Robust and efficient statistical learning algorithms with applications in actuarial science

大数据时代为统计方法提供了一个大放异彩的机会，通过与包括精算科学在内的广泛领域相关的应用。然而，为了抓住并充分利用这一机会，研究人员和从业人员必须有效地管理大数据带来的挑战。首先，数量并不意味着质量，数据也不例外。很难确定存在严重误差的低质量数据的趋势。对离群值具有鲁棒性的方法有助于完成这项任务。我的研究目标之一是介绍稳健的贝叶斯模型的精算师的实际相关性。例如，广义线性模型（GLM）在一般保险索赔建模中无处不在。鲁棒GLM是要提出的一类模型。当异常值越来越远时，它们将具有仅基于极限中的非异常值产生结果的显著特性，而在没有异常值的情况下，它们的表现与传统模型相似。事实上，异常值的影响逐渐消失，这反映出在开始时，当它们离大部分数据不远时，它们是否真的是异常值是不确定的。自动处理这种不确定性的方法在高维和变量选择问题中特别有价值。其次，提出处理粗差的复杂模型是不够的。推理所需的数值方法必须与模型复杂性和数据大小成比例，以确保统计程序是可实施的。为此，我将提出自动不可逆跳转算法。这些是马尔可夫链蒙特卡罗（MCMC）方法，用于近似积分相对于联合后验分布的模型及其参数，它允许同时贝叶斯变量选择和参数估计。与可逆方法相比，不可逆方法在典型情况下具有更好的可扩展性。这是由于马尔可夫链的路径特征在于持续运动，允许更快地遍历状态空间，并防止可逆方案经常表现出的扩散行为。这转化为更少的自相关性，这意味着从后验分布中获得独立样本的迭代次数更少。因此，结果更接近常规蒙特卡罗，这是MCMC的最终目标。强大的统计模型与自动和有效的数值方法一起用于自动推理，导致：强大和有效的统计学习算法。统计过程的优点（例如，与典型的机器学习算法相比）是它们允许风险和不确定性量化。这种量化是精算师作用的核心，可以发布包含有关保险公司支付未来索赔能力的丰富概率描述的报表。

The big data era represents an opportunity for statistical methods to shine, through applications relevant to a wide spectrum of fields, including actuarial science. In order to seize and make the most out of this opportunity, researchers and practitioners must, however, effectively manage the challenges that big data pose. Firstly, quantity surely does not imply quality, and data are no exception to this rule. Identifying trends in poor quality data with gross errors is difficult. Methods that are robust against outliers help in this task. One of my research goals is to introduce robust Bayesian models of practical relevance for actuaries. Generalised linear models (GLMs) are, for instance, ubiquitous in general insurance claim modelling. Robust GLMs are a class of models to be proposed. They will have the remarkable characteristic of producing results based solely on the nonoutliers in the limit when the outliers move further and further away, while performing similarly to the traditional models in the absence of outliers. The impact of the outliers in fact gradually vanishes, reflecting that at the beginning when they are not so far from the bulk of the data, there is an uncertainty about whether they really are outliers or not. Methods automatically dealing with this uncertainty are particularly valuable in high-dimensional and variable selection problems. Secondly, proposing complex models handling gross errors is not enough. The numerical methods required for inference must scale with the model complexity and data size to ensure that the statistical procedures are implementable. For this purpose, I will propose automatic nonreversible jump algorithms. These are Markov chain Monte Carlo (MCMC) methods used for approximating integrals with respect to joint posterior distributions of models and their parameters, which allows simultaneous Bayesian variable selection and parameter estimation. Nonreversible methods are known for their better scalability in typical cases, comparatively to their reversible counterparts. This is due to Markov chains with paths characterised by persistent movement that allow to traverse the state space more quickly and prevent the diffusive behaviour often exhibited by reversible schemes. This translates into less autocorrelations, which implies less iterations to obtain independent samples from the posterior distributions. The results are thus closer to regular Monte Carlo, which is the ultimate MCMC goal. Robust statistical models together with automatic and efficient numerical methods for automated inference result in: robust and efficient statistical learning algorithms. The advantage of statistical procedures (over typical machine learning algorithms for instance) is that they allow risk and uncertainty quantification. This quantification is at the core of actuaries' role and allows to issue statements containing rich probabilistic descriptions about the capacity of insurance firms to pay for future claims.