Non-Reversible Markov Chain Monte Carlo

在过去的几十年里，马尔可夫链蒙特卡罗（MCMC）的应用得到了爆炸式的发展，因为它允许对非常广泛的模型进行高维概率分布的有效采样，例如在统计力学、贝叶斯统计、人工智能、计算化学、数字信号处理中发现的模型。然后将生成的样本应用于推理，例如计算期望或估计概率。为了使这些算法高效和可扩展，它们需要快速收敛到所需的分布。基于Diaconis、Holmes和Neal（2000）的研究，以及最近Sun等人（2010）和Bierkens（2016）对涡度方法的研究，该项目涵盖了统计和应用概率，将尝试通过在模型设计中引入不可逆性来改进基本的MCMC模型，例如，可以通过在探索过程中添加方向矢量来实现。或者通过提升（在适当的意义上）马尔科夫链到一个更大的状态空间，而它相对于期望的概率分布保持不变。直观地说，目标是通过避免在已经访问过的地方花费太多时间来更快地探索一个空间。随着高性能云计算和大量数据的出现，模型复杂性不断增加，项目还需要考虑计算实现，例如设计不增加时间复杂性的方法。比尔肯斯,尤里斯。“不可逆pmmh。”统计与计算26.6(2016):1213-1228。Diaconis, Persi， Susan Holmes和Radford M. Neal。不可逆马尔可夫链取样器的分析应用概率年鉴（2000）：726-752。孙，Yi, Jurgen Schmidhuber, Faustino J. Gomez。通过插入漩涡改善马尔可夫链蒙特卡洛算法的渐近性能。神经信息处理系统进展。2010。

The application of Markov Chain Monte Carlo (MCMC) has exploded in the last decades, as it allows for efficient sampling of high-dimensional probability distributions for a very broad range of models, such as ones found in statistical mechanics, Bayesian statistics, artificial intelligence, computational chemistry, digital signal processing. The samples generated are then applied to make inference, for example calculating expectations or estimating probabilities. To make these algorithms efficient and scalable, they need to quickly converge to the desired distribution. Based on research by Diaconis, Holmes and Neal (2000) and more recently on the vorticity methodology of Sun et al. (2010) and Bierkens (2016), this project, covering both statistics and applied probability, will attempt to improve the fundamental MCMC models by introducing non-reversibility into the model design, which for example can be done by adding a direction vector to the exploratory process, or by lifting (in a suitable sense) the Markov chain to a larger state space while it remains invariant with respect to the desired probability distribution. Intuitively, the goal is to explore a space quicker by avoiding to spend too much time in places already visited. As model complexity grows with the advent of high-performance cloud computing and large amounts of data, the project also needs to consider computational implementations, for example by devising methods that does not increase the time complexity. Bierkens, Joris. "Non-reversible metropolis-hastings." Statistics and Computing 26.6 (2016): 1213-1228.Diaconis, Persi, Susan Holmes, and Radford M. Neal. "Analysis of a nonreversible Markov chain sampler." Annals of Applied Probability (2000): 726-752.Sun, Yi, Jurgen Schmidhuber, and Faustino J. Gomez. "Improving the asymptotic performance of Markov chain Monte-Carlo by inserting vortices." Advances in Neural Information Processing Systems. 2010.