{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:53:35Z","timestamp":1791323615127,"version":"4.3.1"},"reference-count":44,"publisher":"Institute of Mathematical Statistics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,2,1]]},"abstract":"<jats:p>In this paper, we propose a novel Bayesian approach for nonparametric estimation in Wicksell\u2019s problem. This has important applications in astronomy for estimating the distribution of the positions of the stars in a galaxy given projected stellar positions and in materials science to determine the 3D microstructure of a material, using its 2D cross-sections. We deviate from the classical Bayesian nonparametric approach, which would place a Dirichlet Process (DP) prior on the distribution function of the unobservables, by directly placing a DP prior on the distribution function of the observables. Our method offers computational simplicity due to the conjugacy of the posterior and allows for asymptotically efficient estimation by projecting the posterior onto the L2 subspace of increasing, right-continuous functions. Indeed, the resulting Isotonized Inverse Posterior (IIP) satisfies a Bernstein\u2013von Mises (BvM) phenomenon with minimax asymptotic variance g0(x)\/2\u03b3, where \u03b3&gt;1\/2 reflects the degree of H\u00f6lder continuity of the true cdf at x. Since the IIP gives automatic uncertainty quantification, it eliminates the need to estimate \u03b3. Our results provide the first semiparametric Bernstein\u2013von Mises theorem for projection-based posteriors with a DP prior in inverse problems.<\/jats:p>","DOI":"10.1214\/25-aos2571","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T06:32:30Z","timestamp":1772605950000},"page":"383-407","source":"Crossref","is-referenced-by-count":1,"title":["Semiparametric Bernstein\u2013von Mises phenomenon via Isotonized Posterior in Wicksell\u2019s problem"],"prefix":"10.1214","volume":"54","author":[{"given":"Francesco","family":"Gili","sequence":"first","affiliation":[{"name":"Delft Institute of Applied Mathematics (DIAM), Delft University of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geurt","family":"Jongbloed","sequence":"additional","affiliation":[{"name":"Delft Institute of Applied Mathematics (DIAM), Delft University of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aad","family":"van der Vaart","sequence":"additional","affiliation":[{"name":"Delft Institute of Applied Mathematics (DIAM), Delft University of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"6","doi-asserted-by":"publisher","unstructured":"<small>Castillo, I.<\/small> and <small>Nickl, R.<\/small> (2013). 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