Abstract

Approximate Bayesian computation (ABC) is a popular technique for approximating likelihoods and is often used in parameter estimation when the likelihood functions are analytically intractable. In the context of Hidden Markov Models (HMMs), we analyze the asymptotic behavior of the posterior distribution in ABC based Bayesian parameter estimation. In particular we show that Bernstein-von Mises type results still hold but that the resulting posterior is biased in the sense that it concentrates around a point in parameter space that differs from the true parameter value. Furthermore we obtain precise rates for the size of this bias with respect to a natural accuracy parameter of the ABC method. Finally we discuss, via a numerical example, the implications of our results for the practical implementation of ABC.

摘要

近似贝叶斯计算 (ABC) 是一种用于近似似然的流行技术，并且当似然函数在分析上难以处理时，通常用于参数估计。在隐马尔可夫模型 (HMM) 的背景下，我们分析了基于 ABC 的贝叶斯参数估计中后验分布的渐近行为。特别是，我们表明 Bernstein-von Mises 类型的结果仍然成立，但结果后验是有偏差的，因为它集中在参数空间中与真实参数值不同的点周围。此外，我们根据 ABC 方法的自然精度参数获得了这种偏差大小的精确比率。最后，我们通过一个数值例子讨论了我们的结果对 ABC 实际实施的影响。