We consider the asymptotic behavior of posterior distributions and Bayes estimators based on obse... more 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. POSTERIOR CONVERGENCE RATES 193 dressed by Amewou-Atisso, Ghosal, Ghosh and Ramamoorthi [1] and Choudhuri, Ghosal and Roy . The main purpose of the present paper is to obtain a theorem on rates of convergence of posterior distributions in a general framework not restricted to the setup of i.i.d. observations. We specialize this theorem to several classes of non-i.i.d. models including i.n.i.d. observations, Gaussian time series, Markov processes and the white noise model. The theorem applies in every situation where it is possible to test the true parameter versus balls of alternatives with exponential error probabilities and it is not restricted to any particular structure on the joint distribution. The existence of such tests has been proven in many special cases by Le Cam and Birgé , who used them to construct estimators with optimal rates of convergence, determined by the (local) metric entropy or "Le Cam dimension" of the model. Our main theorem uses the same metric entropy measure of the complexity of the model and combines this with a measure of prior concentration around the true parameter to obtain a bound on the posterior rate of convergence, generalizing the corresponding result of Ghosal, Ghosh and van der Vaart . We apply these results to obtain posterior convergence rates for linear regression, nonparametric regression, binary regression, Poisson regression, interval censoring, spectral density estimation and nonlinear autoregression. van der Meulen, van der Vaart and van Zanten have extended the approach of this paper to several types of diffusion models.
We consider the asymptotic behavior of posterior distributions and Bayes estimators based on obse... more 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. POSTERIOR CONVERGENCE RATES 193 dressed by Amewou-Atisso, Ghosal, Ghosh and Ramamoorthi [1] and Choudhuri, Ghosal and Roy . The main purpose of the present paper is to obtain a theorem on rates of convergence of posterior distributions in a general framework not restricted to the setup of i.i.d. observations. We specialize this theorem to several classes of non-i.i.d. models including i.n.i.d. observations, Gaussian time series, Markov processes and the white noise model. The theorem applies in every situation where it is possible to test the true parameter versus balls of alternatives with exponential error probabilities and it is not restricted to any particular structure on the joint distribution. The existence of such tests has been proven in many special cases by Le Cam and Birgé , who used them to construct estimators with optimal rates of convergence, determined by the (local) metric entropy or "Le Cam dimension" of the model. Our main theorem uses the same metric entropy measure of the complexity of the model and combines this with a measure of prior concentration around the true parameter to obtain a bound on the posterior rate of convergence, generalizing the corresponding result of Ghosal, Ghosh and van der Vaart . We apply these results to obtain posterior convergence rates for linear regression, nonparametric regression, binary regression, Poisson regression, interval censoring, spectral density estimation and nonlinear autoregression. van der Meulen, van der Vaart and van Zanten have extended the approach of this paper to several types of diffusion models.
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