让哈密顿蒙特卡洛在无限维概率程序中高效运行
LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs
- 利用惰性求值重构梯度计算，支持无限维模型的自动微分
- 提出多个适配无限维空间的HMC变体与无须调参的自适应采样器
- 适用于贝叶斯非参数建模，如随机过程和变化点检测
哈密顿蒙特卡洛（HMC）是概率编程中一种成功的通用推断方法，但传统形式要求梯度和有限维参数空间。在 Haskell 中，惰性求值使概率程序能够表达隐式定义的无限维空间上的随机过程与非参数贝叶斯模型。本文通过惰性求值，发展了适用于该无限维设置的基于梯度的 HMC 新形式。针对自动微分，提出了“分段解析于柱状解析划分下”（PACAP）的新概念，证明即使程序是无限维且惰性定义的，似然函数的梯度仍具有有限支撑。对于蒙特卡洛方法本身，开发了多个在无限维参数空间上运作的 HMC 变体及无需调参的无须翻转采样器（No-U-Turn Sampler），因惰性求值仍保持可生成性。实验涵盖高斯混合聚类、随机游走以及带有泊松过程变点的分段常数回归。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。