability), which greatly simplifies the task of writing samplers or optimizers. To achieve unbounded support, variables declared with constrained support are transformed to an unconstrained space. For instance, variables declared on [0, 1] are log-odds transformed and non-negative variables declared to fall in [0, ∞) are log transformed.⁹ More complex transforms are required for simplexes (a reverse stick-breaking transform) and covariance and correlation matrices (Cholesky factorization). The dimensionality of the resulting probability function may change as a result of the transform. For example, a K × K covariance matrix requires only $\left(\begin{array}{c}K\\ 2\end{array}\right)+K$ unconstrained parameters, and a K-simplex requires only K − 1 unconstrained parameters.

The unconstrained parameters over which the model is defined are inverse transformed back to satisfy their constraints before executing any statements in the model block. To account for the change of variables, the log absolute Jacobian determinant of the inverse transform is added to the overall log density.¹⁰ The gradients of the log density include the Jacobian term.

There is no validation required for the parameter block because the variable transforms are guaranteed to produce values that satisfy the declared constraints.

4.4. Transformed parameters block

The transformed parameters block allows users to define transforms of parameters within a model. Following the model in (Gelman et al. 2013), the example in Figure 3 uses the transformed parameter block to define transformed parameters alpha and beta for the prior success and failure counts to use in the beta prior for theta.

Following the same convention as the transformed data block, the (optional) transformed parameter block begins with declarations of the transformed parameters, followed by a sequence of statements defining them. Variables from previous blocks as well as the transformed parameters block may be used. In the example, the prior success and failure counts alpha and beta are defined in terms of the prior mean lambda and total prior count kappa.

The transformed parameter block is executed after the parameter block. Constraints are validated after all of the statements defining the transformed parameters have executed. Failure to validate a constraint results in an exception being thrown, which halts the execution of the log density function. The log density function can be defined to return negative infinity or the special not-a-number value, both of which are available through built-in functions and may be passed to the target density increment statement (see below).

If transformed parameters are used on the left-hand side of a sampling statement, it is up to the user to add the appropriate log absolute Jacobian determinant adjustment to the log density accumulator. For instance, a lognormal variate could be generated as follows without the built-in lognormal density function using the normal density as

parameters { 
  real<lower=0> u; 
  ... 
transformed parameters {
  real v = log(u);
} 
model {
  v ~ normal(0, 1); // distribution on transformed parameter
  target += u; // log density Jacobian adjustment
}

The transform is f(u) = log u, the inverse transform is f⁻¹(v) = exp v, so the absolute log Jacobian determinant is $\left|\frac{d}{dv}\exp v\right|=\exp v=u$. Whenever a transformed parameter is used on the left side of a sampling statement, a warning is printed to remind the user of the need for a Jacobian adjustment for the change of variables. The log density increment statement (target +=) is used to add u to the log density defined by the rest of the program.

Values of transformed parameters are saved in the output along with the parameters. As an alternative, local variables may be used to define temporary values in the model block.

4.5. Model block

The purpose of the model block is to define the log density on the constrained parameter space. The example in Figure 3 has a simple model containing four sampling statements. The hyperprior on the prior mean lambda is uniform, and the hyperprior on the prior count kappa is a Pareto distribution with lower-bound of support at 0.1 and shape 1.5, leading to a probability of κ > 0.1 proportional to κ^−5/2. Note that the hierarchical prior on theta is vectorized: each element of theta is drawn independently from a beta distribution with prior success count alpha and prior failure count beta. Both alpha and beta are transformed parameters, but because they are only used on the right-hand side of a sampling statement do not require a Jacobian adjustment of their own. The likelihood function is also vectorized, with the effect that each success count y[i] is drawn from a binomial distribution with number of trials n[i] and chance of success theta[i]. In vectorized sampling statements, scalar values will be repeated as many times as necessary.

The model block is executed after the transformed parameters block every time the log density is evaluated.

