Chapter 4 Brief Introduction to STAN

The engine used for running the Bayesian analyses covered in this course is STAN, as well as the rstan package that allows it to interface with R. STAN requires some programming from the users, but the benefit is that it allows users to fit a lot of different kinds of models. The goal of this lecture is not to make you an expert of STAN; I myself only have used maybe just 1 or 2% of the power of STAN. Instead, the goal is to give you a brief introduction with some sample codes, so that you can study further by yourself, and estimate models that no frequentist estimation exists yet.

4.1 STAN

STAN ( is itself a programming language, just like R. Strictly speaking it is not only for Bayesian methods, as you can actually do penalized maximum likelihood and automatic differentiation; however, it is most commonly used as an MCMC sampler for Bayesian analyses. It is written in C++, which makes it much faster than R (R is actually quite slow as a computational language). You can actually write a STAN program without calling R or other software, although eventually you may want to use statistical software to post-process the posterior samples after running MCMC. There are interfaces of STAN for different programs, including R, Python, MATLAB, Julia, Stata, and Mathematica, and for us we will be using the RStan interface.

4.1.1 STAN code

In STAN, you need to define a model using the STAN language. Below is an example for the Poisson model, which is saved with the file name "poisson_model.stan".

data {
  int<lower=0> N;  // number of observations
  int<lower=0> y[N];  // data array (counts);
parameters {
  real log_lambda;  // log of rate parameter
model {
  y ~ poisson_log(log_lambda);
  // prior
  log_lambda ~ normal(0, 5);
generated quantities {
  real lambda = exp(log_lambda);
  int yrep[N];
  for (i in 1:N) {
    yrep[i] = poisson_log_rng(log_lambda);

In STAN, anything after // denotes comments and will be ignored by the program, and in each blocks (e.g., data {}) a statement needs to be ended by a semicolon (;). There are several blocks in the above STAN code:

  • data: The data for input for STAN is usually not only a data set, but include other information, including sample size, number of predictors, and prior scales. Each type of data has an input type, such as
    • int = integer,
    • real = numbers with decimal places,
    • matrix = 2-dimensional data of real numbers,
    • vector = 1-dimensional data of real numbers, and
    • array = 1- to many-dimensional data. For example y[N] is a one-dimensional array of integers. you can set the lower and upper bounds so that STAN can check the input data
  • parameters: The parameters to be estimated
  • transformed parameters: optional variables that are transformation of the model parameters. It is usually used for more advanced models to allow for more efficient MCMC sampling.
  • model: It includes definition of priors for each parameter, and the likelihood for the data. There are many possible distributions that can be used in STAN.
  • generated quantities: Any quantities that are not part of the model but can be computed from the parameters for every iteration. Examples include posterior generated samples, effect sizes, and log-likelihood (for fit computation).

4.2 RStan

STAN is written in C++, which is a compiled language. This is different from programs like R, which you can input a command and get results right away. In contrast, a STAN program needs to be converted to something that can be executed in your computer. The benefit, however, is that the programs can be run much faster after the compilation process.

To feed data from R to STAN, and import output from STAN to R, you will use the rstan package ( Then, follow the following steps:

We will continue with the red card example:

4.2.1 Assembling data list in R

First, you need to assemble a list of data for STAN input, which should match the specific STAN program. In the STAN program we define two components (N and y) for data, so we need seven elements in an R list:

4.2.2 Call rstan

># Loading required package: StanHeaders
># rstan (Version 2.19.2, GitRev: 2e1f913d3ca3)
># For execution on a local, multicore CPU with excess RAM we recommend calling
># options(mc.cores = parallel::detectCores()).
># To avoid recompilation of unchanged Stan programs, we recommend calling
># rstan_options(auto_write = TRUE)
># Attaching package: 'rstan'
># The following object is masked from 'package:tidyr':
>#     extract

4.2.3 Summarize the results

After you call the stan function in R, it will compile the STAN program, which usually takes a minute or so. Then it starts sampling. You can now see a summary of the results by printing the results:

># Inference for Stan model: poisson_model.
># 4 chains, each with iter=2000; warmup=1000; thin=1; 
># post-warmup draws per chain=1000, total post-warmup draws=4000.
>#             mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
># lambda     27.67       0 0.12 27.45 27.59 27.67 27.75 27.90  1672    1
># log_lambda  3.32       0 0.00  3.31  3.32  3.32  3.32  3.33  1671    1
># Samples were drawn using NUTS(diag_e) at Fri Nov 15 10:55:01 2019.
># For each parameter, n_eff is a crude measure of effective sample size,
># and Rhat is the potential scale reduction factor on split chains (at 
># convergence, Rhat=1).

And you can also use the shinystan package to visualize the results:

4.3 Resources

STAN is extremely powerful and can fit almost any statistical models, but the price is that it takes more effort to code the model. To learn more about STAN, please check out for the manual, examples of some common models, and case studies (which includes more complex models like item response theory). See for a vignettes for working with the rstan package.

As you see, fitting simple models in STAN may sometimes be more work, but as we go further we will use the brms program that simplify the process for many commonly used models, such as regression and multilevel models. On the other hand, for truly complex models, STAN is actually a lifesaver as it would be extremely hard to fit some of those models with other approaches.