## Bayesian Analysis

### Bayesian model assessment using pivotal quantities

Valen E. Johnson

#### Abstract

Suppose that $S({\bf Y},\Theta)$ is a function of data ${\bf Y}$ and a model parameter $\Theta$, and suppose that the sampling distribution of $S({\bf Y},\Theta)$ is invariant when evaluated at $\Theta_0$, the "true" (i.e., data-generating) value of $\Theta$. Then $S({\bf Y},\Theta)$ is a pivotal quantity, and it follows from simple probability calculus that the distribution of $S({\bf Y},\Theta_0)$ is identical to the distribution of $S({\bf Y},\Theta_{\bf Y})$, where $\Theta_{\bf Y}$ is a value of $\Theta$ drawn from the posterior distribution given ${\bf Y}$. This fact makes it possible to define a large number of Bayesian model diagnostics having a known sampling distribution. It also facilitates the calibration of the joint sampling of model diagnostics based on pivotal quantities.

#### Article information

Source
Bayesian Anal. Volume 2, Number 4 (2007), 719-733.

Dates
First available in Project Euclid: 22 June 2012

http://projecteuclid.org/euclid.ba/1340370712

Digital Object Identifier
doi:10.1214/07-BA229

Mathematical Reviews number (MathSciNet)
MR2361972

Zentralblatt MATH identifier
1331.62147

#### Citation

Johnson, Valen E. Bayesian model assessment using pivotal quantities. Bayesian Anal. 2 (2007), no. 4, 719--733. doi:10.1214/07-BA229. http://projecteuclid.org/euclid.ba/1340370712.

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