Probabilistic programming

programming with probability distributions for, e.g. Bayesian inference or other inference

This is apparently what we call Bayesian inference these days. When we say Bayesian programming, we might mean a simple hierarchical model, but we want to emphasise hope that we might even succeed in doing inference for very complicated models indeed, possibly ones without tractable likelihoods of any kind, maybe even Turing-complete. Hope in this context means something like “we provide the programming primitives to in principle express the awful crazy likelihood structure of your complicated problem, although you are on your own in demonstrating any kind of concentration or convergence for your estimates of its posterior likelihood in the light of data.”

Mostly these tools are based on Markov Chain Monte Carlo sampling which turns out to be a startlingly general way to grind out the necessary calculations. There are other ways, such as classic conjugate priors, variational methods or reparameterisation flows, and many hybrids thereof.

See George Ho of PyMC3/PyMC4 for an in-depth introduction into what might be desirable to solve these problems in practice.

A probabilistic programming framework needs to provide six things:

  1. A language or API for users to specify a model
  2. A library of probability distributions and transformations to build the posterior density
  3. At least one inference algorithm, which either draws samples from the posterior (in the case of Markov Chain Monte Carlo, MCMC) or computes some approximation of it (in the case of variational inference, VI)
  4. At least one optimizer, which can compute the mode of the posterior density
  5. An autodifferentiation library to compute gradients required by the inference algorithm and optimizer
  6. A suite of diagnostics to monitor and analyze the quality of inference

George Ho diagrams probabilistic programming frameworks

See also Col Carroll’s overview of several trendy frameworks. This seems fresh, i.e. post-2019?, and includes more than I did here.


Stan is the inference toolbox for broad classes of Bayesian model and the de facto reference point. Usually seen in concert with brms which makes it easier to use for various standard regression models.

Andrew Gelman notes

The basic execution structure of Stan is in the JSS paper (by Bob Carpenter, Andrew Matt Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell) and in the reference manual. The details of autodiff are in the arXiv paper (by Bob Carpenter, Matt Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and Michael Betancourt). These are sort of background for what we’re trying to do.

If you haven’t read Maria Gorinova’s MS thesis and POPL paper (with Andrew Gordon and Charles Sutton), you should probably start there.

Radford Neal’s intro to HMC is nice, as is the one in David McKay’s book. Michael Betancourt’s papers are the thing to read to understand HMC deeply—he just wrote another brain bender on geometric autodiff (all on arXiv). Starting with the one on hierarchical models would be good as it explains the necessity of reparameterizations.

Also I recommend our JEBS paper (with Daniel Lee, and Jiqiang Guo) as it presents Stan from a user’s rather than a developer’s perspective.

Stan is a really good tool that I use all the time and if it solves your problem then it is an unbeatable developer experience. It does so much automatically, and makes it easy to do the right thing even where it is not automatic. It has some sharp boundaries around what it can do, however. Firstly, it cannot handle bounded parameter domains , or at least, ones that cannot easily be mapped onto scalar unbounded domains, so the space of e.g. positive-definite matrices, or discrete parameters will only work if you can find a sweet hack to shoehorn them in. It has no foreign function interface, so you can't plug external tools in, however, and if you want to do unorthodox variational inference it is not ideal. Since it is its own programming language, it is hard to work around this.


From Blei’s lab, leverages trendy deep learning machinery, tensorflow for variational Bayes and such.

This is now baked in to tensorflow as a probabilistic programming interface.

TensorFlow Probability

The tensorflow entrant. Low-level and messy. Used in Edward2, above, but presumably more basic.


pytorch + bayes = pyro. For rationale, see the pyro launch announcment:

We believe the critical ideas to solve AI will come from a joint effort among a worldwide community of people pursuing diverse approaches. By open sourcing Pyro, we hope to encourage the scientific world to collaborate on making AI tools more flexible, open, and easy-to-use. We expect the current (alpha!) version of Pyro will be of most interest to probabilistic modelers who want to leverage large data sets and deep networks, PyTorch users who want easy-to-use Bayesian computation, and data scientists ready to explore the ragged edge of new technology.


Numpyro uses jax for autodiff and reputedly comes from the creators of pyro. It has no second-order derivatives as in HMC but looks sleek.


pyprob: (Le, Baydin, and Wood 2017)

pyprob is a PyTorch-based library for probabilistic programming and inference compilation. The main focus of this library is on coupling existing simulation codebases with probabilistic inference with minimal intervention.

The main advantage of pyprob, compared against other probabilistic programming languages like Pyro, is a fully automatic amortized inference procedure based on importance sampling. pyprob only requires a generative model to be specified. Particularly, pyprob allows for efficient inference using inference compilation which trains a recurrent neural network as a proposal network.

In Pyro such an inference network requires the user to explicitly define the control flow of the network, which is due to Pyro running the inference network and generative model sequentially. However, in pyprob the generative model and inference network runs concurrently. Thus, the control flow of the model is directly used to train the inference network. This alleviates the need for manually defining its control flow.

