Intractable posterior distributions of parameters with intractable normalizing constants depending upon the parameters are known as doubly intractable posterior distributions. The terminology itself indicates that obtaining Bayesian inference from such posteriors is doubly difficult compared to traditional intractable posteriors where the normalizing constants are tractable and admit traditional Markov Chain Monte Carlo (MCMC) solutions. As can be anticipated, a plethora of MCMC-based methods have originated in the literature to deal with doubly intractable distributions. Yet, it remains very much unclear if any of the methods can satisfactorily sample from such posteriors, particularly in high-dimensional setups. In this article, we consider efficient Monte Carlo and importance sampling approximations of the intractable normalizing constant for a few values of the parameters, and Gaussian process interpolations for the remaining values of the parameters, using the approximations. We then incorporate this strategy within the exact iid sampling framework developed in Bhattacharya (2021a) and Bhattacharya (2021b), and illustrate the methodology with simulation experiments comprising a two-dimensional normal-gamma posterior, a two-dimensional Ising model posterior, a two-dimensional Strauss process posterior and a 100-dimensional autologistic model posterior. In each case we demonstrate great accuracy of our methodology, which is also computationally extremely efficient, often taking only a few minutes for generating 10, 000 iid realizations on 80 processors.
翻译:术语本身表明,从这些子宫中获取贝叶斯的推论比传统的难解的后背体要困难得多,因为常态常态常态是可移植的,并接受传统的马可夫链锁蒙特卡洛(MCMCC)解决方案。正如可以预料的那样,许多MCMC方法都源于文献中,以便处理加倍难解的分布。然而,仍然非常不清楚的是,是否有任何方法能够令人满意地从这些后背体中,特别是在高维设置中,从这些后背体中进行抽样。在本篇文章中,我们认为高效的蒙特卡洛和重要抽样近似对一些参数值的难解常态常态的常态,而高斯进程则利用近似值对其余参数值进行相互调。然后,我们将这一战略纳入在Bhattacharya (2021a) 和Bhattattacharya (2021b) 制定的精确的取样框架之中。我们用模拟实验的方法包括两维的正常-伽马马萨马氏-100多级海平面的海平面的测图, 一种两维的模型,我们用来对10维的海平面的海平面的海图进行模拟的模型的模拟的模拟模型的模型。