Markov chain Monte Carlo methods for exponential family models with intractable normalizing constant, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian process surrogate model for the likelihood function, which jointly models the sample means and variances of the sufficient statistics, and uses warping functions to capture covariance nonstationarity in the input parameter space. We show that both the consideration of nonstationarity and the inclusion of gradient information can be leveraged to obtain a surrogate model that outperforms the conventional stationary Gaussian process surrogate model when making inference, particularly in regions where the likelihood function exhibits a phase transition. We also show that the proposed surrogate model can be used to improve the effective sample size per unit time when embedded in exact inferential algorithms. The utility of our approach in speeding up inferential algorithms is demonstrated on simulated and real-world data.

翻译：马尔可夫链蒙特卡洛方法在处理具有难解归一化常数的指数族模型时（如交换算法），需要在马尔可夫链的每次迭代中模拟充分统计量，这往往导致昂贵的计算成本。为此，研究人员开发了似然函数的代理模型以加速此类推断算法。然而，现有代理模型通常灵活性不足，且对真实似然函数的逼近效果较差。本文提出一种扭曲梯度增强高斯过程代理模型用于似然函数建模，该模型联合建模充分统计量的样本均值和方差，并通过扭曲函数捕捉输入参数空间中的协方差非平稳性。研究表明，通过利用非平稳性和梯度信息，所提出的代理模型在推断性能上优于传统平稳高斯过程代理模型，尤其在似然函数表现出相变特性的区域。此外，该代理模型嵌入精确推断算法后，可有效提升单位时间的有效样本量。最后，通过模拟数据和真实数据验证了该方法加速推断算法的实用性。