Robust and scalable Markov chain Monte Carlo for heterogeneous models

很大一部分统计推断任务可以被定义为优化或集成问题。马尔可夫链蒙特卡罗（MCMC）算法可以用来解决这两个问题，但最常用的是后者。它们在诸如引力波观测、传染病传播建模以及根据政治民意调查数据预测选举结果等重要而多样的环境中取得了成功。MCMC算法在统计推断之外也很受欢迎，特别是它们在统计物理学中的分子动力学模拟中的广泛使用。尽管它们取得了许多成功，但当前的MCMC算法具有一些已知的缺点。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个相互依赖水平（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可用，并且随着研究人员开始从他们的数据中提出越来越微妙的问题，需要异质模型，因此越来越迫切地需要提高这种性能。在高维环境中天真地做这件事会付出很大的代价（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的数量是二次方）。此外，目前的最先进的算法，如汉密尔顿和朗之万蒙特卡罗工作特别差的技术相结合，最近已经表明，无论是理论和实验由我自己和其他人。在本建议中，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。我将这样做，设计算法的基础上，最近派生的类马尔可夫过程被称为“局部平衡”，有相当多的证据表明，模型的异质性提高了鲁棒性。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的关键理论属性，然后设计新的算法基于这类过程，每个针对特定的应用领域的已知interests.In第二个工作包，我将开发新的理论接地方法的可扩展的自适应预处理算法。我将从高维数据集协方差矩阵稀疏估计的文献中获得灵感。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，如汉密尔顿蒙特卡罗的高维和异构模型setting.There是一个广泛使用的统计软件中集成的新方法的建议内的强烈关注的适用性。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100，000名用户，并详细计划在R和Python等软件中创建定制的开源软件包。我还概述了与数据科学家密切合作的计划，以便在许多突出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.