Implementing Bayesian inference is often computationally challenging in applications involving complex models, and sometimes calculating the likelihood itself is difficult. Synthetic likelihood is one approach for carrying out inference when the likelihood is intractable, but it is straightforward to simulate from the model. The method constructs an approximate likelihood by taking a vector summary statistic as being multivariate normal, with the unknown mean and covariance matrix estimated by simulation for any given parameter value. Our article makes three contributions. The first shows that if the summary statistic satisfies a central limit theorem, then the synthetic likelihood posterior is asymptotically normal and yields credible sets with the correct level of frequentist coverage. This result is similar to that obtained by approximate Bayesian computation. The second contribution compares the computational efficiency of Bayesian synthetic likelihood and approximate Bayesian computation using the acceptance probability for rejection and importance sampling algorithms with a "good" proposal distribution. We show that Bayesian synthetic likelihood is computationally more efficient than approximate Bayesian computation, and behaves similarly to regression-adjusted approximate Bayesian computation. Based on the asymptotic results, the third contribution proposes using adjusted inference methods when a possibly misspecified form is assumed for the covariance matrix of the synthetic likelihood, such as diagonal or a factor model, to speed up the computation. The methodology is illustrated with some simulated and real examples.
翻译:在涉及复杂模型的应用中,执行贝耶斯的推论往往具有计算上的挑战性,有时还难以计算概率本身。合成可能性是当可能性难以捉摸时进行推论的一种方法,但从模型中可以简单模拟。该方法将矢量摘要统计的计算效率设定为多变性正常,以任何给定参数值的模拟估算出未知的平均值和共变矩阵。我们的文章做出了三项贡献。第一点表明,如果摘要统计满足一个中心限值,合成可能性后继体则过于正常,产生具有正确频率覆盖水平的可靠数据集。这一结果与近似贝耶斯计算得出的结果相似。第二个贡献将巴耶斯合成概率的计算效率与使用拒绝和重要取样算法的接受概率和任何特定参数值的“良好”分布相比。我们发现,贝耶斯合成可能性的计算效率比近似于近似贝亚斯计算值,而合成可能性的后继体概率则与精确度测算结果相似。根据亚伊斯频繁度测测算结果,在假设的模型和精确度模型模型分析中,可能采用某种误算法的模型分析方法时,即采用某种模拟模型推算法的模型推算法。