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Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions

机译：解释具有独立性假设的隐变量模型中联合和边际蒙特卡洛估计的行为

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摘要

In latent variable models parameter estimation can be implemented by using the joint or the marginal likelihood, based on independence or conditional independence assumptions. The same dilemma occurs within the Bayesian framework with respect to the estimation of the Bayesian marginal (or integrated) likelihood, which is the main tool for model comparison and averaging. In most cases, the Bayesian marginal likelihood is a high dimensional integral that cannot be computed analytically and a plethora of methods based on Monte Carlo integration (MCI) are used for its estimation. In this work, it is shown that the joint MCI approach makes subtle use of the properties of the adopted model, leading to increased error and bias in finite settings. The sources and the components of the error associated with estimators under the two approaches are identified here and provided in exact forms. Additionally, the effect of the sample covariation on the Monte Carlo estimators is examined. In particular, even under independence assumptions the sample covariance will be close to (but not exactly) zero which surprisingly has a severe effect on the estimated values and their variability. To address this problem, an index of the sample's divergence from independence is introduced as a multivariate extension of covariance. The implications addressed here are important in the majority of practical problems appearing in Bayesian inference of multi-parameter models with analogous structures.
机译：在潜在变量模型中，可以基于独立性或条件独立性假设，通过使用联合或边际可能性来实现参数估计。关于贝叶斯边际（或综合）可能性的估计，在贝叶斯框架内发生相同的困境，这是模型比较和平均的主要工具。在大多数情况下，贝叶斯边际似然是无法解析计算的高维积分，并且使用了许多基于蒙特卡洛积分（MCI）的方法进行估计。在这项工作中，它表明联合MCI方法巧妙地利用了所采用模型的属性，从而导致有限设置中的误差和偏差增加。此处确定了两种方法下与估计量相关的误差的来源和成分，并以确切的形式提供了误差。此外，还检查了样本协方差对蒙特卡洛估计量的影响。特别是，即使在独立性假设下，样本协方差也将接近（但不完全是）零，这出乎意料地严重影响了估计值及其变异性。为了解决这个问题，引入了样本独立性差异的指数作为协方差的多元扩展。在具有相似结构的多参数模型的贝叶斯推断中出现的大多数实际问题中，此处涉及的含义很重要。

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