A Simple Approach to Parameter Inference in State-Space Models

状态空间模型是许多科学领域的流行工具，包括经济学，生物学或生态学，主要用于从可用数据中预测未观察到的（或潜在的）时间序列。例如，在经济学中，我们通常对从价格中推断资产的风险感兴趣，而在生态学中，人们可能希望使用一些观察来估计给定动物群体数量的演变。与任何统计模型一样，状态空间模型依赖于需要从数据中学习的参数。然而，已知这类模型中的参数推断是具有挑战性的，因为计算相关的对数似然函数或其梯度需要整合出潜在变量，这通常是棘手的任务。粒子滤波算法已被证明是用于估计关于未观察到的变量的期望的强大的蒙特卡罗技术，并且用于状态空间模型中的参数推断的最新方法相当于在“标准”算法（诸如用于贝叶斯推断的Metropolis-Hastings算法，或用于最大似然估计的梯度上升算法）内使用这些蒙特卡罗估计。尽管这些方法的不可否认的有用性，他们通常是难以实现和调整的非专家在蒙特卡洛方法，除了计算昂贵，即使是中等样本量。因此，在实践中，更简单但理论上不合理的算法往往是首选从数据中学习模型参数。这些更简单的方法中最流行的可能是20年前由Liu和West（2001）提出的方法，其中未知的模型参数被简单地视为额外的潜在变量。如果这种策略的优点是简单的实现，但它的缺点是没有得到任何理论结果的支持。根据Liu和West（2001）的思想，本研究的目标是提出一种理论上合理的方法，将状态空间模型的参数作为潜变量来处理，以了解其值。为了实现这一目标，本研究将建立在Gerber和Douc（2021）开发的静态模型参数估计方法的基础上，并利用部分观测马尔可夫模型中出现的贝叶斯后验分布的浓度特性的结果（见Douc等人，2020年）。通过提供一种易于实现但理论上合理的方法，在广泛使用的状态空间模型类参数推断，研究将有利于广泛的研究人员和从业者的科学结论和预测依赖于这些模型。 该项目属于EPSRC统计和应用概率研究领域的福尔斯。

State-space models are popular tools in many areas of science, including economics, biology or ecology, and are mainly used to predict an unobserved (or latent) time series of interest from the available data. For instance, in economics we are often interested in inferring the risk of an asset from its price, while in ecology one may want to use some observations to estimate the evolution of the population size of a given group of animals. As in any statistical models, state-space models depend on parameters that need to be learnt from the data. Parameter inference in this class of models is however known to be challenging, because computing the associated log-likelihood function, or its gradient, requires to integrate out the latent variables, a task which is usually intractable. Particle filter algorithms have proven to be powerful Monte Carlo techniques for estimating expectations with respect to the unobserved variables, and state-of the art methods for parameter inference in state-space models amount to using these Monte Carlo estimates within "standard" algorithms (such as the Metropolis-Hastings algorithm for Bayesian inference, or the gradient ascent algorithm for maximum likelihood estimation). Despite the undeniable usefulness of these approaches, they are typically difficult to implement and to tune for non-experts in Monte Carlo methods, in addition to be computationally expensive even for moderate sample sizes. Consequently, in practice, simpler but theoretically unjustified algorithms are often preferred to learn the model parameter from the data. The most popular of these simpler approaches is probably the one proposed 20 years ago by Liu and West (2001), in which the unknown model parameter is simply treated as an additional latent variable. If this strategy has the merit to be simple to implement it however has the drawback of not being supported by any theoretical results. Following the idea of Liu and West (2001), the objective of this research is to propose a theoretically justified way of treating the parameter of a state-space model as a latent variable in order to learn its value. To achieve this goal this research will build on the approach developed by Gerber and Douc (2021) for parameter estimation in static models, and leverage results on the concentration properties of the Bayesian posterior distributions that arise in partially observed Markov models (see Douc et al., 2020). By providing an easy to implement but theoretically justified approach to parameter inference in the widely used class of state-space models, the research will benefit to a broad range of researchers and practitioners whose scientific conclusions and predictions rely on these models. This project falls within the EPSRC Statistics and Applied Probability research area.