Dirichlet forms and Markov Chain Monte Carlo

考虑一下测量一堆沙子中的材料数量的问题。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。如果表面是用公式给出的，那么用积分学计算体积通常是可行的。然而，计算可能会很苛刻。此外，可能没有公式；可能只可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，但如果曲面非常不规则，这可能会很棘手。这个问题对于更高维度的类似问题就更成问题了。在实践中，使用统计方法往往更可取：人们测量随机选择的点的高度，从该区域统一抽取，然后对得到的测量结果进行平均。乘以区域的面积，这给出了一个体积的统计估计，其精度随着一个样本中随机选择的点越多而增加。如果所讨论的区域相对简单，这种方法效果很好，被称为蒙特卡罗方法。即使是从该区域绘制随机点的任务也可能是具有挑战性的；对于重要的高维类似物，可能会非常困难。因此，取而代之的是，人们设计了一个随机的点序列，从长远来看，这些点是单独均匀分布的。一个简单的例子是，将每个点依次建议为相对于上一个点的随机位移(使用正态分布)，如果建议落在区域之外，则拒绝该建议。第一个点是随意选择的，所以最初的几个位置是有偏见的，但从长远来看，这一点会消失。测量的平均值乘以区域的面积，现在得到体积的统计估计，其精度随着使用的点的增加而提高。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，例如：统计物理(MCMC产生的原始背景)，随机计算机算法，生物分子研究，数值积分，以及(本项目的重点)应用统计学。在每一种情况下，不同的关注点导致不同的侧重点和不同的方法。在应用统计学中，人们处理的是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，从本质上讲，问题是如上所述。问题包括：“老化”(长期平均值必须持续多长时间，才能避免因第一个点的选择而产生偏差)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方案来快速混合的问题，特别是建议位移分布的最佳比例的问题。1997年，Roberts等人提出了关于调整比例的明确实用建议：在高维情况下，大约只有四分之一的建议应该被接受。随后的许多工作探索了概括性。然而，使用随机微分方程(SDE)理论的证明方法有其局限性。简而言之，它使用了快速进行小跳跃的随机行走的限制，导致了严重的平滑限制。最近有人建议使用基于跳跃平方平均值的“狄利克雷形式”，因此没有那么多的限制。此外，SDE方法只处理目标概率分布的一个方面，而Dirichlet形式一次处理所有内容。该项目将开发Dirichlet Form方法。

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.