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The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition -- existence of a solvable Lie algebra -- being satisfied. Numerical illustrations are provided.
We consider Bayesian inference for stochastic differential equation mixed effects models (SDEMEMs) exemplifying tumor response to treatment and regrowth in mice. We produce an extensive study on how a SDEMEM can be fitted using both exact inference b
Infectious diseases on farms pose both public and animal health risks, so understanding how they spread between farms is crucial for developing disease control strategies to prevent future outbreaks. We develop novel Bayesian nonparametric methodolog
Identifying the most deprived regions of any country or city is key if policy makers are to design successful interventions. However, locating areas with the greatest need is often surprisingly challenging in developing countries. Due to the logistic
This is the collection of solutions for all the exercises proposed in Bayesian Essentials with R (2014).
Julian Besag was an outstanding statistical scientist, distinguished for his pioneering work on the statistical theory and analysis of spatial processes, especially conditional lattice systems. His work has been seminal in statistical developments ov