New approaches to Bayesian consistency

By Stephen G. Walker. 2004

179 citations recorded by the source corpus.

We use martingales to study Bayesian consistency. We derive sufficient conditions for both Hellinger and Kullback-Leibler consistency, which do not rely on the use of a sieve. Alternative sufficient conditions for Hellinger consistency are also found and demonstrated on examples.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/80ff49826266b0f74e454fc68d30c1cb124f1bfd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Dynamics of Bayesian Updating with Dependent Data and Misspecified Models (2009, background): “For mis-specified models, that is settings where (a) above fails, important early results were obtained by Berk [5; 6] for IID data, albeit under rather strong restrictions on likelihood functions and parameter spaces, showing that the posterior distribution concentrates on an “asymptotic carrier”, ”
POSTERIOR CONSISTENCY IN CONDITIONAL DENSITY ESTIMATION BY COVARIATE DEPENDENT MIXTURES (2011, background): “Quite general weak and strong posterior consistency theorems were established (Schwartz, 1965; Barron, 1988; Barron et al., 1999; Ghosal, Ghosh, and Ramamoorthi, 1999; and Walker, 2004).”
On Bayesian supremum norm contraction rates (2013, background): “Possible choices for ϕ cover several commonly used classes of prior distributions, such as so-called exponential power (EP) distributions, see e.g. Choy and Smith [8], Walker and Gutiérrez-Peña [39] and references therein, as well as some of the univariate Kotz-type distibutions, see e.g. Nadaraja”
Dirichlet Process Mixtures of Generalized Linear Models (2009, background): “Most current literature centers around consistency of the posterior density for DP Gaussian mixture models (Barron et al. 1999, Ghosal et al. 1999, Ghosh and Ramamoorthi 2003, Walker 2004, Tokdar 2006) and semi-parametric linear regression models (Amewou-Atisso et al. 2003, Tokdar 2006).”
Remarks on consistency of posterior distributions (2008, background): “Similar results to Theorem 7.1 along with regression problems have been discussed in Walker [33].”
A general framework for updating belief distributions (2013, background): “The topic is dealt with by White (1982) and gives conditions under which θ̂ → θ0 a.”
Posterior convergence rates of Dirichlet mixtures at smooth densities (2007, background): “The trade-off between entropy and summability of the square roots of prior probabilities is interesting and requires further investigation; see [15] for a consistency result based on the summability condition.”
On rates of convergence for posterior distributions in infinite-dimensional models (2007, unknown): “The aim of the present paper is to tackle the problem based on the approach of Walker [ 14], which leads to improvements in the examples we consider.”
Convergence rates for Bayesian density estimation of infinite-dimensional exponential families (2006, methodology): “Preliminary ascertainment of consistency is based on results by Barron, Schervish and Wasserman [2], Walker and Hjort [23] and Walker [22], who have addressed the issue of consistency of posterior distributions for infinitedimensional exponential families generated by orthonormal systems of bounded ”