State Space Models: A New Look at Smoothing, Parameter Inference, and Model Choice

时间序列是对一段时间内物体的重复观察；例如，金融资产的每小时价格、海洋中浮游生物的每日种群数量，或疾病爆发期间的每月病例数量。所有这些例子通常使用状态空间模型来研究，在状态空间模型下，观测值被视为未观察到的随机过程的噪声测量。可以使用被认为与手头的对象相关的任何方程来模拟潜在的过程，但灵活性是以计算成本为代价的。例如，如果使用非线性微分方程式对过程进行建模，则相关的统计计算可能会变得难以进行。对于长和高维的时间序列来说，这一点尤其严重。该项目介绍了状态空间模型中三个重要问题的新方法：如何估计未观测过程，如何估计模型参数，以及如何比较多个模型。这些历史问题将在并行计算硬件、具有大量缺失值的大型数据集以及有关参数的有限先验信息的背景下重新讨论。正在开发的工具箱将参与一项全球努力，量化时间序列科学模型的不确定性。状态空间模型形成了一类非常灵活的时间序列模型，在该模型下，假设观测值在给定潜在随机过程的情况下是条件独立的。该项目涉及这些模型中的三个基本问题：潜在过程估计、参数估计和模型选择。对于潜在过程估计(1)，提出并研究了平滑期望的无偏估计。该方法依赖于粒子过滤器的一种新的耦合，使用了最大耦合和最优传输的思想。无偏特性允许估计过程的完全并行化，并构造可靠的置信度区间。对于参数估计(2)，可以利用粒子过滤器的类似耦合来显著降低分数估计器和Metropolis接受率的方差，特别是对于长时间序列。对于模型比较(3)，在预测序贯方法之后研究了一种新的准则。它是为参数的先验信息有限的情况而设计的，与贝叶斯因子相反。该项目旨在帮助状态空间模型的当前用户，例如在计量经济学、遗传学、神经科学、生态学和海洋生物地质化学领域，并帮助目前可用的方法不足的科学家。在更基本的层面上，将耦合和最优传输的思想用于计算方法是独立的兴趣，并为广泛的新采样和优化方法铺平了道路。

Time series are repeated observations of objects over time; for example, hourly prices of a financial asset, daily population sizes of plankton in the ocean, or monthly case counts during a disease outbreak. All of these examples are routinely studied using state space models, under which observations are treated as noisy measurements of an unobserved stochastic process. The underlying process can be modeled using any equation considered relevant for the object at hand, but flexibility comes at a computational cost. For instance, if the process is modeled using nonlinear differential equations, the associated statistical computations might become prohibitive. This is particularly acute for long and high-dimensional time series. This project introduces new methods for three important questions regarding state space models: how to estimate the unobserved process, how to estimate the model parameters, and how to compare multiple models. These historical questions will be revisited in the contexts of parallel computing hardware, of large datasets with vast amounts of missing values, and of limited prior information on the parameters. The toolbox under development will participate in a global effort to quantify uncertainty in scientific models for time series. It will be implemented in efficient and user-friendly software packages, made publicly available during the project.State space models form a very flexible class of time series models, under which observations are assumed conditionally independent given a latent stochastic process. The project concerns three fundamental questions in these models: latent process estimation, parameter estimation and model choice. For latent process estimation (1), an unbiased estimator of smoothing expectations is developed and investigated. The approach relies on a novel coupling of particle filters, using maximal couplings and ideas from optimal transport. The unbiasedness property allows a complete parallelization of the estimation procedure, and the construction of reliable confidence intervals. For parameter estimation (2), similar couplings of particle filters can be leveraged to drastically reduce the variance in score estimators and in Metropolis acceptance ratios, especially for long time series. For model comparison (3), a novel criterion is investigated following a predictive sequential approach. It is designed for situations where prior information on parameters is limited, contrarily to Bayes factors. The project aims at helping current users of state-space models, for instance in econometrics, genetics, neuroscience, ecology, and marine biogeochemistry, and to assist scientists for whom currently available methods are insufficient. On a more fundamental level, leveraging ideas from coupling and optimal transport for computational methods is of independent interest and paves the way to a wide range of new sampling and optimization methods.