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On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods

机译：关于具有难以处理的似然似然的贝叶斯推理的俄罗斯轮盘赌估计。

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A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approximations to the posterior have been suggested, and these are reviewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors. In addition, taking inspiration from the Physics literature, we study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher Bingham distributions on the d-Sphere and a large-scale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.
机译：大量统计模型是“双重难处理的”：可能性归一化项以及模型的边际可能性都是难解的，它是模型参数的函数。这意味着不能使用从后部采样的标准推理技术，例如马尔可夫链蒙特卡洛（MCMC）。示例包括但不限于大规模高斯马尔可夫随机场，自物流模型和指数随机图模型。已经提出了许多基于MCMC技术的近似方案，近似贝叶斯计算（ABC）或对后验的解析近似，这里对它们进行了综述。精确的MCMC方案，可以应用于双难分布的子集，也已经开发出来并在本文中进行了描述。迄今为止，尚不存在可应用于具有双重难处理后代的所有类别模型的通用方法。此外，从物理学文献中汲取灵感，我们研究了一种将难解似然表示为无限级数的替代方法。然后，可以通过俄罗斯轮盘赌抽样，通过有限时间随机截断该系列，来获得似然估计值的无偏估计，尽管该估计不一定是正数。利用量子色动力学文献的结果，可以在伪边际MCMC方案中使用可能的负估计，从而保留对后验分布的期望。对该方法进行了综述，例如Ising模型中的参数，d球上Fisher Bingham分布的后验以及描述臭氧柱数据的大规模高斯马尔可夫随机场模型。这导致对该方法的优缺点进行了严格的评估，并指出了正在进行的研究。

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