In Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture-of-parameters approximations that exploit the parametric structure of Bayesian models perform particularly well. Under the continuous ranked probability score, the empirical distribution function is a simple and appealing alternative option.

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基于马尔可夫链蒙特卡罗输出的预测推理

在贝叶斯推理中，预测分布通常采用通过马尔可夫链蒙特卡罗或相关算法生成的样本的形式。在本文中，我们对如何根据此类模拟输出做出和评估概率预测进行了系统分析。基于适当的评分规则，我们开发了一个一致性概念，允许评估用于估计模拟输出基础的平稳分布的方法的充分性。然后，我们提供渐近结果，这些结果解释了贝叶斯后验模拟器的显着特征，并推导出文献中的选择满足我们的一致性概念的条件。重要的是，这些条件取决于所使用的评分规则，因此近似方法和评分规则的选择是交织在一起的。虽然对数规则需要相当严格的条件，但连续排序的概率分数在最小假设下产生一致的近似值。这些结果在模拟研究和经济数据示例中得到了说明。总的来说，利用贝叶斯模型参数结构的混合参数近似表现得特别好。在连续排名概率分数下，经验分布函数是一个简单而有吸引力的替代选项。