Open Access
February, 1966 Limiting Behavior of Posterior Distributions when the Model is Incorrect
Robert H. Berk
Ann. Math. Statist. 37(1): 51-58 (February, 1966). DOI: 10.1214/aoms/1177699597


The large sample behavior of posterior distributions is examined without the assumption that the model is correct. Under certain conditions it is shown that asymptotically, the posterior distribution for a parameter $\theta$ is confined to a set (called the asymptotic carrier) which may, in general, contain more than one point. The asymptotic carrier depends on the model, the carrier of the prior distribution and the actual distribution of the observations. An example shows that, in general, there need be no convergence (in any sense) of the posterior distribution to a limiting distribution over the asymptotic carrier. This is in contrast to the (known) asymptotic behavior when the model is correct; see e.g. [7], p. 304: the asymptotic carrier then contains only one point, the "true value" of $\theta$ and the posterior distribution converges in distribution to the distribution degenerate at the "true value."


Download Citation

Robert H. Berk. "Limiting Behavior of Posterior Distributions when the Model is Incorrect." Ann. Math. Statist. 37 (1) 51 - 58, February, 1966.


Published: February, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0151.23802
MathSciNet: MR189176
Digital Object Identifier: 10.1214/aoms/1177699597

Rights: Copyright © 1966 Institute of Mathematical Statistics

Vol.37 • No. 1 • February, 1966
Back to Top