Many modern statistical applications involve inference for complicated stochastic models for which the likelihood function is difficult or even impossible to calculate, and hence conventional likelihood-based inferential techniques cannot be used. In such settings, Bayesian inference can be performed using Approximate Bayesian Computation (ABC). However, in spite of many recent developments to ABC methodology, in many applications the computational cost of ABC necessitates the choice of summary statistics and tolerances that can potentially severely bias the estimate of the posterior.We propose a new “piecewise” ABC approach suitable for discretely observed Markov models that involves writing the posterior density of the parameters as a product of factors, each a function of only a subset of the data, and then using ABC within each factor. The approach has the advantage of side-stepping the need to choose a summary statistic and it enables a stringent tolerance to be set, making the posterior “less approximate”. We investigate two methods for estimating the posterior density based on ABC samples for each of the factors: the first is to use a Gaussian approximation for each factor, and the second is to use a kernel density estimate. Both methods have their merits. The Gaussian approximation is simple, fast, and probably adequate for many applications. On the other hand, using instead a kernel density estimate has the benefit of consistently estimating the true piecewise ABC posterior as the number of ABC samples tends to infinity. We illustrate the piecewise ABC approach with four examples; in each case, the approach offers fast and accurate inference.

中文翻译：

---

分段近似贝叶斯计算：使用分解后验分布对离散观察的马尔可夫模型进行快速推断。

许多现代统计应用涉及对复杂随机模型的推理，其似然函数难以甚至不可能计算，因此不能使用传统的基于似然的推理技术。在这种情况下，可以使用近似贝叶斯计算 (ABC) 执行贝叶斯推理。然而，尽管 ABC 方法最近取得了许多进展，但在许多应用中，ABC 的计算成本需要选择汇总统计量和容差，这可能会严重影响后验的估计。我们提出了一种新的“分段”ABC 方法，适用于离散观察的马尔可夫模型，涉及将参数的后验密度写为因子的乘积，每个因子仅是数据子集的函数，然后在每个因子中使用 ABC。该方法具有回避选择汇总统计量的需要的优势，并且可以设置严格的容差，使后验“不太近似”。我们研究了两种基于每个因子的 ABC 样本估计后验密度的方法：第一种是对每个因子使用高斯近似，第二种是使用核密度估计。两种方法都有其优点。高斯近似简单、快速，并且可能适用于许多应用。另一方面，使用核密度估计的好处是一致地估计真正的分段 ABC 后验，因为 ABC 样本的数量趋于无穷大。我们用四个例子来说明分段 ABC 方法；在每种情况下，该方法都提供了快速准确的推理。