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。它们被用于多个应用领域，如天气预报，医学，物理，安全等，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，除了导致更好地了解采样器，往往指向改进的采样计划。在方法方面，主要的挑战是提出新的抽样方案，需要最少的投入，并保持计算负担得起。第三个目标是通过在特定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，也是所提出研究的基础方法。它在无数方面得到了增强，生产出了尖端的采样器。在每次迭代中，从选定的建议分布中抽取候选项，然后根据特定的接受概率将其接受为样本的合适值。为了获得代表目标的样本，需要仔细调整建议分布。最近，人们对从一次迭代到下一次迭代的局部调优产生了兴趣;这些以低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区域中产生更智能的候选，减少接受/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、局部调谐和梯度信息采样器之间的联系。我打算提出一个灵活的建议分布，这是一个功能的维度，收敛到一个局部平衡的建议，随着维度的增加。一般来说，我希望这种有前途的局部平衡概念能够为MH采样器的变体中选择各种调谐参数提供一种明智的方式（例如，具有候选池的采样器中的权重函数，可逆跳变MCMC采样器中模型之间的转换等）*研究采样器理论行为的传统方法是限制性的，因为它一次只关注一个目标成分，使得很难研究具有相关性的目标。最近，我们提出了一种新的方法，允许同时研究所有组件;我打算调查的程度采样器和/或目标复杂性（与copula），这个证明可以处理。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性处理。计划实施两个项目。第一个涉及在建模金融环境中使用统计框架;第二个涉及生态学，将根据对一段时间内阳光强度的测量，使用光的真实的地理定位数据对候鸟的轨迹进行建模。

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.