## The Annals of Mathematical Statistics

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

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

#### 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.Project Euclid: euclid.aoms/1177699477