Implicit uniform priors

The default distribution for a parameter is uniform over its declared (constrained) support. For instance, a variable declared with a lower bound of 0 and an upper bound of 1 implicitly receives a Uniform(0, 1) distribution. These implicit uniform priors are improper if the variable has unbounded support. For instance, the uniform distributions over real values with upper and lower bounds, simplexes, and correlation matrices is proper, but the uniform distribution over unconstrained or one-side constrained reals, ordered vectors or covariance matrices are not proper.

Stan does not require proper priors, but if the posterior is improper, Stan will halt with an error message.¹¹

4.6. Generated quantities block

The (optional) generated quantities block allows values that depend on parameters and data, but do not affect estimation, to be defined efficiently. It may be used to calculate predictive inferences as well as to carry out forward simulation for posterior predictive checks; see (Gelman et al. 2013) for examples. Pseudorandom number generators are also available in the generated quantities block. The generated quantities block is called only once per iteration, not once per log density evaluation. Calculations in the generated quantities block are also more efficient because they do not require gradients.

The BUGS surgical example explored the ranking of institutions in terms of surgical mortality (Lunn et al. 2000, Examples, Volume 1). This is coded in the example in Figure 3 using the generated quantities block. The generated quantity variable rnk[j] will hold the rank of institution j from 1 to J in terms of mortality rate theta[j]. The ranks are extracted using the rank function. The posterior summary will print average rank and deviation. (Lunn et al. 2000) illustrated posterior inference by plotting posterior rank histograms.

Posterior comparisons can be carried out directly or using rankings. For instance, the model in Figure 3 sets highest[j] to 1 if hospital j has the highest estimated mortality rate. For a discussion of multiple comparisons and hierarchical models, see (Gelman, Hill, and Yajima 2012; Efron 2010).

As a second illustration, the generated quantities block in Figure 3 calculates the (posterior) probability that a given institution is above average in terms of mortality rate. This is done for each institution j with the usual plug-in estimate of theta[j] > mean(theta), which returns a binary (0 or 1) value. The posterior mean of above_avg[j] calculates the posterior probability $\mathrm{Pr}\left[{\theta }_j>\overline{\theta }\mid y,n\right]$.

4.7. Initialization

Stan’s samplers and optimizers all start from either random or user-supplied values for each parameter. User-supplied initial values are validated and transformed to the underlying unconstrained space; if a parameter value does not satisfy its declared constraints, the program exits and an informative error message is printed. For each variable that is not initialized, the built-in pseudorandom number generator is called once per unconstrained variable dimension. The default initialization is to randomly generate values uniformly on [−2, 2]; another symmetric interval around zero may be configured. This supplies diffuse starting points when transformed back to the constrained scale, facilitating convergence diagnostics (Gelman et al. 2013). Models with more data or more elaborate structure require narrower intervals for initialization to ensure the sampler is able to quickly locate the high mass region of the posterior.

Although Stan is quite effective at converging from diffuse random initializations, the user may supply their own initial values for sampling, optimization, or diagnosis. The top-level command line option configures a file from which to read initial values for parameters in the same R dump format used for data.

5. Statements

5.1. Assignment and sampling

Stan supports the same two basic statements as BUGS, assignment and sampling, examples of which were introduced earlier. In BUGS, these two kinds of statement define a directed acyclic graphical model; in Stan, they define a log density function.

Log density accumulator

There is an implicitly defined log density accumulator, the current value of which is available through the nullary function target().

Sampling statements

A sampling statement is nothing more than shorthand for incrementing the log density accumulator. For example, if beta is a parameter of type real, the sampling statement

beta ~ normal(0, 1);

has the exact same effect (up to dropping constant terms) as the incrementing the log density directly with the value of the log probability density function for the normal distribution using the target increment statement

target += normal_lpdf(beta | 0, 1);

Define variables before sampling statements

The translation of sampling statements to log density function evaluations explains why variables must be defined before they are used. In particular, a sampling statement does not draw the left-hand side variable from the right-hand side distribution.

