Elizabeth Louise Baker

Stefan Sommer, Gefan Yang, Elizabeth Louise Baker

Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.

Elizabeth Louise Baker, Gefan Yang, Michael L. Severinsen et al.

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.