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Bayesian estimation via Stan’s HMC-NUTS sampler

To exercise the essential functions of CmdStanPy we show how to run Stan’s HMC-NUTS sampler to estimate the posterior probability of the model parameters conditioned on the data, using the example Stan model bernoulli.stan and corresponding dataset bernoulli.data.json which are distributed with CmdStan.

This is a simple model for binary data: given a set of N observations of i.i.d. binary data y[1] … y[N], it calculates the Bernoulli chance-of-success theta.

data {
   int<lower=0> N;
   int<lower=0,upper=1> y[N];
 parameters {
   real<lower=0,upper=1> theta;
 model {
   theta ~ beta(1,1);  // uniform prior on interval 0,1
   y ~ bernoulli(theta);

The data file specifies the number of observations and their values.

 "N" : 10,
 "y" : [0,1,0,0,0,0,0,0,0,1]

Instantiate the Stan model, assemble the data

The CmdStanModel class manages the Stan program and its corresponding compiled executable. It provides properties and functions to inspect the model code and filepaths. By default, the Stan program is compiled on instantiation.

# import packages
import os
from cmdstanpy import cmdstan_path, CmdStanModel

# specify Stan program file
bernoulli_stan = os.path.join(cmdstan_path(), 'examples', 'bernoulli', 'bernoulli.stan')

# instantiate the model; compiles the Stan program as needed.
bernoulli_model = CmdStanModel(stan_file=bernoulli_stan)

# inspect model object

Run the HMC-NUTS sampler

The CmdStanModel method sample is used to do Bayesian inference over the model conditioned on data using using Hamiltonian Monte Carlo (HMC) sampling. It runs Stan’s HMC-NUTS sampler on the model and data and returns a CmdStanMCMC object. The data can be specified either as a filepath or a Python dict; in this example, we use the example datafile bernoulli.data.json:

By default, the sample command runs 4 sampler chains. This is a set of per-chain Stan CSV files The filenames follow the template ‘<model_name>-<YYYYMMDDHHMM>-<chain_id>’ plus the file suffix ‘.csv’. There is also a correspondingly named file with suffix ‘.txt’ which contains all messages written to the console. If the output_dir argument is omitted, the output files are written to a temporary directory which is deleted when the current Python session is terminated.

# specify data file
bernoulli_data = os.path.join(cmdstan_path(), 'examples', 'bernoulli', 'bernoulli.data.json')

# fit the model
bern_fit = bernoulli_model.sample(data=bernoulli_data, output_dir='.')

# printing the object reports sampler commands, output files

Access the sample

The CmdStanMCMC object provides properties and methods to access, summarize, and manage the sample and its metadata.

The sampler and model outputs from each chain are written out to Stan CSV files. The CmdStanMCMC object assembles these outputs into a numpy.ndarray which contains all across all chains arranged as (draws, chains, columns). The draws method returns the draws array. By default, it returns the underlying 3D array. The optional boolean argument concat_chains, when True, will flatten the chains resulting in a 2D array.


To work with the draws from all chains for a parameter or quantity of interest in the model, use the stan_variable method to obtains a numpy.ndarray which contains the set of draws in the sample for the named Stan program variable by flattening the draws by chains into a single column:

draws_theta = bern_fit.stan_variable(name='theta')

The draws array contains both the sampler variables and the model variables. Sampler variables report the sampler state and end in __. To see the names and output columns for all sampler and model variables, we call accessor functions sampler_vars_cols and stan_vars_cols:

sampler_variables = bern_fit.sampler_vars_cols
stan_variables = bern_fit.stan_vars_cols
print('Sampler variables:\n{}'.format(sampler_variables))
print('Stan variables:\n{}'.format(stan_variables))

The NUTS-HMC sampler reports 7 variables. The Bernoulli example model contains a single variable theta.

Summarize the results

CmdStan is distributed with a posterior analysis utility stansummary that reads the outputs of all chains and computes summary statistics for all sampler and model parameters and quantities of interest. The CmdStanMCMC method summary runs this utility and returns summaries of the total joint log-probability density lp__ plus all model parameters and quantities of interest in a pandas.DataFrame:


CmdStan is distributed with a second posterior analysis utility diagnose which analyzes the per-draw sampler parameters across all chains looking for potential problems which indicate that the sample isn’t a representative sample from the posterior. The diagnose method runs this utility and prints the output to the console.


Save the Stan CSV files

The save_csvfiles function moves the CmdStan CSV output files to a specified directory.