PyMC (formerly PyMC3) is a Python package for Bayesian statistical modeling focusing on advanced Markov chain Monte Carlo (MCMC) and variational inference (VI) algorithms. Its flexibility and extensibility make it applicable to a large suite of problems.

Check out the PyMC overview, or one of the many examples! For questions on PyMC, head on over to our PyMC Discourse forum.

Features

• Intuitive model specification syntax, for example, x ~ N(0,1) translates to x = Normal('x',0,1)
• Powerful sampling algorithms, such as the No U-Turn Sampler, allow complex models with thousands of parameters with little specialized knowledge of fitting algorithms.
• Variational inference: ADVI for fast approximate posterior estimation as well as mini-batch ADVI for large data sets.

• Relies on PyTensor which provides:

  • Computation optimization and dynamic C or JAX compilation
  • NumPy broadcasting and advanced indexing
  • Linear algebra operators
  • Simple extensibility

• Transparent support for missing value imputation

Linear Regression Example

Plant growth can be influenced by multiple factors, and understanding these relationships is crucial for optimizing agricultural practices.

Imagine we conduct an experiment to predict the growth of a plant based on different environmental variables.

import pymc as pm

# Taking draws from a normal distribution
seed = 42
x_dist = pm.Normal.dist(shape=(100, 3))
x_data = pm.draw(x_dist, random_seed=seed)

# Independent Variables:
# Sunlight Hours: Number of hours the plant is exposed to sunlight daily.
# Water Amount: Daily water amount given to the plant (in milliliters).
# Soil Nitrogen Content: Percentage of nitrogen content in the soil.


# Dependent Variable:
# Plant Growth (y): Measured as the increase in plant height (in centimeters) over a certain period.


# Define coordinate values for all dimensions of the data
coords={
 "trial": range(100),
 "features": ["sunlight hours", "water amount", "soil nitrogen"],
}

# Define generative model
with pm.Model(coords=coords) as generative_model:
   x = pm.Data("x", x_data, dims=["trial", "features"])

   # Model parameters
   betas = pm.Normal("betas", dims="features")
   sigma = pm.HalfNormal("sigma")

   # Linear model
   mu = x @ betas

   # Likelihood
   # Assuming we measure deviation of each plant from baseline
   plant_growth = pm.Normal("plant growth", mu, sigma, dims="trial")


# Generating data from model by fixing parameters
fixed_parameters = {
 "betas": [5, 20, 2],
 "sigma": 0.5,
}
with pm.do(generative_model, fixed_parameters) as synthetic_model:
   idata = pm.sample_prior_predictive(random_seed=seed) # Sample from prior predictive distribution.
   synthetic_y = idata.prior["plant growth"].sel(draw=0, chain=0)


# Infer parameters conditioned on observed data
with pm.observe(generative_model, {"plant growth": synthetic_y}) as inference_model:
   idata = pm.sample(random_seed=seed)

   summary = pm.stats.summary(idata, var_names=["betas", "sigma"])
   print(summary)

From the summary, we can see that the mean of the inferred parameters are very close to the fixed parameters

Params	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
betas[sunlight hours]	4.972	0.054	4.866	5.066	0.001	0.001	3003	1257	1
betas[water amount]	19.963	0.051	19.872	20.062	0.001	0.001	3112	1658	1
betas[soil nitrogen]	1.994	0.055	1.899	2.107	0.001	0.001	3221	1559	1
sigma	0.511	0.037	0.438	0.575	0.001	0	2945	1522	1

# Simulate new data conditioned on inferred parameters
new_x_data = pm.draw(
   pm.Normal.dist(shape=(3, 3)),
   random_seed=seed,
)
new_coords = coords | {"trial": [0, 1, 2]}

with inference_model:
   pm.set_data({"x": new_x_data}, coords=new_coords)
   pm.sample_posterior_predictive(
      idata,
      predictions=True,
      extend_inferencedata=True,
      random_seed=seed,
   )

pm.stats.summary(idata.predictions, kind="stats")

The new data conditioned on inferred parameters would look like:

