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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.

考虑一下测量一堆沙子中物质的数量的问题。实际上，人们必须计算在给定区域上延伸的表面下的总体积。如果表面由公式给出，那么通常可以使用积分来计算体积。然而，计算可能是苛刻的。此外，可能没有公式;可能只能在特定位置通过实验确定高度。在这种情况下，可以进行数值近似，但如果表面非常不规则，这可能很棘手。在更高维度的类似问题中，这个问题变得更加棘手。在实践中，通常最好采用统计方法：在随机选择的点上测量高度，从区域中均匀地抽取，然后对所得测量结果进行平均。乘以该区域的面积，这给出了体积的统计估计，其准确性随着随机选择的点的采样而增加。这种方法适用于相对简单的区域，被称为蒙特卡罗方法。甚至从区域中随机抽取点的任务也可能具有挑战性;对于重要的高维类似物可能非常困难。因此，人们设计了一个随机的点序列，从长远来看，这些点是均匀分布的。一个简单的例子是依次提出每个点作为前一个点的随机位移（使用正态分布），如果它福尔斯在区域之外，则拒绝该建议。第一个点是任意选择的，所以前几个位置是有偏差的，但从长远来看，这会消失。测量的平均值乘以区域的面积，现在产生体积的统计估计，其精度随着使用更多点而提高。这个想法有很多变化，通常使用更一般的随机过程，如合适的马尔可夫链，但上面说明了重要的想法。这种技术被称为马尔可夫链蒙特卡罗（MCMC），现在被用于各种不同的背景下，例如：统计物理（MCMC产生的最初背景），随机计算机算法，生物分子研究，数值积分和（该项目的重点）应用统计。在每一种情况下，不同的关注点导致不同的重点和不同的方法。在应用统计学中，人们处理事件的概率而不是材料的体积，并计算与观测条件下的概率分布有关的不同函数的长期平均值。然而，问题的实质是如上所述。问题包括：“老化”（为了避免第一个点的选择而产生偏差，长期平均值必须有多长）和“快速混合”（一旦达到平衡，样本点序列在样本空间中移动的速度有多快）。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建议，使混合快速的问题，特别是建议displacement.In 1997年罗伯茨等人产生的理论，给出明确的实际建议调整规模：在高维的情况下，大约只有四分之一的建议应被接受的分布的最佳规模的问题。随后的许多工作探索了普遍性。然而，使用随机微分方程理论的证明方法具有局限性。简而言之，它使用了快速进行小跳跃的随机游走的限制，导致了严重的平滑限制。最近有人提出使用“狄利克雷形式”，基于跳跃平方的平均值，因此没有那么严格。此外，Dirichlet方法适用于目标概率分布的单个方面，而Dirichlet形式同时适用于所有方面。该项目将开发狄利克雷形式的方法。