Approximate Bayesian computation (ABC) has become one of the major tools of likelihood-free statistical inference in complex mathematical models. Simultaneously, stochastic differential equations (SDEs) have developed to an established tool for modelling time-dependent, real-world phenomena with underlying random effects. When applying ABC to stochastic models, two major difficulties arise: First, the derivation of effective summary statistics and proper distances is particularly challenging, since simulations from the stochastic process under the same parameter configuration result in different trajectories. Second, exact simulation schemes to generate trajectories from the stochastic model are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To obtain summaries that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method (e.g. the choice of the summary statistics) on the underlying structural properties of the model. Here, we focus on the existence of an invariant measure and we map the data to their estimated invariant density and invariant spectral density. Then, to ensure that these model properties are kept in the synthetic data generation, we adopt measure-preserving numerical splitting schemes. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian type SDEs, both with simulated data and with real electroencephalography data. The derived summaries are particularly robust to the model simulation, and this fact, combined with the proposed reliable numerical scheme, yields accurate ABC inference. In contrast, the inference returned using standard numerical methods (Euler–Maruyama discretisation) fails. The proposed ingredients can be incorporated into any type of ABC algorithm and directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.

中文翻译：

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基于光谱密度和保留度量的ABC，用于部分观察到的扩散过程。哈密​​顿SDE的图解

近似贝叶斯计算（ABC）已成为复杂数学模型中无可能性统计推断的主要工具之一。同时，随机微分方程（SDE）已发展成为一种用于建模具有潜在随机效应的时间相关的现实世界现象的工具。将ABC应用于随机模型时，会遇到两个主要困难：第一，有效汇总统计数据和适当距离的推导尤其具有挑战性，因为在相同参数配置下对随机过程进行的仿真会产生不同的轨迹。第二，从随机模型生成轨迹的精确仿真方案很少，需要为合成数据生成推导合适的数值方法。为了获得对模型的内在随机性较不敏感的摘要，我们建议针对模型的基础结构属性建立统计方法（例如，选择摘要统计）。在这里，我们关注不变度量的存在，并将数据映射到它们的估计不变密度和不变谱密度。然后，为了确保在合成数据生成中保留这些模型属性，我们采用了保留度量的数值拆分方案。在部分观察到的哈密顿型SDE的广泛类别中，通过模拟数据和真实脑电图数据，都说明了基于属性的，保存度量的ABC方法。导出的摘要对于模型仿真特别健壮，因此，结合提出的可靠数值方案，可以得出准确的ABC推论。相反，使用标准数值方法（Euler-Maruyama离散化）返回的推论失败。所提出的成分可以被合并到任何类型的ABC算法中，并直接应用于以不变分布为特征的所有SDE，并且可以推导出保量数值方法。