In Bayesian inference, we are usually interested in the numerical approximation of integrals that are posterior expectations or marginal likelihoods (a.k.a., Bayesian evidence). In this paper, we focus on the computation of the posterior expectation of a function $f(\x)$. We consider a \emph{target-aware} scenario where $f(\x)$ is known in advance and can be exploited in order to improve the estimation of the posterior expectation. In this scenario, this task can be reduced to perform several independent marginal likelihood estimation tasks. The idea of using a path of tempered posterior distributions has been widely applied in the literature for the computation of marginal likelihoods. Thermodynamic integration, path sampling and annealing importance sampling are well-known examples of algorithms belonging to this family of methods. In this work, we introduce a generalized thermodynamic integration (GTI) scheme which is able to perform a target-aware Bayesian inference, i.e., GTI can approximate the posterior expectation of a given function. Several scenarios of application of GTI are discussed and different numerical simulations are provided.

翻译：在贝叶斯推断中，我们通常关注后验期望或边缘似然（即贝叶斯证据）的数值近似积分计算。本文重点研究函数 $f(\x)$ 后验期望的计算问题。我们考虑一种**目标感知**场景，其中 $f(\x)$ 可预先获知并用于改进后验期望的估计。在此场景下，该任务可转化为多个独立的边缘似然估计任务。利用温度调节后验分布路径的思想在边缘似然计算领域已得到广泛应用，热力学积分、路径采样与退火重要性采样等算法均属于此类方法的典型代表。本研究提出一种广义热力学积分方案，能够实现目标感知的贝叶斯推断，即该方案可近似计算给定函数的后验期望。文中讨论了GTI的多种应用场景，并提供了不同数值模拟结果。