This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X, theta)f(X | theta)f(theta), where Y is the (set of) observed data, theta is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X((t+1)) | X((t)), theta), where X((t)) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given theta, the transition model f(X((t+1)) | X((t)), theta) is known but the distribution of the stochastic process in equilibrium, that is f(X | theta), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time.We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model.As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.

中文翻译：

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潜在均衡过程下的模型拟合和推断。

本文介绍了一种在类型为f（Y | X，theta）f（X | theta）f（theta）的贝叶斯模型的背景下进行模型拟合和推断的方法，其中Y是观测数据（theta）的集合是一组模型参数，并且X是由一阶转换模型f（X（（t + 1））| X（（t）），theta）引起的不可观察的（潜在）平稳随机过程，其中X（（t ））表示时间（或生成时间）t处的过程状态。上述类型模型的关键特征是，给定theta，过渡模型f（X（（t + 1））| X（（t）），theta是已知的，但是随机过程在平衡状态下的分布， f（X | theta）除外，在非常特殊的情况下很难处理，因此未知。还要注意的另一点是，假设基本过程处于平衡状态时，可以观察到数据Y。换一种说法，数据不是随时间动态收集的。我们将此规范称为潜在平衡过程（LEP）模型。它受种群遗传学问题的启发（尽管讨论了其他应用），在给定一个或多个基因座等位基因频率的样本的情况下，有兴趣了解诸如突变和迁移率以及种群大小等参数是很有意义的。在这样的问题中，自然可以假设观察到的等位基因频率的分布取决于真实的（未观察到的）种群等位基因频率，而真实的等位基因频率的分布仅通过过渡模型间接指定。使LEP适合贝叶斯框架是很自然的。通常通过马尔可夫链蒙特卡洛（MCMC）拟合此类模型。但是，我们证明了这一点，对于LEP模型，MCMC的实现远非简单易行。本文的主要贡献是提供一种为LEP模型实施MCMC的方法。我们用模拟和真实数据集展示了我们在人口遗传学问题中的方法。结果模型拟合的计算量很大，因此，在特殊情况下，我们还将讨论该过程的并行实现。