Parameters are all defined externally by the sampler; local variables must be explicitly defined with an assignment statement before being used.

Direct definition of probability functions

Because computation is only up to a proportionality constant (an additive constant on the log scale), this sampling statement in turn has the same effect as the direct implementation in terms of basic arithmetic,

target += −0.5 * betâ2;

If beta is of type vector, replace the square with the vector product beta’ * beta, or the more efficient dot_self(beta). Distributions whose probability functions are not built directly into Stan can be implemented directly in this fashion.

5.2. Sequences of statements and execution order

Stan allows sequences of statements wherever statements may occur. Unlike BUGS, in which statements define a directed acyclic graph, in Stan, statements are executed imperatively in the order in which they occur in a program.

Blocks and variable scope

Sequences of statements surrounded by curly braces ({ and }) form blocks. Blocks may start with local variable declarations. The scope of a local variable (i.e., where it is available to be used) is that of the block in which it is declared.

Variables declared in the top-level blocks (data, transformed data, parameters, transformed parameters, generated quantities), may only be assigned to in the block in which they are declared. They may be used at any point after they are declared, including subsequent blocks.

5.3. Whitespace, semicolons, and comments

Following the convention of C++, statements are separated with semicolons in Stan so that the content of whitespace (outside of comments) is irrelevant. This is in contrast to BUGS and R, in which carriage returns are special and may indicate the end of a statement.

Stan supports the line comment style of C++, using two forward slashes (//) to comment out the rest of a line; this is the one location where the content of whitespace matters. Stan also supports C++-style block comments, with everything between the start-comment (/*) and end-comment (*/) markers being ignored.

The preferred style follows that of C++, with line comments used for everything but multiline comments. Stan follows the C++ convention of separating words in variable names using underbars (_), rather than dots (.), as used in R and BUGS, or camel case as used in Java. Camel case is valid Stan syntax, but dots may not be used in variable names.

5.4. Control structures

Stan supports the same kind of explicitly bounded for loops as found in BUGS and R. Like R, but unlike BUGS, Stan supports while loops and conditional (if-then-else) statements, as well as break and continue statements.¹² Stan provides the usual comparison operators and boolean operators to help define conditionals and condition-controlled while loops.

5.5. Print and reject statements

Stan provides print statements which take arbitrarily many arguments consisting of expressions or string literals consisting of sequences of characters surrounded by double quotes (“). These statements may be used for debugging purposes to report on intermediate states of variables or to indicate how far execution has proceeded before an error.

As an example, suppose a user’s program raises an error at run time because a covariance matrix defined in the transformed parameters block fails its symmetry constraint.

transformed parameters {
  cov_matrix[K] Sigma;
  for (m in 1:M)
    for (n in m:M)
      Sigma[m, n] <- Omega[m, n] * sigma[m] * sigma[n];
  print("Sigma=", Sigma);
}

The print statement added at the last line will print the values in the matrix before the validation occurs at the end of the transformed parameters block.

Stan also supports reject statements which may be used to halt execution and return a meaningful error message. Like print statements, they take any number of string and expression arguments.

if (n > size(x))
  reject("Index out of bounds, n = ", n,
         "; required n < size(x) = ", size(x));

6. Function and distribution library

Stan is translated to C++ code that depends on the Stan math library to compute special functions, probability functions, matrix arithmetic and linear algebra, and solutions to ordinary differential equation (Carpenter, Hoffman, Brubaker, Lee, Li, and Betancourt 2015). In order to support the efficient algorithmic differentiation required to calculate gradients, Hessians, and higher-order derivatives in Stan, C++ functions must be templated separately for each argument. In order for these functions to be efficient in computing both values and derivatives, they need to operate directly on vectors of arguments so that shared computations can be reused. For example, if y is a vector and sigma is a scalar, the logarithm of sigma need only be evaluated once in order to compute the normal density for every member of y in

y ~ normal(mu, sigma);

6.1. Basic operators

Stan supports all of the basic C++ arithmetic operators, boolean operators, and comparison operators. In addition, it extends the arithmetic operators to matrices and includes elementwise matrix operators, left and right matrix division, and transposition.¹³

6.2. Special functions

Stan provides an especially rich set of special functions. This includes all of the C++ math library functions, as well as numerous more specialized functions such as Bessel functions, gamma and digamma functions, and generalized linear model link functions and their inverses. There are also many compound functions, such as log1m(x), which is more stable arithmetically for values of x near 0 than log(1 - x).

