# Tim Sullivan

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### Postdoc position: Analysis of MAP estimators

Within a few weeks a full-time two-year position for a postdoctoral researcher in the UQ group will be advertised by the Freie Universität Berlin. This position will be associated to the project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems” funded by the DFG.

This project aims to advance the state of the art in rigorous mathematical understanding of MAP estimators in infinite-dimensional statistical inverse problems. In particular, the research in this project will connect the “small balls” approach of Dashti, Law, Stuart, and Voss to the calculus of variations and hence properly link the variational and fully Bayesian points of view on inverse problems.

Precise details of the application process will appear soon, at which point this news item will be updated accordingly. In the meantime, prospective candidates are welcome to contact me with informal enquiries.

Published on Monday 21 October 2019 at 08:00 UTC #group #job #fu-berlin #inverse-problems #dfg #map-estimators

### Weak and strong modes in Inverse Problems

The paper “Equivalence of weak and strong modes of measures on topological vector spaces” by Han Cheng Lie myself has now appeared in Inverse Problems. This paper addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in an infinite-dimensional space $$X$$. Such modes can be defined either strongly (a la Dashti et al. (2013), via a global maximisation) or weakly (a la Helin & Burger (2015), via a dense subspace $$E \subset X$$). The question is, when are strong and weak modes equivalent? The answer turns out to be rather subtle: under reasonable uniformity conditions, the two kinds of modes are indeed equivalent, but finite-dimensional counterexamples exist when the uniformity conditions fail.

H. C. Lie and T. J. Sullivan. “Equivalence of weak and strong modes of measures on topological vector spaces.” Inverse Problems 34(11):115013, 2018. doi:10.1088/1361-6420/aadef2

(See also H. C. Lie and T. J. Sullivan. “Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013).” Inverse Problems 34(12):129601, 2018. doi:10.1088/1361-6420/aae55b )

Abstract. A strong mode of a probability measure on a normed space $$X$$ can be defined as a point $$u \in X$$ such that the mass of the ball centred at $$u$$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $$E$$ of $$X$$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with non-atomic measures that are finite on bounded sets, the density of $$E$$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.

Published on Saturday 22 September 2018 at 12:00 UTC #publication #inverse-problems #modes #map-estimators #lie

### Preprint: Weak and strong modes

Han Cheng Lie and I have just uploaded a revised preprint of our paper, “Equivalence of weak and strong modes of measures on topological vector spaces”, to the arXiv. This addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in an infinite-dimensional space $$X$$. Such modes can be defined either strongly (a la Dashti et al. (2013), via a global maximisation) or weakly (a la Helin & Burger (2015), via a dense subspace $$E \subset X$$). The question is, when are strong and weak modes equivalent? The answer turns out to be rather subtle: under reasonable uniformity conditions, the two kinds of modes are indeed equivalent, but finite-dimensional counterexamples exist when the uniformity conditions fail.

Abstract. A strong mode of a probability measure on a normed space $$X$$ can be defined as a point $$u \in X$$ such that the mass of the ball centred at $$u$$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $$E$$ of $$X$$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with non-atomic measures that are finite on bounded sets, the density of $$E$$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.

Published on Monday 9 July 2018 at 08:00 UTC #publication #preprint #inverse-problems #modes #map-estimators #lie