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

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.

• Publication year

  2007

• Venue

  Annals of Statistics

• Publication date

  2007-02-01

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1214/009053606000001172arXiv 0708.0491
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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  2007influential reference
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