Dirichlet process-based inference for Wicksell's inverse problem.
Consider these two problems. The first one: astronomers are interested in the study of the way stars are distributed in the universe. In this setting, globular clusters --- spherical aggregations of stars held together by gravity --- are a topic of particular focus. But how to determine the distribution of stars within these three-dimensional structures when given only a 2D photo of the clusters? The second one: having material that presents a globular microstructure and that cannot be scanned (e.g. steel) but only sectioned, how can we determine the distribution of particles' size inside the material when given only a limited number of 2D cross sections? Despite their seemingly distant nature, both of these problems share a common thread: the inherent structure of a inverse problem, originally conceptualized by Wicksell. There, the setting of the opaque material having globular microstructure is formalized by assuming the embedded particles' radii are i.i.d. realization from a random variable X with cdf F. In this talk, we explore the properties of nonparametric Bayesian estimators based on the Dirichlet process prior for F in this problem, which is the first time the Dirichlet process is studied in an inverse setting. In particular, we illustrate contraction rates results and uncertainty quantification (Bernstein-von Mises type of results) for our methodology and we compare it to the state-of-the-art nonparametric frequentist efficient estimator. These results have also theoretical relevance because they provide an instance of a semiparametric setting, embedded in an inverse problem, where the nonparametric Bayesian procedure asymptotically behaves like the nonparametric frequentist estimators.
Area: CS15 - Recent Advances in Bayesian Nonparametric Statistics (Marta Catalano and Beatrice Franzolini)
Keywords: Nonparametric Bayesian inverse problems, Wicksell's problem, Dirichlet process prior.
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