Constructing summary statistics from composite likelihoods for approximate Bayesian computation

为了从后验分布生成实现，马尔可夫链蒙特卡罗和顺序蒙特卡罗等方法依赖于似然函数的评估。然而，对于许多复杂的模型，可能性在计算上是难以处理的。近似贝叶斯计算（ABC）和复合似然是有用的工具，可以在似然不可用时进行推理。当数据可以很容易地从模型中模拟出来时，ABC通过测量观测数据和模拟数据之间的相似性，提供了对后验分布的近似，避免了对似然的评估。基本ABC拒绝方案使用从先验分布中采样的参数值来模拟数据，如果观测数据的汇总统计向量与模拟数据之间的距离小于某个预定义的公差，则接受参数值。在ABC中使用的汇总统计的选择可以极大地影响近似推理的质量。如果完全似然不可用，但是对某些数据子集的似然的评估是直接的，那么可以使用复合似然来代替完全似然。复合似然是有效似然项的加权乘积，对应于一组边缘事件或条件事件。复合似然中的权重可以设置为相等并忽略。但是，仔细选择一组权重可以提高统计效率。如果在贝叶斯定理中直接使用复合似然，则后验的可变性往往被大大低估，复合后验过于集中。在贝叶斯定理中，提出了校正复合似然的方法。然而，这种方法可能导致校准的复合后验过于分散。以前的工作结合复合似然和ABC建议使用综合得分作为ABC的汇总统计。要实现这种方法，首先必须选择一个合适的组合似然，包括相关的权重。对于某些模型，可能没有明确的权重选择，或者可能有几个候选的组合可能性可供选择。该项目的目的是：探索从复合似然构建ABC汇总统计的新方法，开发自动选择复合似然中相关权重的构建汇总统计的方法，开发将多个复合似然组合在一起的技术，为汇总统计做出贡献；并最终开发一种彻底的方法，允许在ABC中自动构建汇总统计数据，以便对至少可以定义一种复合似然的模型进行有效和准确的推断。这种新方法的可能应用包括空间极值和时空模型。

To generate realisations from a posterior distribution, methods such as Markov chain Monte Carlo and sequential Monte Carlo rely on evaluation of the likelihood function. However, for many complex models the likelihood is computationally intractable. Approximate Bayesian computation (ABC) and composite likelihoods can be useful tools to allow inference to proceed when the likelihood is not available. When data can easily be simulated from a model, ABC provides an approximation to the posterior distribution, avoiding evaluation of the likelihood, by measuring the similarity between the observed data and simulated data. A basic ABC rejection scheme simulates data using parameter values sampled from the prior distribution, and accepts the parameter values if the distance between vectors of summary statistics for the observed data and simulated data is smaller than some pre-defined tolerance. The choice of summary statistics used within ABC can greatly affect the quality of the approximate inference. If the full likelihood is unavailable, but evaluation of the likelihood for some subsets of the data is straightforward, then a composite likelihood can be used to replace the full likelihood. A composite likelihood is a weighted product of valid likelihood terms, corresponding to a collection of marginal or conditional events. The weights in a composite likelihood can be set equal and ignored. However, a carefully selected set of weights can improve statistical efficiency. If a composite likelihood is used directly in Bayes' theorem the variability in the posterior is often greatly underestimated, with the composite posterior being excessively concentrated. Calibration methods have been proposed to adjust the composite likelihood for use in Bayes' theorem. However, such methods can result in the calibrated composite posterior being too dispersed. Previous work combining composite likelihoods and ABC suggests using the composite score as the summary statistic for ABC. To implement this approach a suitable composite likelihood, including the associated weights, must first be chosen. For some models there may not be a clear choice for the weights, or there may be several candidate composite likelihoods to choose from. The aims of this project are: to explore new methods for constructing summary statistics for ABC from composite likelihoods, to develop methods for constructing summary statistics such that the associated weights in the composite likelihood can be chosen automatically, to develop techniques that combine multiple composite likelihoods for contribution to summary statistics, and to ultimately develop a thorough methodology that will allow automatic construction of a summary statistic to use within ABC to allow efficient and accurate inference for models for which at least one composite likelihood can be defined. Possible applications for this new methodology include spatial extremes and space-time models.