Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology, borrowing ideas from statistical physics and computational chemistry, for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at https://doi.org/10.5281/zenodo.5796148
翻译:在离散时观测的非线性扩散的贝叶斯推论是一件艰巨的任务,它促使人们主要在计算统计界内部开发若干算法。我们提出了一个新的方向和配套方法,从统计物理和计算化学中借用思想,根据对过程的观察,推断潜在扩散路径和模型参数的后方分布; 潜在过程噪音和参数的联合配置,根据观察结果绘制扩散路径,形成一个隐含定义的多元体。然后,通过在嵌入的元体上使用一个受限制的汉密尔顿·蒙特卡洛算法,我们能够对一组离散观测的传播模型进行高效的计算推断。与文献中提议的其他方法相比,我们的方法非常自动化,需要最低限度的用户干预,并在一系列环境中应用类似的方法,包括:液态或低电动系统; 与或无噪音的观测; 线性或非线性观测操作员。探索Markovianity,我们提出了一个方法的变式,该方法在解析路径/分线性解中具有复杂性的线性推算法。961/10号。