Convergence rates of posterior distributions for non-i.i.d. observations

By Subhashis Ghosal, A. van der Vaart. 2007

438 citations recorded by the source corpus.

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model. © Institute of Mathematical Statistics, 2007.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8b098d324cc53bf602ef224be9348e958949d817

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Citations in the corpus

Rates of contraction of posterior distributions based on Gaussian process priors (2008, background): “For the case where the prior on σ is degenerate at the true value, it is shown in [11] that the rate of posterior contraction is faster than εn, for which there exist sets Wn satisfying logN(εn,Wn,‖ · ‖n) ≤ nε2 n, n(W c n) ≤ e−2nε 2 n,”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, background): “[12] and Ghosal and van der Vaart [13], so that there is a large literature on such types of conditions which can be applied in the present setting.”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “We need the design ( x (n) i ) n≥1,1≤i≤n to be sufficiently regular but, as stressed in [10], the spacial separation property of B-splines permits to express the precise condition in terms of the covariance matrix ΦΦ.”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “ity qf with respect to Lebesgue measure. The transition density is pf(y|x) = φ(y− f(x)). Denoting r(y) = (φ(y− L) + φ(y+L))/2, the transition density satisﬁes pf(y|x) ≍ r(y) for all x,y∈ R. Following [22], deﬁne the norms, for any s≥2, kf−f0ks,r= Z R |f(x)−f0(x)| sr(x)dx 1/s . As in [22], we conside”
Bayesian inference with rescaled Gaussian process priors (2007, methodology): “e sampled from a ﬁxed true distribution, a prior on an inﬁnite-dimensional model has to be carefully chosen in order to obtain optimal rates of contraction of the posterior (cf. e.g. [3], [17], [19], [4], [25], [23]). Even if posterior consistency is ensured, the rate of contraction of the posterior”
On Bayesian supremum norm contraction rates (2013, background): “Next, one applies Lemma 1 in [14].”
Thomas Bayes’ walk on manifolds (2012, background): “1 in [14]) Suppose there exists tests ψn as in (36)-(37), measurable sets Fn and C, d > 0, such that, for some εn → 0, ε̄n → 0 and nε(2)n → +∞, nε̄(2)n → +∞, (N) logN(ε̄n,Fn, dn) ≤ dnε̄(2)n (S) Π(F\Fn) ≤ e−(C+4)nε 2 n”