Novel Markov chain Monte Carlo methods for high-dimensional statistics.

马尔可夫链蒙特卡罗（MCMC）是探索复杂非标准概率分布的首选工具。这些算法在60多年前就被引入了，但它仍然是一个非常活跃的研究领域，因为我们现在面临着越来越困难的挑战。也就是说，现在期望这些算法在高维环境和存在非常大的数据集的情况下工作。该项目的目的是通过开发新的马尔可夫链蒙特卡罗（MCMC）算法来解决这些挑战，该算法可扩展到数据丰富环境中的高维场景。还将对这些新颖的MCMC方案进行尖锐的理论分析，并将在各种具有挑战性的统计应用中进行演示。在目标分布为对数凹的情况下，对未调整Langevin算法进行了大量的研究。然而，对数凹性假设具有很强的限制性，未经调整的Langevin算法引入了一些不良的偏差。我们将开发提供一致估计的新方案，并将致力于开发绕过对数凹性假设的理论分析。特别是，我们计划将重点放在非可逆方案的开发上。该项目的长期效益也与RCUK数字经济计划密切相关。MCMC被广泛用于分析复杂的数据集，并可用于开发新的协同过滤和主题建模技术。因此，预计一般公众将在中期受益；例如，更强大的搜索引擎和推荐系统的发展，更好的信用卡评分技术，改进的身份欺诈检测方法等。计算金融的许多方面也可以很容易地从中受益。

Markov chain Monte Carlo (MCMC) are the tools of choice to explore complex non-standard probability distributions.These algorithms have been introduced over 60 years ago, yet it remains a very active research area as we now face increasingly difficult challenges. Namely it is now expected for these algorithms to work in high-dimensional settingsand in the presence of very large datasets.The aims and objectives of this project is to address these challenges by developing novel Markov chain Monte Carlo (MCMC)algorithms which scale to high-dimensional scenarios in a data rich enviromnent. A sharp theoretical analysis of these novel MCMC schemes will also been provided and they will be demonstrated on a variety of challenging statistical applications.Much work has been recently done on the analysis of the unadjusted Langevin algorithm in scenarios where the target distributions are log-concave.However, the log-concavity assumption is very restrictive and the unadjusted Langevin algorithm introduces some undesirable bias.We will develop novel schemes which provide consistent estimates and will aim to develop a theoretical analysis that bypasses the log-concavity assumption.In particular, we plan to focus on the development of non-reversible schemes.The longer term benefits of this project are also closely linked to the RCUK Digital Economy programme. MCMC are widely used to analyze complex datasets and can be used to develop novel collaborativefiltering and topic modelling techniques for example. It is thus expected that benefits will be experienced in the medium term by the general public; e.g. the development of more powerful search engines and recommender systems, better credit card scoring techniques, improved methods for identity fraud detection etc. Many aspect of computational finance could also readily benefit from them.