Robust and scalable Markov chain Monte Carlo for heterogeneous models

很大一部分统计推理任务可以被定义为优化问题或集成问题。马尔可夫链蒙特卡罗（MCMC）算法可用于解决这两个问题，但最常用的是后者。它们在诸如观测引力波、建立传染病传播模型以及根据政治投票数据预测选举结果等重要和多样化的环境中取得了成功。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广泛使用。尽管取得了许多成功，但目前的MCMC算法仍有一些已知的缺点。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表现出多级相互依赖（异构模型）时，它们的性能。随着大容量和高度异构的数据集变得越来越多，研究人员开始从他们的数据中提出越来越细微的问题，这些问题需要异构模型，因此提高这种性能的需求越来越迫切。异构模型MCMC的标准方法是通过算法的自适应预处理。在高维设置中天真地这样做会付出巨大的代价（每个算法步骤所需的操作数量通常是模型参数数量的三次方，而要学习的算法调优参数数量是二次的）。此外，目前最先进的算法，如哈密顿算法和朗格万蒙特卡罗算法，在与该技术结合时效果特别差，正如我和其他人最近在理论和实验中所显示的那样。在这个建议中，我将从两个方面着手解决这个问题。在第一个工作包中，我将开发和研究一套新的MCMC算法，专门针对异构模型进行定制。我将通过设计基于最近导出的一类称为“局部平衡”的马尔可夫过程的算法来做到这一点，因为有大量证据表明模型异质性的鲁棒性得到了改善。我将为这类马尔可夫过程提供一个严格的基础，建立收敛到平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进行定制。在第二个工作包中，我将为算法的可扩展自适应预处理开发新的理论基础方法。我将从高维数据集的协方差矩阵稀疏估计的文献中获得部分灵感。我将设计既可扩展到高维设置，又可以建立理论保证的方法，以提供预期性能增益的明确指示。这将提高现有的最先进的方法，如哈密顿蒙特卡罗方法对高维和异构模型设置的适用性。在提案中广泛使用的统计软件中集成新方法是一个敏锐的焦点。为此，我计划与“Stan”统计编程语言的创始开发人员合作，该语言拥有超过10万用户，以及在R和Python等软件中创建定制开源包的详细计划。我还概述了与数据科学家密切合作的计划，以便将新方法应用于许多突出的应用领域。

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.