The joint RSS Highland group / St Andrews meeting takes place in St Andrews (lecture theatre C, Maths building, North Haugh) on Wednesday 26th November 2014

Programme:
2.00pm -2.05pm
Welcome

2.05pm – 2.55pm
Mark Girolami (University of Warwick)
“Quantifying epistemic uncertainty in ODE and PDE solutions using Gaussian measures and Feynman-Kac path integrals”
2.55pm – 3.30pm
Coffee Break

3.30pm -4.20pm
Natalia Bochkina (University of Edinburgh) 
“Statistical inference in possibly misspecified nonregular models”
http://www.st-andrews.ac.uk/~statistics/seminars.html

Abstracts:
Mark Girolami (University of Warwick)
Diaconis and O'Hagan originally set out a programme of research suggesting the evaluation of a functional can be viewed as an inference problem. This perspective naturally leads to construction of a probability measure describing the epistemic uncertainty associated with the evaluation of functions solving for systems of Ordinary Differential Equations (ODE) or a Partial Differential Equation (PDE). By defining a joint Gaussian Measure on the Hilbert space of functions and their derivatives appearing in an ODE or PDE a stochastic process can be constructed. Realisations of this process, conditional upon the ODE or PDE, can be sampled from the associated measure defining "Global" ODE/PDE solutions conditional on a discrete mesh. The sampled realisations are consistent estimates of the function satisfying the ODE or PDE system and the associated measure quantifies our uncertainty in these solutions given a specific discrete mesh. Likewise an unbiased estimate of the "Local" solutions of certain classes of PDEs, along with the associated probability measure, can be obtained by appealing to the Feynman-Kac identities and 'Bayesian Quadrature' which has advantages over the construction of a Global solution for inverse problems. In this talk I will describe the quantification of uncertainty using the methodology above and illustrate with various examples of ODEs and PDEs in specific inverse problems.

Natalia Bochkina (University of Edinburgh) 
Finite dimensional statistical models are called nonregular if it is possible to construct an estimator with the rate of convergence that is faster than the parametric root-n rate. I will give an overview of such models with the corresponding rates of convergence in the frequentist setting under the assumption that they are well-specified. In a Bayesian approach, I will consider a special case where the “true” value of the parameter for a well-specified model, or the parameter corresponding to the best approximating model from the considered parametric family for a misspecified model, occurs on the boundary of the parameter space. I will show that in this case the posterior distribution (a) asymptotically concentrates around the ``true’’ value of the parameter (or the best approximating value under a misspecified model), (b) has not only Gaussian components as in the case of regular models (the Bernstein–von Mises theorem) but also Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary, and (c) has a faster rate of convergence in the directions of the Gamma distribution components. One implication of this result is that for some models, there appears to be no lower bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. I will discuss how this result can be used for identifying misspecification in regular models. The results will be illustrated on a problem from emission tomography. This is joint work with Peter Green (University of Bristol).

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