In addition to special functions, Stan includes distributions with alternative parameterizations, such as bernoulli_logit, which takes a parameter on the log odds (i.e., logit) scale. This allows a more concise notation for generalized linear models as well as more efficient and arithmetically stable execution.

6.3. Matrix and linear algebra functions

Rows, columns, and subblocks of matrices can be accessed using row, col, and block functions. Slices of arrays can be accessed using the head, tail, and segment functions. There are also special functions for creating a diagonal matrix from a vector and accessing the diagonal of a vector.

Various reductions are provided for arrays and matrices, such as sums, means, standard deviations, and norms. Replications are also available to copy a value into every cell of a matrix.

Matrix operators use the types of their operands to determine the type of the result. For example, multiplying a vector by a (column) row vector returns a matrix, whereas multiplying a row vector by a (column) vector returns a real. A postfix apostrophe (‘) is used for matrix and vector transposition. For example, if y and mu are vectors and Sigma is a square matrix, all of the same dimensionality, then y - mu is a vector, (y - mu)’ is a row vector, (y - mu)’ * Sigma is a row vector, and (y - mu)’ * Sigma * (y - mu) will be a real value. Matrix division is provided, which is much more arithmetically stable than inversion, e.g., (y - mu)’ / Sigma computes the same function as (y - mu)’ * inverse(Sigma). Stan also supports elementwise multiplication (.*) and division (./).

Linear algebra functions are provided for trace, left and right division, Cholesky factorization, determinants and log determinants, inverses, eigenvalues and eigenvectors, and singular value decomposition. All of these operations may be applied to matrices of parameters or constants. Various functions are specialized for speed, such as quadratic products, diagonal specializations, and multiply by self transposed; e.g., the previous example (y - mu)’ * Sigma * (y - mu) could be coded as as quad_form_diag(Sigma, y - mu).

6.4. Probability functions

Stan supports a growing collection of built-in univariate and multivariate probability density and mass functions. These probability functions share various features of their declarations and behavior.

All probability functions are defined on the log scale to avoid underflow. They are all named with the suffix _lpdf or _lpmf depending on whether they are density or mass functions, e.g., normal_lpdf is the log-scale normal distribution probability density function and poisson_lpmf is the log scale Poisson probability mass function.

All probability functions check that their arguments are within the appropriate constrained support and are configured to throw exceptions and print error messages for out-of-domain arguments (the behavior of positive and negative infinity and not-a-number values are built into floating-point arithmetic). For example, normal_lpdf(y | mu, sigma) requires the scale parameter sigma to be non-negative. Exceptions that are raised by functions will be caught and their warning messages will be printed for the user. Log density evaluations in which exceptions are raised are treated as if they had evaluated to negative infinity, and are thus rejected by the sampler or optimizer.

Up to a proportion calculations

All probability functions support calculating results up to a constant proportion, which becomes an additive constant on the log scale. Constancy here refers to being a numeric literal such as 1 or 0.5, a constant function such as pi(), data and transformed data variables, or a function that only depends on literals, constant functions or data variables.

Non-constants include parameters, transformed parameters, local variables declared in the transformed parameters or model statements, as well as any expression involving a non-constant.

Constant terms are dropped from probability function calculations at the time the model is compiled, so there is no run-time overhead to decide which expressions denote constants.¹⁴ For example, executing y ~ normal(0, sigma) only evaluates log(sigma) if sigma is a parameter, transformed parameter, or a local variable in the transformed parameters or model block; that is, log(sigma) is not evaluated if sigma is constant as defined above.

