# “Hello, World”¶

## 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. Do do this we use 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]
}
```

### Specify a Stan model¶

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 os
from cmdstanpy import cmdstan_path, CmdStanModel

bernoulli_stan = os.path.join(cmdstan_path(), 'examples', 'bernoulli', 'bernoulli.stan')
bernoulli_model = CmdStanModel(stan_file=bernoulli_stan)
bernoulli_model.name
bernoulli_model.stan_file
bernoulli_model.exe_file
bernoulli_model.code()
```

### 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.

```bernoulli_data = os.path.join(cmdstan_path(), 'examples', 'bernoulli', 'bernoulli.data.json')
bern_fit = bernoulli_model.sample(data=bernoulli_data, output_dir='.')
```

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.

### Access the sample¶

The CmdStanMCMC object stores the CmdStan config information and the names of the the per-chain output files. It manages and retrieves the sampler outputs as Python objects.

```print(bern_fit)
```

The resulting set of draws produced by the sampler is lazily instantiated as a 3-D `numpy.ndarray` (i.e., a multi-dimensional array) over all draws from all chains arranged as draws X chains X columns. Instantiation happens the first time that any of the information in the posterior is accesed via properties: `draws`, `metric`, or `stepsize` are accessed. At this point the stan-csv output files are read into memory. For large files this may take several seconds; for the example dataset, this should take less than a second.

```bern_fit.draws().shape
```

Python’s index slicing operations can be used to access the information by chain. For example, to select all draws and all output columns from the first chain, we specify the chain index (2nd index dimension). As arrays indexing starts at 0, the index ‘0’ corresponds to the first chain in the CmdStanMCMC:

```chain_1 = bern_fit.draws()[:,0,:]
chain_1.shape       # (1000, 8)
chain_1[0]          # first draw:
# array([-7.99462  ,  0.578072 ,  0.955103 ,  2.       ,  7.       ,
# 0.       ,  9.44788  ,  0.0934208])
```

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:

```bern_fit.stan_variable('theta')
```

### Summarize or save 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:

```bern_fit.summary()
```

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.

```bern_fit.diagnose()
```

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

```bern_fit.save_csvfiles(dir='some/path')
```