标题： 哈密顿蒙特卡罗的动态猜测与嵌入数值求根法 显示英文标题

标题： Dynamic guessing for Hamiltonian Monte Carlo with embedded numerical root-finding

摘要： 哈密顿蒙特卡罗（Hamiltonian Monte Carlo, HMC）及相关马尔可夫链蒙特卡罗（Markov Chain Monte Carlo, MCMC）算法的现代实现支持嵌入数值求根算法的概率函数采样，从而允许拟合涉及解析上不可处理的代数约束的统计模型。然而，这些模型在实际应用中的使用受到计算大量数值解的成本限制。 我们指出了以前针对嵌入求根算法的HMC方法的一个关键局限性，即要求在相同的模拟哈密顿轨迹上的所有点具有相同的起始猜测值。我们证明了这一要求可以放宽，使得起始猜测值依赖于前一个积分器状态。 为了利用这些信息选择一个好的猜测值，我们提出了两种启发式方法：使用前一个解，并通过隐式微分外推前一个解。这两种启发式方法在一系列代表性模型上比静态猜测表现出显著的性能提升。 此外，我们还介绍了grapevine，这是一个基于JAX的Python包，提供了对改进的No-U-Turn采样器的便捷访问，该采样器结合了动态猜测功能。 显示英文摘要

摘要： Modern implementations of Hamiltonian Monte Carlo and related MCMC algorithms support sampling of probability functions that embed numerical root-finding algorithms, thereby allowing fitting of statistical models involving analytically intractable algebraic constraints. However the application of these models in practice is limited by the computational cost of computing large numbers of numerical solutions. We identify a key limitation of previous approaches to HMC with embedded root-finding, which require the starting guess to be the same at all points on the same simulated Hamiltonian trajectory. We demonstrate that this requirement can be relaxed, so that the starting guess depends on the previous integrator state. To choose a good guess using this information we propose two heuristics: use the previous solution and extrapolate the previous solution using implicit differentiation. Both heuristics yield substantial performance improvements on a range of representative models compared with static guessing. We also present grapevine, a JAX-based Python package providing easy access to an implementation of the No-U-Turn sampler augmented with dynamic guessing.