Metropolis-Hastings estimates intractable expectations - can differentiating the algorithm estimate their gradients? The challenge is that Metropolis-Hastings trajectories are not conventionally differentiable due to the discrete accept/reject steps. Using a technique based on recoupling chains, our method differentiates through the Metropolis-Hastings sampler itself, allowing us to estimate gradients with respect to a parameter of otherwise intractable expectations. Our main contribution is a proof of strong consistency and a central limit theorem for our estimator under assumptions that hold in common Bayesian inference problems. The proofs augment the sampler chain with latent information, and formulate the estimator as a stopping tail functional of this augmented chain. We demonstrate our method on examples of Bayesian sensitivity analysis and optimizing a random walk Metropolis proposal.

翻译：Metropolis-Hastings算法能够估计难以直接计算的期望——那么对该算法进行微分能否估计其梯度？核心挑战在于，由于离散的接受/拒绝步骤，Metropolis-Hastings采样轨迹在传统意义上是不可微的。通过采用基于链重耦合的技术，我们的方法实现了对Metropolis-Hastings采样器本身的可微化处理，从而能够对原本难以处理的期望关于某参数的梯度进行估计。我们的主要贡献在于，针对常见贝叶斯推断问题中满足的假设条件，证明了所提估计量的强相合性及中心极限定理。证明过程通过为采样器链添加潜信息，并将估计量构造为该增广链的停时尾泛函。我们在贝叶斯敏感性分析与随机游走Metropolis提案分布优化等示例中验证了所提方法的有效性。