Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery
| Autor: | Markus Reiss, Anselm Johannes Schmidt-Hieber |
|---|---|
| Přispěvatelé: | Mathematics of Operations Research |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
62C10 Boundary (topology) Mathematics - Statistics Theory Statistics Theory (math.ST) posterior contraction Poisson distribution compound Poisson process symbols.namesake Frequentist inference Compound Poisson process Poisson point process Prior probability FOS: Mathematics Applied mathematics 62G05 62C10 62G05 60G55 Bernstein–von Mises theorem Mathematics subordinator prior boundary detection Function (mathematics) Frequentist Bayes analysis symbols 60G55 Statistics Probability and Uncertainty |
| Zdroj: | Annals of statistics, 48(3), 1432-1451. Institute of Mathematical Statistics Ann. Statist. 48, no. 3 (2020), 1432-1451 |
| ISSN: | 0090-5364 |
| Popis: | Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to H\"older smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing parameter dimension. As an intermediate result the limiting shape of the posterior for random histogram type priors is obtained. In both settings, it is shown that the marginal posterior of the functional $\vartheta =\int f$ performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, $(1-\alpha)$-credible sets are also asymptotic $(1-\alpha)$-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions indicating that credible sets only have frequentist coverage for priors that are specifically constructed to match properties of the underlying true function. Comment: The first version of arXiv:1703.08358 has been expanded and rewritten. We decided to split it in two separate papers, a new version of arXiv:1703.08358 and this article |
| Databáze: | OpenAIRE |
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