Robust and scalable Markov chain Monte Carlo for heterogeneous models

很大一部分统计推断任务可以被框定为优化或集成问题。马尔可夫链蒙特卡罗(MCMC)算法可以用来解决这两个问题，但最常用于后者。它们在重要和多样化的环境中取得了成功，如观测引力波、对传染病的传播进行建模，以及根据政治民调数据预测选举结果。MCMC算法在统计推断之外也很受欢迎，特别是在统计物理中的分子动力学模拟中得到了广泛的应用。一个突出的例子是，当模型参数在非常不同的尺度上变化并显示出多个水平的相互依赖(异类模型)时，它们的表现。随着海量和高度异质的数据集变得越来越多，以及研究人员开始从他们的数据中提出越来越多需要异质模型的细微差别的问题，人们越来越迫切地需要提高这种性能。针对异质模型的MCMC的标准方法是通过算法的自适应预条件处理。在高维设置中幼稚地执行此操作会付出巨大的代价(每个算法步骤所需的运算次数通常是模型参数数量的立方，而要学习的算法调优参数的数量是二次)。此外，当前最先进的算法，如哈密尔顿和朗之万蒙特卡洛，与该技术结合使用时尤其糟糕，正如我和其他人最近在理论和实验上证明的那样。在这项提案中，我将从两个方面解决这个问题。在第一个工作包中，我将开发和研究一套专门为异类模型量身定做的新的MCMC算法。我将通过设计基于最近派生的马尔可夫过程类的算法来实现这一点，这类马尔可夫过程被称为“局部平衡”，有相当多的证据表明，对模型异质性的稳健性得到了改善。我将为这类马尔可夫过程提供一个严格的基础，建立关于收敛到均衡和最优性的关键理论性质，然后基于这类过程设计新的算法，每个算法都针对已知兴趣的特定应用领域而定制。在第二个工作包中，我将为算法的可伸缩自适应预条件处理开发新的理论基础方法。我将通过从高维数据集的协方差矩阵稀疏估计的文献中获得灵感来部分地做到这一点。我将设计既可扩展到高维设置，又可建立理论保证的方法，以提供预期性能收益的明确指示。这将提高现有最先进的方法，如哈密顿蒙特卡罗对高维和异质模型设置的适用性。在该提案中，重点是将新方法整合到广泛使用的统计软件中。为此，我计划与‘Stan’统计编程语言的创始开发人员进行合作，以及详细计划在R和Python等软件中创建定制的开放源码包。我还概述了与数据科学家密切合作的计划，以便在许多重要的应用领域应用新方法。

A large proportion of statistical inference tasks can be framed as either an optimisation or an integration problem. Markov chain Monte Carlo (MCMC) algorithms can be used to solve both, but are most commonly used for the latter. They have been successful in such important and diverse settings as the observation of gravitational waves, modelling the spread of infectious diseases, and predicting the results of elections from political polling data. MCMC algorithms are also popular outside of statistical inference, in particular their use is widespread for molecular dynamics simulations in statistical physics.Despite their numerous successes, current MCMC algorithms has some known drawbacks. A prominent example is their performance when model parameters vary over very different scales and exhibit multiple levels of inter-dependence (heterogeneous models). There is an increasingly urgent need to improve this performance as high volume and highly heterogeneous datasets become more and more available, and as researchers begin to ask progressively more nuanced questions from their data for which heterogenous models are needed.The standard approach to MCMC for heterogeneous models is through adaptive pre-conditioning of algorithms. Doing this naively in a high-dimensional setting comes at a significant cost (the required number of operations per algorithm step is often cubic in the number of model parameters, and the number of algorithmic tuning parameters to learn is quadratic). In addition, current state of the art algorithms such as Hamiltonian and Langevin Monte Carlo work particularly poorly in combination with the technique, as has recently been shown both theoretically and experimentally by myself and others. In this proposal I will attack this problem on two fronts. In the first work package I will develop and study a new suite of MCMC algorithms that are specifically tailored to heterogeneous models. I will do this by designing algorithms based on the recently derived class of Markov processes termed 'locally-balanced', for which there is considerable evidence of improved robustness to model heterogeneity. I will provide a rigorous foundation for this class of Markov processes, establish key theoretical properties on convergence to equilibrium and optimality, and then design new algorithms based on this class of processes, each tailored towards specific application areas of known interest.In the second work package I will develop new theoretically grounded methodology for scalable adaptive pre-conditioning of algorithms. I will do this in part by taking inspiration from the literature on sparse estimation of covariance matrices for high-dimensional datasets. I will design methods that are both scalable to high-dimensional settings and for which theoretical guarantees can be established, to provide a clear indication of expected performance gains. This should improve the applicability of existing state of the art methods such as Hamiltonian Monte Carlo to the high-dimensional and heterogeneous model setting.There is a keen focus on integrating new methods within widely used statistical software within the proposal. To this end, I have planned collaborations with the founding developers of the 'Stan' statistical programming language, which has over 100,00 users, as well as detailed plans to create bespoke open source packages in software such as R and Python. I also outline plans to work closely with data scientists to apply the new methodology in many prominent application areas.