The flagship application seems to be etalumis (Baydin et al. 2019) a probablistic programming ramework with emphasis AFAICT on Bayesian inverse problems.



Mamba is an open platform for the implementation and application of MCMC methods to perform Bayesian analysis in julia. The package provides a framework for (1) specification of hierarchical models through stated relationships between data, parameters, and statistical distributions; (2) block-updating of parameters with samplers provided, defined by the user, or available from other packages; (3) execution of sampling schemes; and (4) posterior inference. It is intended to give users access to all levels of the design and implementation of MCMC simulators to particularly aid in the development of new methods.

Several software options are available for MCMC sampling of Bayesian models. Individuals who are primarily interested in data analysis, unconcerned with the details of MCMC, and have models that can be fit in JAGS, Stan, or OpenBUGS are encouraged to use those programs. Mamba is intended for individuals who wish to have access to lower-level MCMC tools, are knowledgeable of MCMC methodologies, and have experience, or wish to gain experience, with their application. The package also provides stand-alone convergence diagnostics and posterior inference tools, which are essential for the analysis of MCMC output regardless of the software used to generate it.



Turing.jl is a Julia library for (universal) probabilistic programming. Current features include:

  • Universal probabilistic programming with an intuitive modelling interface
  • Hamiltonian Monte Carlo (HMC) sampling for differentiable posterior distributions
  • Particle MCMC sampling for complex posterior distributions involving discrete variables and stochastic control flows
  • Gibbs sampling that combines particle MCMC and HMC

It is one of many julia options, and includes flashy MCMC, called AdvancedHMC.jl



Gen simplifies the use of probabilistic modeling and inference, by providing modeling languages in which users express models, and high-level programming constructs that automate aspects of inference.

Like some probabilistic programming research languages, Gen includes universal modeling languages that can represent any model, including models with stochastic structure, discrete and continuous random variables, and simulators. However, Gen is distinguished by the flexibility that it affords to users for customizing their inference algorithm.

Gen’s flexible modeling and inference programming capabilities unify symbolic, neural, probabilistic, and simulation-based approaches to modeling and inference, including causal modeling, symbolic programming, deep learning, hierarchical Bayesiam modeling, graphics and physics engines, and planning and reinforcement learning.

It has an impressive talk demonstrating how you would interactively clean data using it.

Miscellaneous julia options

DynamicHMC.jl does Hamiltonian/NUTS sampling in a raw likelihood setting.

Possibly it is a competitor of Klara.jl, the Juliastats MCMC.

Miletus is a financial product adn ter-structure modeling package that is available for quant stuff in Julia as part of the paid packages offerings in finance. Although it looks like it is also freely available?


Pymc3 is python+Theano. PyMC4 will depend upon Tensorflow.

See Chris Fonnesbeck’s example in python.

Thomas Wiecki, Bayesian Deep Learning shows how to some variants with PyMC3.



greta models are written right in R, so there’s no need to learn another language like BUGS or Stan

greta uses Google TensorFlow

I wonder how it uses Google Tensorflow.



Soss is a library for probabilistic programming.

Let’s jump right in with a simple linear model:

using Soss

m = @model X begin
    β ~ Normal() |> iid(size(X,2))
    y ~ For(eachrow(X)) do x
        Normal(x’ * β, 1)

In Soss, models are first-class and function-like, and “applying” a model to its arguments gives a joint distribution.

Just a few of the things we can do in Soss:

  • Sample from the (forward) model
  • Condition a joint distribution on a subset of parameters
  • Have arbitrary Julia values (yes, even other models) as inputs or outputs of a model
  • Build a new model for the predictive distribution, for assigning parameters to particular values


InferPy seems to be a higher-level competitor to Edward2?



ZhuSuan is a python probabilistic programming library for Bayesian deep learning, which conjoins the complimentary advantages of Bayesian methods and deep learning. ZhuSuan is built upon Tensorflow. Unlike existing deep learning libraries, which are mainly designed for deterministic neural networks and supervised tasks, ZhuSuan provides deep learning style primitives and algorithms for building probabilistic models and applying Bayesian inference. The supported inference algorithms include:

  • Variational inference with programmable variational posteriors, various objectives and advanced gradient estimators (SGVB, REINFORCE, VIMCO, etc.).
  • Importance sampling for learning and evaluating models, with programmable proposals.
  • Hamiltonian Monte Carlo (HMC) with parallel chains, and optional automatic parameter tuning.


Church is a general-purpose Turing-complete Monte Carlo lisp-derivative, which is unbearably slow but does some reputedly cute tricks with modeling human problem-solving, and other likelihood-free methods, according to creators Noah Goodman and Joshua Tenenbaum.

See also Anglican, which is the same but different, being built in clojure, and hence also leveraging browser Clojurescript.


