Misspecification in infinite-dimensional Bayesian statistics

Abstract

We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution $P_0$, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback--Leibler divergence with respect to $P_0$. An entropy condition and a prior-mass condition determine the rate of convergence. The method is applied to several examples, with special interest for infinite-dimensional models. These include Gaussian mixtures, nonparametric regression and parametric models.

Published at http://dx.doi.org/10.1214/009053606000000029 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (51)

• Philosophy and the practice of Bayesian statistics
• Rates of contraction of posterior distributions based on Gaussian process priors
• Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Gibbs posterior for variable selection in high-dimensional classification and data mining
• Dynamics of Bayesian Updating with Dependent Data and Misspecified Models
• The semiparametric Bernstein-von Mises theorem
• Bayesian variable selection for high dimensional generalized linear models: convergence rates of the fitted densities
• Considering discrepancy when calibrating a mechanistic electrophysiology model
• Testing hypotheses via a mixture estimation model
• Quasi-Bayesian analysis of nonparametric instrumental variables models
• On the Brittleness of Bayesian Inference
• Brittleness of Bayesian Inference Under Finite Information in a Continuous World
• Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems
• A comparison of learning rate selection methods in generalized Bayesian inference
• On Posterior Concentration in Misspecified Models
• Bayesian Median Autoregression for Robust Time Series Forecasting
• Gibbs posterior concentration rates under sub-exponential type losses
• Gibbs posterior inference on the minimum clinically important difference
• The Bayesian Analysis of Complex, High-Dimensional Models: Can It Be CODA?
• Model-free posterior inference on the area under the receiver operating characteristic curve
• Some models are useful, but how do we know which ones? Towards a unified Bayesian model taxonomy
• Asymptotic equivalence of empirical likelihood and Bayesian MAP
• Shared kernel Bayesian screening
• Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness
• A Bayesian Approach to Multiple-Output Quantile Regression
• Optimal Bayesian estimation in random covariate design with a rescaled Gaussian process prior
• Applications of an algorithm for solving Fredholm equations of the first kind
• On nonparametric estimation of a mixing density via the predictive recursion algorithm
• Nonparametric Bayesian model selection and averaging
• Semiparametric Bernstein-von Mises Theorem: Second Order Studies
• General Robust Bayes Pseudo-Posterior: Exponential Convergence results with Applications
• Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation
• L-Divergence Consistency for a Discrete Prior
• Frequentist Consistency of Generalized Variational Inference
• Bernstein - von Mises theorem and misspecified models: a review
• Approximate Bayesian Computation via Classification
• A Diffusion Process Perspective on Posterior Contraction Rates for Parameters
• Criteria for posterior consistency
• Revisiting consistency of a recursive estimator of mixing distributions
• On Semi-parametric Bernstein-von Mises Theorems for BART
• Calibrating generalized predictive distributions
• Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
• Comparing the Value of Labeled and Unlabeled Data in Method-of-Moments Latent Variable Estimation
• Anytime valid and asymptotically optimal inference driven by predictive recursion
• Equivalence of Convergence Rates of Posterior Distributions and Bayes Estimators for Functions and Nonparametric Functionals
• Nonasymptotic Laplace approximation under model misspecification
• Mixtures of Gaussian Processes for regression under multiple prior distributions
• Bayesian Shrinkage towards Sharp Minimaxity
• Semiparametric posterior limits
• Posterior consistency for the spectral density of non-Gaussian stationary time series