Topics in nonparametric inference and generative modelling

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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.