Output	mean	sd	hdi_3%	hdi_97%
plant growth[0]	14.229	0.515	13.325	15.272
plant growth[1]	24.418	0.511	23.428	25.326
plant growth[2]	-6.747	0.511	-7.740	-5.797

# Simulate new data, under a scenario where the first beta is zero
with pm.do(
 inference_model,
 {inference_model["betas"]: inference_model["betas"] * [0, 1, 1]},
) as plant_growth_model:
   new_predictions = pm.sample_posterior_predictive(
      idata,
      predictions=True,
      random_seed=seed,
   )

pm.stats.summary(new_predictions, kind="stats")

The new data, under the above scenario would look like:

Output	mean	sd	hdi_3%	hdi_97%
plant growth[0]	12.149	0.515	11.193	13.135
plant growth[1]	29.809	0.508	28.832	30.717
plant growth[2]	-0.131	0.507	-1.121	0.791

Getting started

If you already know about Bayesian statistics:

• API quickstart guide
• The PyMC tutorial
• PyMC examples and the API reference

Learn Bayesian statistics with a book together with PyMC

• Bayesian Analysis with Python (third edition) by Osvaldo Martin: Great introductory book.
• Probabilistic Programming and Bayesian Methods for Hackers: Fantastic book with many applied code examples.
• PyMC port of the book "Doing Bayesian Data Analysis" by John Kruschke as well as the first edition.
• PyMC port of the book "Statistical Rethinking A Bayesian Course with Examples in R and Stan" by Richard McElreath
• PyMC port of the book "Bayesian Cognitive Modeling" by Michael Lee and EJ Wagenmakers: Focused on using Bayesian statistics in cognitive modeling.

Audio & Video

• Here is a YouTube playlist gathering several talks on PyMC.
• You can also find all the talks given at PyMCon 2020 here.
• The "Learning Bayesian Statistics" podcast helps you discover and stay up-to-date with the vast Bayesian community. Bonus: it's hosted by Alex Andorra, one of the PyMC core devs!

Installation

To install PyMC on your system, follow the instructions on the installation guide.

Citing PyMC

Please choose from the following:

• PyMC: A Modern and Comprehensive Probabilistic Programming Framework in Python, Abril-Pla O, Andreani V, Carroll C, Dong L, Fonnesbeck CJ, Kochurov M, Kumar R, Lao J, Luhmann CC, Martin OA, Osthege M, Vieira R, Wiecki T, Zinkov R. (2023)
• A DOI for all versions.
• DOIs for specific versions are shown on Zenodo and under Releases

Contact

We are using discourse.pymc.io as our main communication channel.

To ask a question regarding modeling or usage of PyMC we encourage posting to our Discourse forum under the “Questions” Category. You can also suggest feature in the “Development” Category.

You can also follow us on these social media platforms for updates and other announcements:

• LinkedIn @pymc
• YouTube @PyMCDevelopers
• X @pymc_devs
• Mastodon @pymc@bayes.club

To report an issue with PyMC please use the issue tracker.

Finally, if you need to get in touch for non-technical information about the project, send us an e-mail.

License

Apache License, Version 2.0

Software using PyMC

General purpose

• Bambi: BAyesian Model-Building Interface (BAMBI) in Python.
• calibr8: A toolbox for constructing detailed observation models to be used as likelihoods in PyMC.
• gumbi: A high-level interface for building GP models.
• SunODE: Fast ODE solver, much faster than the one that comes with PyMC.
• pymc-learn: Custom PyMC models built on top of pymc3_models/scikit-learn API

Domain specific

• Exoplanet: a toolkit for modeling of transit and/or radial velocity observations of exoplanets and other astronomical time series.
• beat: Bayesian Earthquake Analysis Tool.
• CausalPy: A package focussing on causal inference in quasi-experimental settings.

Please contact us if your software is not listed here.

Papers citing PyMC

See Google Scholar here and here for a continuously updated list.

Contributors

See the GitHub contributor page. Also read our Code of Conduct guidelines for a better contributing experience.

Support

PyMC is a non-profit project under NumFOCUS umbrella. If you want to support PyMC financially, you can donate here.

Professional Consulting Support

You can get professional consulting support from PyMC Labs.

Sponsors

Thanks to our contributors