Constant terms are not dropped in explicit function evaluations, such as normal_lpdf(y | 0, sigma).

Vector arguments and shared computations

All of the univariate probability functions in Stan are implemented so that they accept arrays or vectors as arguments. For example, although the basic signature of the probability function normal_lpdf(y | mu, sigma) involves real y, mu and sigma, it supports calls in which any any or all of y, mu and sigma contain more than one element. A typical use case would be for linear regression, such as y ~ normal(X * beta, sigma), where y is a vector of observed data, X is a predictor matrix, beta is a coefficient vector, and sigma is a real value for the noise scale.

The advantage of using vectors is twofold. First, the models are more concise and closer to mathematical notation. Second, the vectorized versions are much faster. They reduce the number of times expensive operations need to be evaluated and stored as well as reducing the number of virtual function calls required in the compiled C++ executable for calculating gradients and higher-order derivatives. For example, if sigma is a parameter, then in evaluating y ~ normal(X * beta, sigma), the logarithm of sigma need only be computed once; if either y or beta is an N-vector, it also reduces the number of virtual function calls in C++ from N to 1.

7. Built-in inference engines

Stan includes Markov chain Monte Carlo (MCMC) samplers and optimizers. Others may be straightforwardly implemented within Stan’s C++ framework for sampling and optimization using the log density and derivative functions supplied by a model.

7.1. Markov chain Monte Carlo samplers

Hamiltonian Monte Carlo

The MCMC samplers provided include Euclidean Hamiltonian Monte Carlo (EHMC, which in much of the literature is referenced as simply HMC, Duane, Kennedy, Pendleton, and Roweth 1987; Neal 1994, 2011) and the no-U-turn sampler (NUTS, Hoffman and Gelman 2014; Betancourt 2016). Both the basic and NUTS versions of HMC allow estimation or specification of unit, diagonal, or full mass matrices. NUTS, the default sampler for Stan, automatically adapts the number of leapfrog steps, eliminating the need for user-specified tuning parameters. Both algorithms take advantage of gradient information in the log density function to generate coherent motion through the posterior that dramatically reduces the autocorrelation of the resulting transitions.

7.2. Optimizers

In addition to performing full Bayesian inference via posterior sampling, Stan also can perform optimization (i.e., computation of the posterior mode). We are currently working on implementing other optimization-based inference approaches including variational Bayes, expectation propagation, and and marginal inference using approximate integration. All these algorithms require optimization steps.

L-BFGS

The default optimizer in Stan is the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer (Nocedal and Wright 2006). L-BFGS is a quasi-Newton optimizer that evaluates gradients directly, then uses a limited history of gradients to update an approximation to the Hessian.

Conjugate gradient

Stan provides a standard form of conjugate gradient optimization (Nocedal and Wright 2006). As its name implies, conjugate gradient optimization requires gradient evaluations.

Newton

Additionally, Stan implements a straightforward version of Newton’s algorithm (Nocedal and Wright 2006), using gradients and Hessians.

8. Conclusion

Stan is a probabilistic programming language which allows users to specify a broad range of statistical models involving continuous parameters by coding their log posteriors (or penalized maximum likelihood) up to a proportion. Random variables are first-class objects and behave as expected under transformations. Stan provides full Bayesian inference for posterior expectations including parameter estimation and posterior predictive inference by defining appropriate derived quantities of interest. Stan implements full Bayesian inference with adaptive Hamiltonian Monte Carlo sampling and penalized maximum likelihood estimation with quasi-Newton optimization. Stan is implemented in standards-compliant C++, runs on all major computer platforms, and can be used interactively through interface languages including R and Python.

Appendix A describes the development process and Appendix B describes the library dependencies for Stan.

