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18 October 2022 — by Reuben Cohn-Gordon
Improving the Probabilistic Programming Language monad-bayes

I’ve spent some time this summer extending Monad-Bayes, a probabilistic programming library in Haskell. This was done in the context of a Tweag open source fellowship, under the supervision of Dominic Steinitz. These improvements are part of the upcoming 1.0 release of the library.

Briefly, what’s Monad-Bayes about? Like Church, Gen and Pyropyro (among other probabilistic programming languages), it allows a user to specify a Bayesian model as a program, and automates the process of inferring the specified distribution. Unlike other languages, it is a first-class library of its host language (Haskell), rather than an embedded language. This means that any language feature in the host language is available to you. It also means that you can have distributions over absolutely anything. For instance, you can visualize a distribution by transforming it into a distribution over histograms or even over plots, and then just sampling one!

What’s new?

What’s new in version 1.0 of Monad-Bayes? This post outlines the changes, and then some of the things I learned from the fellowship.

Website and documentation

I used Hakyll to make a website for Monad-Bayes, including tutorials, examples and documentation.

Fairly thorough documentation of both the API and the architecture is now available. The documentation is stored in the same repository as the library, so contributions are welcome.

Choose your own RNG

Monad-Bayes was previously bound to the mwc random number generator, but now the sampler is parametric in the choice of RNG (although a default from the random package is provided). Dominic did this.

Terminal Based API

You can now run MCMC with an improved API that ensures a non-zero initial state, and shows a progress bar. This is accessible in the Control.Monad.Bayes.Inference.TUI module.


Inference by numerical integration is now possible, which you can express for arbitrary probabilistic programs using the continuation monad, as described nicely here. The code borrows from this implementation, with the author’s permission.

For now, its purpose is mostly didactic, since it’s impractical to numerically calculate the normalizing constant of large probabilistic programs. But used as a part of a more complex inference method, it may well be quite useful in future. For example, one might want to integrate out discrete variables from a model.

Lazy sampling

Standard samplers are not lazy, and hang if you try to sample e.g. an infinite list. The LazyPPL language has a clever implementation of a lazy sampler, and Monad-Bayes now supports this interpretation of probabilistic programs. This means you can sample an entire infinite list. Some accompanying inference methods built on top of this sampler are also provided, also taken more or less wholesale from LazyPPL.

The repo now hosts a variety of Jupyter notebooks, which can be run using Nix. This allows them to rely on a much wider range of packages than the library requires to install, so it’s where I showcase uses of Monad-Bayes that rely on more heavy-weight libraries.

These include physics simulations, probabilistic uses of lenses (randomly updating JSON), probabilistic parser combinators (currently very simple, but an exciting area to explore), comonads with probability for Ising models, probabilistically generating diagrams, and more.


A second set of nix-runnable notebooks are tutorials, inspired by the similar ones for Gen, another probabilistic programming language. These show how to use all of the inference methods available in Monad-Bayes, and visualize the results. They are available in the website.


Monadic streaming libraries interact very naturally with probability: a probabilistic stream is a random walk, which is useful for defining Markov Chain Monte Carlo, as well as a variety of models. To avoid heavy dependencies, streamly is not present in the library, but pipes is. This is accessible through the mcmcP function.

Future work here could use streamly for parallel chains (this worked when testing on an experimental branch).

Histograms and plotting

I use histogram-fill to easily form plots from weighted samples. I use hvega to easily plot those histograms in notebooks. These are available in notebooks.

Clean up

  • consistent naming conventions
  • core inference methods smc and mcmc now take config objects, so that its easier to work out what their parameters are
  • removed or fixed broken models
  • removed unused helper functions
  • added many tests
  • removed broken badges on the site

Future work

Three future directions are particularly interesting to me:

  • Implement something similar to Reactive Probabilistic Programming, using a Haskell functional reactive programming library like dunai. This is now underway and is extremely cool.

  • Implement Hamiltonian Monte Carlo, using Ed Kmett’s ad package. This is somewhat underway (

  • Use the continuation monad transformer to implement numerical integration on top of other MonadSample instances, to allow marginalization of only certain variables in a probabilistic program. Particularly for marginalizing out discrete variables when doing HMC.

What I learned

First and foremost, how Monad-Bayes works under the hood. It’s a very beautiful library. At its core is the idea that there are many representations of the probability monad, which can be built compositionally. Inference consists of transformations within and between these representations. The developer guide part of the documentation explains this in depth.

It’s also a library that uses some advanced concepts. In fact, Monad-Bayes is a pretty excellent tour of a range of more advanced level Haskell constructions, from continuations and coroutines to the (Church transformed!) free monad transformer.

I also learned a little about the limitations of Haskell. Some of the tracking that Gen does with execution traces is difficult in Haskell because of the static types, although one can always use Dynamic or similar to circumvent this (or fancy dependent types). Performance-wise, I had the familiar experience that most of the times, things were fast enough, but that engineering for performance would have required me to learn more about Haskell’s compiler and runtime, and how to use a profiler. So not impossible, but too much of a commitment for a summer.

Finally, thanks to everyone at Tweag for their help, especially Dominic Steinitz and Matthias Meschede.

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