Adaptive Bayesian density estimation with location-scale mixtures

By W. Kruijer, Judith Rousseau, A. van der Vaart. 2010

127 citations recorded by the source corpus.

We study convergence rates of Bayesian density estimators based on finite location-scale mixtures of exponential power distributions. We construct approximations of �-Holder densities be continuous mix- tures of exponential power distributions, leading to approximations of the �-Holder densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a logn term) and since the priors are inde- pendent of the smoothness the rates are adaptive to the smoothness.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d5e48f0d6dfaf46fc38b31291b3977009b01e634

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Citations in the corpus

POSTERIOR CONSISTENCY IN CONDITIONAL DENSITY ESTIMATION BY COVARIATE DEPENDENT MIXTURES (2011, methodology): “, & MacEachern, S. N. (2004). An anova model for dependent random measures.”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, methodology): “This would mean that a common variation of prior model 10 Judith Rousseau (9), namely the location - scale mixture written as fP,σ(x) = Z Rd ϕσ(x− µ)dP (µ, σ), (10) might not be the best suited prior model for estimating a smooth density.”
Consistency of mixture models with a prior on the number of components (2022, methodology): “Many theoretical advances have been made in establishing posterior consistency and contraction rates for density estimation when using nonparametric mixture models (see Ghosal and Van der Vaart, 2017 and the many references therein) or finite mixture models with a prior on the number of components (”
Bayesian fractional posteriors (2016, background): “Assumption (31) allows heavy-tailed prior distributions on the component specific means μjs, which is not permissible in the existing theory due to the additional requirement [36, 52] on the tail behavior 1−Π(μj ∈ [−r, r]) ≤ exp(−a13r3), for some a13, τ3 > 0 and all sufficiently large r > 0.”
On adaptive posterior concentration rates (2013, background): “For instance, in the case of density estimation, the renown class of Dirichlet process mixtures or related types of mixtures lead to adaptive posterior concentration rates over collections of Hölder balls of regularity β , up to a logn term, see, for instance, [19, 24, 28] and [27] under the Helling”