Markov chain Monte Carlo algorithms and locally informed proposal distributions

马尔可夫链蒙特卡罗（MCMC）方法允许从高度复杂的概率分布（目标）生成数据。它们被应用于天气预报、医学、物理、安全等多个领域。我的目标是通过不同的项目为理论、方法和应用MCMC文献做出贡献。理论结果，除了导致更好地理解采样器，往往指向改进采样方案。在方法方面，主要的挑战是提出新的采样方案，需要最小的输入和保持计算负担得起。第三个目标是通过在特定的建模环境中提出有效的采样器来促进跨学科研究。Metropolis-Hastings （MH）算法是MCMC工具箱中最流行的采样器，也是本研究的基础方法。它以无数的方式得到了增强，产生了尖端的采样器。在每次迭代中，从选定的提案分布中抽取一个候选，然后根据特定的接受概率将其作为样本的合适值接受。为了获得代表目标的样本，需要仔细调整提案分布。最近，人们对从一个迭代到下一个迭代的本地调优产生了兴趣；它们以较低的计算成本提供了有趣的效率增益。MH采样器中的局部平衡建议分布在高维状态下产生更聪明的候选，减少了接受/拒绝步骤的影响，并以有效的方式产生高质量的样本。我希望研究局部平衡建议，局部调优和梯度通知采样器之间的联系。我打算提出一个灵活的提案分布，它是维度的函数，随着维度的增加，向局部平衡的提案收敛。一般来说，我希望这种有前途的局部平衡概念能够导致一种明智的方式来选择MH采样器变体中的各种调谐参数（例如，具有候选池的采样器中的权重函数，可逆跳跃MCMC采样器中模型之间的转换等）。传统的研究采样器理论行为的方法是限制性的，因为它一次只关注一个目标组件，使得难以研究具有相关性的目标。最近，我们提出了一种允许同时研究所有组件的新方法；我打算研究这个证明可以处理的采样器和/或目标复杂性（使用copula）的程度。这可能会导致明智的设计方案，并在MCMC采样器中打开一个全新的相关处理。计划申请两个项目。第一个涉及在建模金融环境中使用统计框架；第二个，在生态学，将模拟候鸟的轨迹使用真实的地理位置数据的光，基于测量的阳光强度随时间的变化。

Markov chain Monte Carlo (MCMC) methods allow for data generation from highly complex probability distributions (the target). They are used in several fields of application such as weather forecasting, medicine, physics, security, etc. I aim at contributing to the theoretical, methodological, and applied MCMC literature through different projects. Theoretical results, besides leading to a better understanding of samplers, often point towards an improvement of the sampling scheme. In terms of methodology, the main challenge is to propose new sampling schemes that require minimum input and remain computationally affordable. A third objective is contributing to interdisciplinary research by proposing efficient samplers in specific modeling contexts. The Metropolis-Hastings (MH) algorithm is the most popular sampler in the MCMC toolbox and the underlying method in the proposed research. It has been enhanced in countless ways, producing cutting edge samplers. At each iteration, a candidate is drawn from a selected proposal distribution and then accepted as a suitable value for the sample according to a specific acceptance probability. To obtain a sample that is representative of the target, careful tuning of the proposal distribution is required. Lately, there has been interest in local tunings that evolve from one iteration to the next; these offer interesting efficiency gains at low computational costs. Locally-balanced proposal distributions in MH samplers produce smarter candidates in high-dimensional regimes, reducing the impact of the accept/reject step and producing quality samples in an efficient way. I wish to study the connection between locally-balanced proposals, local tunings, and gradient-informed samplers. I intend to propose a flexible proposal distribution that is a function of the dimension, converging towards a locally-balanced proposal as the dimensionality increases. Generally, I expect this promising local balance concept to lead to an informed way of selecting the various tuning parameters in variants of the MH sampler (e.g. weight function in samplers with pools of candidates, transitions between models in the reversible-jump MCMC sampler, etc.) The traditional approach to study the theoretical behavior of samplers is restrictive as it focuses on one target component at a time, making it difficult to study targets with correlation. Recently, we proposed a new approach that allows studying all components simultaneously; I intend to investigate the extent of the sampler and/or target complexity (with copulas) this proof can handle. This could lead to informed design schemes and open a whole new handling of correlation in MCMC samplers. Two applied projects are planned. The first involves the use of a statistical framework in a modeling financial context; the second one, in Ecology, will model the trajectory of migratory birds using real geolocation data by light, based on measurements of sunlight intensity over time.