# An Introduction to Probabilistic Programming

arXiv: https://arxiv.org/abs/1809.10756 · HTML: https://mlsanity.com/paper/1809.10756/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2018.

## Built on by (most cited first, 30 of 63)

- [Language Model Cascades](https://mlsanity.com/paper/2207.10342/) (2022)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Scallop: A Language for Neurosymbolic Programming](https://mlsanity.com/paper/2304.04812/) (2023)
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- [Automated Statistical Model Discovery with Language Models](https://mlsanity.com/paper/2402.17879/) (2024)
- [A Structural Causal Model for MR Images of Multiple Sclerosis](https://mlsanity.com/paper/2103.03158/) (2021)
- [Designing Perceptual Puzzles by Differentiating Probabilistic Programs](https://mlsanity.com/paper/2204.12301/) (2022)
- [Graphically Structured Diffusion Models](https://mlsanity.com/paper/2210.11633/) (2022)
- [One Life to Learn: Inferring Symbolic World Models for Stochastic Environments from Unguided Exploration](https://mlsanity.com/paper/2510.12088/) (2025)
- [ULLER: A Unified Language for Learning and Reasoning](https://mlsanity.com/paper/2405.00532/) (2024)
- [The General Theory of General Intelligence: A Pragmatic Patternist Perspective](https://mlsanity.com/paper/2103.15100/) (2021)
- [Recursive Monte Carlo and Variational Inference with Auxiliary Variables](https://mlsanity.com/paper/2203.02836/) (2022)
- [Learning Proposals for Probabilistic Programs with Inference Combinators](https://mlsanity.com/paper/2103.00668/) (2021)
- [Modularity in Reinforcement Learning via Algorithmic Independence in Credit Assignment](https://mlsanity.com/paper/2106.14993/) (2021)
- [Variational Marginal Particle Filters](https://mlsanity.com/paper/2109.15134/) (2021)
- [PGMax: Factor Graphs for Discrete Probabilistic Graphical Models and Loopy Belief Propagation in JAX](https://mlsanity.com/paper/2202.04110/) (2022)
- [Foundation Posteriors for Approximate Probabilistic Inference](https://mlsanity.com/paper/2205.09735/) (2022)
- [Smoothness Analysis for Probabilistic Programs with Application to Optimised Variational Inference](https://mlsanity.com/paper/2208.10530/) (2022)
- [PriorCVAE: scalable MCMC parameter inference with Bayesian deep generative modelling](https://mlsanity.com/paper/2304.04307/) (2023)
- [Scaling Integer Arithmetic in Probabilistic Programs](https://mlsanity.com/paper/2307.13837/) (2023)
- [CriticAL: Critic Automation with Language Models](https://mlsanity.com/paper/2411.06590/) (2024)
- [RefineStat: Efficient Exploration for Probabilistic Program Synthesis](https://mlsanity.com/paper/2509.01082/) (2025)
- [Nonparametric Hamiltonian Monte Carlo](https://mlsanity.com/paper/2106.10238/) (2021)
- [A Survey on Uncertainty Toolkits for Deep Learning](https://mlsanity.com/paper/2205.01040/) (2022)
- [Automatically Marginalized MCMC in Probabilistic Programming](https://mlsanity.com/paper/2302.00564/) (2023)
- [Exact Bayesian Inference on Discrete Models via Probability Generating Functions: A Probabilistic Programming Approach](https://mlsanity.com/paper/2305.17058/) (2023)
- [Sparser, Better, Faster, Stronger: Sparsity Detection for Efficient Automatic Differentiation](https://mlsanity.com/paper/2501.17737/) (2025)
- [Lobster: A GPU-Accelerated Framework for Neurosymbolic Programming](https://mlsanity.com/paper/2503.21937/) (2025)
- [Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling](https://mlsanity.com/paper/2110.06021/) (2021)

## Abstract (arXiv)

This book is a graduate-level introduction to probabilistic programming. It not only provides a thorough background for anyone wishing to use a probabilistic programming system, but also introduces the techniques needed to design and build these systems. It is aimed at people who have an undergraduate-level understanding of either or, ideally, both probabilistic machine learning and programming languages. We start with a discussion of model-based reasoning and explain why conditioning is a foundational computation central to the fields of probabilistic machine learning and artificial intelligence. We then introduce a first-order probabilistic programming language (PPL) whose programs correspond to graphical models with a known, finite, set of random variables. In the context of this PPL we introduce fundamental inference algorithms and describe how they can be implemented. We then turn to higher-order probabilistic programming languages. Programs in such languages can define models with dynamic computation graphs, which may not instantiate the same set of random variables in each execution. Inference requires methods that generate samples by repeatedly evaluating the program. Foundational algorithms for this kind of language are discussed in the context of an interface between program executions and an inference controller. Finally we consider the intersection of probabilistic and differentiable programming. We begin with a discussion of automatic differentiation, and how it can be used to implement efficient inference methods based on Hamiltonian Monte Carlo. We then discuss gradient-based maximum likelihood estimation in programs that are parameterized using neural networks, how to amortize inference using by learning neural approximations to the program posterior, and how language features impact the design of deep probabilistic programming systems.