# LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs

arXiv: https://arxiv.org/abs/2608.08588 · HTML: https://mlsanity.com/paper/2608.08588/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 37 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [An Introduction to Probabilistic Programming](https://mlsanity.com/paper/1809.10756/) (2018)
- [A New Approach to Probabilistic Programming Inference](https://mlsanity.com/paper/1507.00996/) (2015)
- [On Correctness of Automatic Differentiation for Non-Differentiable Functions](https://mlsanity.com/paper/2006.06903/) (2020)
- [On the Geometric Ergodicity of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1601.08057/) (2016)
- [Involutive MCMC: a Unifying Framework](https://mlsanity.com/paper/2006.16653/) (2020)
- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [Correctness of Automatic Differentiation via Diffeologies and Categorical Gluing](https://mlsanity.com/paper/2001.02209/) (2020)
- [Divide, Conquer, and Combine: a New Inference Strategy for Probabilistic Programs with Stochastic Support](https://mlsanity.com/paper/1910.13324/) (2019)
- [A general perspective on the Metropolis-Hastings kernel](https://mlsanity.com/paper/2012.14881/) (2020)
- [Efficient Differentiable Programming in a Functional Array-Processing Language](https://mlsanity.com/paper/1806.02136/) (2018)
- [On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem](https://mlsanity.com/paper/2002.08489/) (2020)
- [Affine Monads and Lazy Structures for Bayesian Programming](https://mlsanity.com/paper/2212.07250/) (2022)
- [Nonparametric Hamiltonian Monte Carlo](https://mlsanity.com/paper/2106.10238/) (2021)
- [Forward- or Reverse-Mode Automatic Differentiation: What's the Difference?](https://mlsanity.com/paper/2212.11088/) (2022)

## Abstract (arXiv)

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.