New challenges in high-dimensional statistical inference

作为一个社会，我们认为越来越多的活动很大程度上依赖于尖端技术，依赖于对大量数据的快速和高效处理。明显的例子包括互联网搜索引擎和移动电话的使用。同样，医疗保健领域最近的进步在一定程度上是由于医院改进了高度数据密集型的扫描设备，以及新的有效药物治疗的开发，这是以数据为核心的广泛科学研究的结果。然而，这些进步只能通过开发适当的统计模型和方法来实现，使从业者能够从这些海量数据中提取有用的信息。为了捕捉数据生成过程的复杂性，这些模型不可避免地是高维的，在过去15年左右的时间里，这些模型一直是统计学中大量研究的主题。这个建议解决了在处理上述应用程序中经常出现的巨大数据集以及其他许多问题时的一些基本和重要的挑战。例如，在高维模型中，稀疏估计器对于稳定性和可解释性至关重要。但这些只给出了参数的一个点估计，而且通常实践者需要更复杂的推论语句来评估不确定性。我们将展示如何通过基于这些稀疏估计提出易于使用和稳健的p值。稀疏估计最重要的应用之一是在生物技术中。事实上，我们将把上述方法应用于丹麦生物统计学家使用微阵列技术进行的高维癌症研究。我们将选择几个弥漫性大B细胞淋巴瘤的稳定区分基因，并对不确定性进行相关量化，从而加强我们对这些癌症的了解。另一个面临高维挑战的应用领域是神经科学，我们将致力于使用脑电(EEG)的脑成像技术来研究计算障碍。计算障碍是一种阻碍正常算术功能的数学障碍。在这里，实验心理学家在这一领域使用的现有统计技术是不充分的，现代高维方法有可能极大地提高我们对这种障碍的理解。在分类问题中，挑战是根据观察值与来自每个类别的(标记的)数据的相似性来将其分配到两个或更多类别中的一个。它们是一些最常遇到的高维统计问题，特别是在机器学习等领域以及计算机视觉和机器人等计算机科学领域。我们将对可能最流行的方法(k近邻分类器)提供一个简单而有力的证明，方法是以最优的方式加权最近的近邻。我们还将对改进情况进行量化。我们要研究的一个相关问题是对由训练数据构造的分类器的不确定性进行量化。例如，这可以用来为医生提供诊断中的不确定性度量。我们将解决的最后一个主要问题涉及模型错误指定。这在高维统计问题中是一个特别重要的问题，在高维统计问题中，我们的模型几乎不可避免地遗漏了一些重要的影响，或者没有以正确的方式对它们进行建模。我们将提供对统计程序在这种情况下如何执行的理解，并开发对模型错误指定具有健壮性的新程序。一个特别重要的应用将是独立分量分析模型，这些模型是非常流行的统计信号处理，用于分析来自多个来源的数据，包括微阵列和脑成像数据。

As a society, more and more of the activities that we take for grantedrely on sophisticated technology, and are dependent on the fast andefficient handling of large quantities of data. Obvious examplesinclude the use of internet search engines and mobile telephones.Similarly, recent advances in healthcare are partly due to improved,highly data-intensive scanning equipment in hospitals, and thedevelopment of new, effective drug treatments, which have been theresult of extensive scientific study with data at its core.Nevertheless, such advances can only be achieved through thedevelopment of appropriate statistical models and methods which enablepractitioners to extract useful information from these vast quantitiesof data. In order to capture the complexity of the data generatingprocesses, these models are inevitably high-dimensional, and have beenthe topic of an enormous amount of research in Statistics over thelast 15 years or so.This proposal addresses some of the fundamental and important challenges in handling the huge data sets that routinely arise in the applications above, as well as many others. For instance, in high-dimensional models, sparse estimators are crucial for stability andinterpretability. But these give only a point estimate of aparameter, and typically practioners require more sophisticatedinferential statements to assess uncertainty. We will show how thisby done by proposing easy-to-use and robust p-valuesbased on these sparse estimators.One of the most important applications of sparse estimators is inbiotechnology. Indeed, we will apply our methodology described abovein a high-dimensional cancer study carried out by Danishbiostatisticians that uses microarray techniques. We will select,with an associated quantification of uncertainty, a handful of stabledistinguishing genes for diffuse large B-cell lymphomas, therebyenhancing our understanding of these cancers.Another application area facing high-dimensional challenges isneuroscience, and we will work on a study of dyscalculia that uses thebrain imaging technique of Electroencephalography (EEG). Dyscalculiais a mathematical disability that prevents normal arithmetic function.Here, existing statistical techniques used by experimentalpsychologists in this area are inadequate, and modern high-dimensionalmethods have the potential to improve dramatically our understandingof this disability.In classification problems, the challenge is to assign an observationto one of two or more classes based on its similarity to (labelled)data from each of these classes. They are some of the most frequentlyencountered high-dimensional statistical problems, particularly infields such as machine learning and areas of computer science such ascomputer vision and robotics. We will provide a simple and robustimprovement to perhaps the most popular method (the k-nearestneighbour classifier), by weighting the nearest neighbours in anoptimal fashion. We will also give a quantification of theimprovement. A related problem we will study is to quantify theuncertainty of a classifier constructed from training data. As anexample this could be used to give a doctor a measure of uncertaintyin a diagnosis.The final main issue we will address concerns model misspecification.This is a particularly important issue in high-dimensional statisticalproblems, where it is almost inevitable that our model misses someimportant effects, or does not model them in the correct way. We willprovide understanding of how statistical procedures perform in suchcircumstances and develop new ones that are robust to modelmisspecification. A particularly important application will be toIndependent Component Analysis models that are very popular instatistical signal processing for analysing data arising from multiplesources, including microarray and brain imaging data.