Dirichlet forms and Markov Chain Monte Carlo

狄利克雷形式和马尔可夫链蒙特卡罗

基本信息

批准号：
EP/R022100/1
负责人：
Wilfrid Kendall
金额：
$ 42.86万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR022100%2F1
关键词：
Dirichlet forms Markov Chain Monte

项目摘要

Consider the problem of measuring the amount of material in a heap of sand. In effect one has to calculate the total volume lying under a surface extending over a given region. If the surface is given by a formula then it is often feasible to calculate the volume using integral calculus. However, the calculations may be demanding. Moreover, there may be no formula; it may only be possible to determine the heights experimentally at specified locations. In that case numerical approximations can be made, but this may be tricky if the surface is very irregular. This issue becomes even more problematic for analogous problems in higher dimensions.In practice it is often preferable to employ a statistical method: one measures the heights at randomly chosen points, drawn uniformly from the region, and then averages the resulting measurements. Multiplied by the area of the region, this gives a statistical estimate of the volume, the accuracy of which increases as one samples more randomly chosen points. This method works well if the region in question is relatively simple, and is colourfully known as the Monte Carlo method.Even the task of drawing random points from the region can be challenging; and may be overwhelmingly difficult for important higher dimensional analogues. So instead one designs a random sequence of points which, in the long run, are individually uniformly distributed. A simple example is to propose each point in turn as a random displacement from the previous point (using the normal distribution), rejecting the proposal if it falls outside the region. The first point is chosen arbitrarily, so the first few locations are biased, but this goes away in the long run. The average of measurements, multiplied by the area of the region, now yields a statistical estimate of the volume whose accuracy improves as more points are used. There are many variations on this idea, often using more general random processes such as suitable Markov chains, but the above illustrates the important ideas.This technique is known as Markov chain Monte Carlo (MCMC), and is now used in a wide variety of different contexts, for example: statistical physics (the original context in which MCMC arose), randomised computer algorithms, study of bio-molecules, numerical integration, and (the focus of this project) applied statistics. In each case a different set of concerns lead to different emphases and different approaches.In applied statistics one deals with probabilities of events rather then volumes of material, and with calculating long run means of different functions relating to probability distribution conditional on observations. However in essence the problem is as above. Issues include: "burn-in" (how long must a long-run average be, to avoid being biased by the choice of the first point) and "fast mixing" (how quickly does the sequence of sample points move around the sample space once equilibrium is attained). The project focuses on the question of fast mixing, and the question of how to choose the right kind of jump proposal to make mixing fast, and specifically the question of the best scale for the distribution of proposed displacements.In 1997 Roberts et al produced theory giving clear practical advice on tuning the scale: in high dimensional situations approximately only a quarter of the proposals should be accepted. Much subsequent work has explored generalisations. However the method of proof, using stochastic differential equation (SDE) theory, has limitations. In a nutshell, it uses limits of random walks rapidly making small jumps, leading to severe smoothness restrictions. Recently it has been proposed to use "Dirichlet forms", based on averages of squares of jumps, thus not so restrictive. Additionally, the SDE approach works with a single aspect of the target probability distribution, while Dirichlet forms work with everything at once. The project will develop the Dirichlet form approach.

考虑测量一堆沙子中材料量的问题。实际上，必须计算出在给定区域延伸的表面下的总体积。如果表面由公式给出，则通常可以使用积分计算来计算体积。但是，计算可能要求。而且，可能没有公式。只能在指定位置实验确定高度。在这种情况下，可以进行数值近似值，但是如果表面非常不规则，这可能很棘手。对于更高维度的类似问题，这个问题变得更加有问题。在实践中，通常可以采用统计方法：一个人在随机选择点处测量高度，从该区域均匀地绘制，然后平均得出的测量值。乘以该区域的面积，给出了体积的统计估计，其准确性随着一个样本而更随机选择的点而增加。如果所讨论的区域相对简单，并且被称为Monte Carlo方法，则此方法可以很好地工作。即使从该地区绘制随机点的任务也可能具有挑战性。对于重要的更高维度类似物来说，可能非常困难。因此，相反，人们设计了一个随机的点序列，从长远来看，该点是单独分布的。一个简单的示例是依次提出每个点作为从上点开始的随机位移（使用正态分布），如果该提案落在区域之外，则拒绝该建议。第一个点是任意选择的，因此前几个位置是有偏见的，但是从长远来看，这消失了。现在，测量的平均值乘以该区域的面积，现在得出了量的统计估计值，其精度随着更多的点而提高。 There are many variations on this idea, often using more general random processes such as suitable Markov chains, but the above illustrates the important ideas.This technique is known as Markov chain Monte Carlo (MCMC), and is now used in a wide variety of different contexts, for example: statistical physics (the original context in which MCMC arose), randomised computer algorithms, study of bio-molecules, numerical integration, and (the focus of this项目）应用统计。在每种情况下，一组不同的关注点会导致不同的重点和不同的方法。在应用统计中，一个人涉及事件的概率，而不是材料的量，并计算出与观测值有条件的概率分布有关的不同功能的长期手段。但是，从本质上讲，问题如上所述。问题包括：“燃烧”（要避免选择第一个点的选择偏见）和“快速混合”（一旦达到平衡，必须绕过样品空间的序列，必须多长时间。该项目的重点是快速混合的问题，以及如何选择正确的跳跃提案以使混合的问题的问题，尤其是最佳的问题，即提议流离失所的分布的最佳规模问题。随后的许多工作探索了概括。但是，使用随机微分方程（SDE）理论的证明方法具有局限性。简而言之，它使用随机步行的限制，迅速进行小跳跃，从而导致严重的平滑度限制。最近，有人提议根据跳跃的平均值来使用“ dirichlet形式”，因此并不是那么限制。此外，SDE方法与目标概率分布的一个方面一起使用，而Dirichlet形成的形式同时使用所有内容。该项目将开发Dirichlet形式的方法。