### Abstract

Let (X_{1}, . . . , X_{n}) be independently and identically distributed observations from an exponential family p_{θ} equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ_{0}, admits an asymptotic expansion in terms of the Fisher information I(θ_{0}), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^{d/2}C_{1}(d, θ_{0}) and the second order term is of the form n^{d/2-1}c_{2}(d, θ_{0}, w), with an error term that is o(n^{d/2-1}). We identify the functions C_{1} and C_{2} explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ_{0}. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Original language | English (US) |
---|---|

Pages (from-to) | 163-185 |

Number of pages | 23 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1999 |

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### Keywords

- Asymptotics
- Bayes factor
- Chi-squared distance
- Expected posterior
- Relative entropy

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Annals of the Institute of Statistical Mathematics*,

*51*(1), 163-185. https://doi.org/10.1023/A:1003891404142

**Asymptotics of the expected posterior.** / Clarke, Bertrand S; Sun, Dongchu.

Research output: Contribution to journal › Article

*Annals of the Institute of Statistical Mathematics*, vol. 51, no. 1, pp. 163-185. https://doi.org/10.1023/A:1003891404142

}

TY - JOUR

T1 - Asymptotics of the expected posterior

AU - Clarke, Bertrand S

AU - Sun, Dongchu

PY - 1999/1/1

Y1 - 1999/1/1

N2 - Let (X1, . . . , Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2C1(d, θ0) and the second order term is of the form nd/2-1c2(d, θ0, w), with an error term that is o(nd/2-1). We identify the functions C1 and C2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

AB - Let (X1, . . . , Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2C1(d, θ0) and the second order term is of the form nd/2-1c2(d, θ0, w), with an error term that is o(nd/2-1). We identify the functions C1 and C2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

KW - Asymptotics

KW - Bayes factor

KW - Chi-squared distance

KW - Expected posterior

KW - Relative entropy

UR - http://www.scopus.com/inward/record.url?scp=6544265019&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=6544265019&partnerID=8YFLogxK

U2 - 10.1023/A:1003891404142

DO - 10.1023/A:1003891404142

M3 - Article

VL - 51

SP - 163

EP - 185

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 1

ER -