WebPPL is a successor to Church designed as a teaching language for probabilistic reasoning in the browser. If you like Javascript ML.


See also BAT the Bayesian Analysis Toolkit, which does sophisticated Bayes modelling although AFAICT uses a fairly basic Metropolis-Hasting Sampler?

Baydin, Atılım Güneş, Lei Shao, Wahid Bhimji, Lukas Heinrich, Lawrence Meadows, Jialin Liu, Andreas Munk, et al. 2019. “Etalumis: Bringing Probabilistic Programming to Scientific Simulators at Scale.” In.

Carroll, Colin. n.d. “A Tour of Probabilistic Programming Language APIs.” Https://

Cusumano-Towner, Marco, Benjamin Bichsel, Timon Gehr, Martin Vechev, and Vikash K. Mansinghka. 2018. “Incremental Inference for Probabilistic Programs.” In Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation, 571–85. PLDI 2018. New York, NY, USA: ACM.

Cusumano-Towner, Marco F., and Vikash K. Mansinghka. 2017. “Encapsulating Models and Approximate Inference Programs in Probabilistic Modules,” May.

———. 2018. “Using Probabilistic Programs as Proposals,” January.

Cusumano-Towner, Marco F., Feras A. Saad, Alexander K. Lew, and Vikash K. Mansinghka. 2019. “Gen: A General-Purpose Probabilistic Programming System with Programmable Inference.” In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation, 221–36. PLDI 2019. New York, NY, USA: ACM.

Cusumano-Towner, Marco, and Vikash K. Mansinghka. 2018. “A Design Proposal for Gen: Probabilistic Programming with Fast Custom Inference via Code Generation.” In Proceedings of the 2Nd ACM SIGPLAN International Workshop on Machine Learning and Programming Languages, 52–57. MAPL 2018. New York, NY, USA: ACM.

Gelman, Andrew, Daniel Lee, and Jiqiang Guo. 2015. “Stan: A Probabilistic Programming Language for Bayesian Inference and Optimization.” Journal of Educational and Behavioral Statistics 40 (5): 530–43.

Goodrich, Ben, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Bob Carpenter, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. “Stan : A Probabilistic Programming Language.” Journal of Statistical Software 76 (1).

Gorinova, Maria I., Andrew D. Gordon, and Charles Sutton. 2019. “Probabilistic Programming with Densities in SlicStan: Efficient, Flexible and Deterministic.” Proceedings of the ACM on Programming Languages 3 (POPL): 1–30.

Kochurov, Max, Colin Carroll, Thomas Wiecki, and Junpeng Lao. 2019. “PyMC4: Exploiting Coroutines for Implementing a Probabilistic Programming Framework,” September.

Lao, Junpeng. 2019. “A Hitchhiker’s Guide to Designing a Bayesian Library in Python.” Presentation Slides presented at the PyData Córdoba, Córdoba, Argentina, September 29.

Le, Tuan Anh, Atılım Güneş Baydin, and Frank Wood. 2017. “Inference Compilation and Universal Probabilistic Programming.” In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS), 54:1338–48. Proceedings of Machine Learning Research. Fort Lauderdale, FL, USA: PMLR.

Moore, Dave, and Maria I. Gorinova. 2018. “Effect Handling for Composable Program Transformations in Edward2,” November.

Pradhan, Neeraj, Jonathan P. Chen, Martin Jankowiak, Fritz Obermeyer, Eli Bingham, Theofanis Karaletsos, Rohit Singh, Paul Szerlip, Paul Horsfall, and Noah D. Goodman. 2018. “Pyro: Deep Universal Probabilistic Programming,” October.

PyMC Development Team. 2019. “PyMC3 Developer Guide.”

Rainforth, Tom. 2017. “Automating Inference, Learning, and Design Using Probabilistic Programming.” PhD Thesis, University of Oxford.

Salvatier, John, Thomas V. Wiecki, and Christopher Fonnesbeck. 2016. “Probabilistic Programming in Python Using PyMC3.” PeerJ Computer Science 2 (April): e55.

Ścibior, Adam Michał. 2018. Formally Justified and Modular Bayesianinference for Probabilistic Programs.

Tran, Dustin, Matthew D. Hoffman, Rif A. Saurous, Eugene Brevdo, Kevin Murphy, and David M. Blei. 2017. “Deep Probabilistic Programming.” In ICLR.

Tran, Dustin, Alp Kucukelbir, Adji B. Dieng, Maja Rudolph, Dawen Liang, and David M. Blei. 2016. “Edward: A Library for Probabilistic Modeling, Inference, and Criticism,” October.

Vasudevan, Srinivas, Ian Langmore, Dustin Tran, Eugene Brevdo, Joshua V. Dillon, Dave Moore, Brian Patton, Alex Alemi, Matt Hoffman, and Rif A. Saurous. 2017. “TensorFlow Distributions,” November.