A. Developer process

A.1. Version control and source repository

Stan’s source code is hosted on GitHub and managed using the Git version control system (Chacon 2009). To manage the workflow with so many developers working at any given time, the project follows the GitFlow process (Driessen 2010). All developer submissions are managed through pull requests and we have gratefully received patches from numerous sources outside the core development team.

A.2. Continuous integration

Stan uses continuous integration, meaning that the entire program and set of tests are run automatically as code is pushed to the Git repository. Each pull request is tested for compatibility with the development branch, and the development branch itself is tested for stability. Stan uses Jenkins (Smart 2011), an open-source continuous integration server.

A.3. Testing framework

Stan includes extensive unit tests for low-level C++ code. Unit tests are implemented using the open-source googletest framework (Google 2016). These unit tests evaluate every function for accuracy of values and derivatives, as well as error checking. This requires an extensive meta-testing framework for the probability distributions due to their high degree of configurability as to argument types. The testing portion of the make file also runs tests of all of the built-in models, including almost all of the BUGS example models. Models are tested for both convergence and posterior mean estimation to within MCMC standard error.

A.4. Builds

The build process for Stan is highly automated through a cross-platform series of make files. The top-level make file builds the Stan-to-C++ translator command bin/stanc and posterior analysis command bin/print. It also builds the library archive bin/libstan.a. Great care was taken to avoid complicated platform-dependent configuration requirements that place a high burden on user system knowledge for installation. All that is needed is a relatively recent C++ compiler and version of make.

As exemplified in the introduction, the make file is automated enough to build an executable form of a Stan model in a single command. All libraries and other executables will be built as a side effect.

The top-level make file also supplies targets to build all of the documentation C++ API documentation is generated using the doxygen package (Van Heesch 2016). The Stan manual (Stan Development Team 2016) is typeset using the LATEX package (Mittelbach, Goossens, Braams, Carlisle, and Rowley 2004).

The make file also has targets for all of the unit and functional testing, for building the source-level distribution, and for cleaning any temporary files that it creates.

B. Library dependencies

Stan’s programming language translates to C++ code that depends on the following three external libraries.

B.1. Boost

Stan depends on several of the Boost C++ libraries (Schäling 2011). Stan makes extensive use of Boost’s template metaprogramming facilities including the Enable if package, the Type Traits library, and the Lexical Cast library. The Stan language is parsed using Boost’s Spirit parser, which itself depends on the binding libraries Phoenix, Bind, and Lambda, the variant type library Variant, and the container library Fusion. Exceptions are largely handled and configured through the error checking facilities in the Math and Exception packages. Output formatting and ad-hoc input parsing for various formats is facilitated with the Format library. Stan relies heavily on the special functions defined in the Math subpackages Special Functions and Statistical Distributions. Random number generation is carried out using the Random package. Ordinary differential equation solving for non-stiff systems is performed using the odeint package. The posterior analysis framework and some built-in functions depend on the Accumulators package.

B.2. Eigen

Stan’s handling of matrices and linear algebra is implemented through the Eigen C++ template library (Guennebaud, Jacob et al. 2010). Eigen uses template metaprogramming to achieve state-of-the-art performance for matrix and linear algebra operations with a great deal of flexibility with respect to input types. Unfortunately, many of the expression templates that Eigen uses for efficient static analysis and lazy evaluation are short-circuited because of Stan’s need to have mixed type operations (i.e., multiplying a constant predictor matrix of double values by a parameter vector of algorithmic differentiation values). To make up for this in some important cases, Stan has provided compound functions such as the quadratic form, which allow speedups of both the matrix operations and their derivatives compared to a direct implementation using Stan’s built-in operators.

B.3. Sundials

Stan uses the SUNDIALS suite of nonlinear differential equation solvers for solving stiff differential equations (Hindmarsh, Brown, Grant, Lee, Serban, Shumaker, and Woodward 2005). The subpackage CVODES implements the backward differentiation formula (BDF) algorithm with sensitivities (Cohen and Hindmarsh 1996).