Topics in nonparametric inference and generative modelling

建模和从数据中推断是统计学和应用数学的基本任务。与应用相关的许多类型的数据最自然地被认为是位于无限维空间中的函数，而本项目的重点是在这种非参数设置中的理论和算法。虽然数值算法本质上是离散的，但一个成功的理念是直接在无限维水平上设计模型。这导致了在任何分辨率下都可以工作的方法，并避免了分辨率细化时的缩放问题。仅举一个这种策略的例子，Cotter等人（2013）的预处理Crank-Nicolson MCMC算法允许在非常高的维度上从概率密度中采样，而不会降低与经典MCMC算法相关的性能。本项目将考虑的主要任务是生成建模和贝叶斯推理，后者主要是由贝叶斯反问题的应用驱动的。生成模型试图从样本中近似未知的数据分布，对数据分布的结构施加很少的先验假设。另一方面，推理问题寻求将（可能不确定的）统计模型与数据结合起来，使用贝叶斯规则以连贯的方式吸收数据。受函数空间之间可学习神经算子的最新发展的启发，一个目标是将现有的生成模型（如归一化流）适应连续体设置，以允许在任何分辨率下进行采样和密度估计。这补充了其他最近提出的基于生成对抗网络（gan）和扩散模型的函数空间生成模型。在连续体水平上设计算法的使用提出了许多关于适定性和收敛性的新问题。围绕非参数生成模型的理论尤其不成熟，关于这些模型的适定性和收敛性的许多问题仍然存在。还有许多与无限维贝叶斯推理相关的开放问题，而一个特别的焦点-由反问题的需要驱动-是贝叶斯推理中的后验分布在非参数设置中是否具有定义良好的最大后验估计量。这种模式是否存在是微妙的，并导致了一种抽象理论的发展，许多问题仍未解决。这一理论也影响了算法的发展：贝叶斯反问题的后验模式可以被看作是完全后验的可处理的变分近似，并且与反问题中经典方法给出的解相一致。

Modelling and making inferences from data are fundamental tasks in statistics and applied mathematics. Many types of data relevant in applications are most naturally thought of as functions lying in an infinite-dimensional space, and the focus of this project is on theory and algorithms in this nonparametric setting.While numerical algorithms are inherently discrete, a successful philosophy is to design models directly at the infinite-dimensional level. This leads to methods which work at any resolution and avoid scaling problems as the resolution is refined. To give just one example of this strategy, the preconditioned Crank-Nicolson MCMC algorithm of Cotter et al. (2013) allows for sampling from probability densities in very high dimension without the degradation of performance associated with classical MCMC algorithms.The main tasks that will be considered in this project are generative modelling and Bayesian inference, with the latter motivated primarily by applications to Bayesian inverse problems. Generative models seek to approximate an unknown data distribution from samples, imposing few prior assumptions on the structure of the data distribution. Inference problems, on the other hand, seek to combine a (possibly uncertain) statistical model with data, using Bayes' rule to assimilate data in a coherent way. Inspired by recent developments in learnable neural operators between function spaces, one goal is to adapt existing generative models such as normalising flows to the continuum setting to allow for sampling and density estimation at any resolution. This complements other recently proposed generative models on function spaces based on generative adversarial networks (GANs) and diffusion models.The use of algorithms designed at the continuum level raises many new questions about well-posedness and convergence. The theory surrounding nonparametric generative models is particularly immature, and many questions about the well-posedness and convergence properties of these models remain open. There are also many open questions of relevance to Bayesian inference in infinite dimensions, and a particular focus - motivated by the needs of inverse problems - is on whether the posterior distribution in Bayesian inference has a well-defined maximum a posteriori estimator in the nonparametric setting. Whether such a mode exists is subtle and has led to the development of an abstract theory, for which many questions remain unresolved. This theory also informs algorithmic developments: modes of the posterior of a Bayesian inverse problem can be viewed as tractable variational approximations to the full posterior and coincide with the solution given by classical methods in inverse problems.