### Abstract

In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance Dn between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate Dn/n converges to zero at rate (log n)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection.

Original language | English (US) |
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Pages (from-to) | 453-471 |

Number of pages | 19 |

Journal | IEEE Transactions on Information Theory |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - May 1990 |

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### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Cite this

*IEEE Transactions on Information Theory*,

*36*(3), 453-471. https://doi.org/10.1109/18.54897

**Information-Theoretic Asymptotics of Bayes Methods.** / Clarke, Bertrand S.; Barron, Andrew R.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 36, no. 3, pp. 453-471. https://doi.org/10.1109/18.54897

}

TY - JOUR

T1 - Information-Theoretic Asymptotics of Bayes Methods

AU - Clarke, Bertrand S.

AU - Barron, Andrew R.

PY - 1990/5

Y1 - 1990/5

N2 - In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance Dn between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate Dn/n converges to zero at rate (log n)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection.

AB - In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance Dn between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate Dn/n converges to zero at rate (log n)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection.

UR - http://www.scopus.com/inward/record.url?scp=0025430804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0025430804&partnerID=8YFLogxK

U2 - 10.1109/18.54897

DO - 10.1109/18.54897

M3 - Article

AN - SCOPUS:0025430804

VL - 36

SP - 453

EP - 471

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 3

ER -