## The Annals of Mathematical Statistics

### Limiting Behavior of Posterior Distributions when the Model is Incorrect

Robert H. Berk

#### Abstract

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."

#### Article information

Source
Ann. Math. Statist., Volume 37, Number 1 (1966), 51-58.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699597

Digital Object Identifier
doi:10.1214/aoms/1177699597

Mathematical Reviews number (MathSciNet)
MR189176

Zentralblatt MATH identifier
0151.23802

JSTOR
links.jstor.org

#### Citation

Berk, Robert H. Limiting Behavior of Posterior Distributions when the Model is Incorrect. Ann. Math. Statist. 37 (1966), no. 1, 51--58. doi:10.1214/aoms/1177699597. https://projecteuclid.org/euclid.aoms/1177699597

#### Corrections

• See Correction: Robert H. Berk. Correction Notes: Correction to Limiting Behavior of Posterior Distributions when the Model is Incorrect. Ann. Math. Statist., Volume 37, Number 3 (1966), 745--746.