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**A Bayesian \(\chi^2\) test for goodness-of-fit.**
*(English)*
Zbl 1068.62028

Summary: This article describes an extension of classical \(\chi^2\) goodness-of-fit tests to Bayesian model assessment. The extension, which essentially involves evaluating Pearson’s goodness-of-fit statistic at a parameter value drawn from its posterior distribution, has the important property that it is asymptotically distributed as a \(\chi^2\) random variable on \(K-1\) degrees of freedom, independently of the dimension of the underlying parameter vector.

By examining the posterior distribution of this statistic, global goodness-of-fit diagnostics are obtained. Advantages of these diagnostics include ease of interpretation, computational convenience and favorable power properties. The proposed diagnostics can be used to assess the adequacy of a broad class of Bayesian models, essentially requiring only a finite-dimensional parameter vector and conditionally independent observations.

By examining the posterior distribution of this statistic, global goodness-of-fit diagnostics are obtained. Advantages of these diagnostics include ease of interpretation, computational convenience and favorable power properties. The proposed diagnostics can be used to assess the adequacy of a broad class of Bayesian models, essentially requiring only a finite-dimensional parameter vector and conditionally independent observations.

### MSC:

62F15 | Bayesian inference |

62G10 | Nonparametric hypothesis testing |

62C10 | Bayesian problems; characterization of Bayes procedures |

62E20 | Asymptotic distribution theory in statistics |

### Keywords:

Pearson’s chi-squared statistic; posterior-predictive diagnostics; \(p\)-value; Bayes factor; intrinsic Bayes factor; discrepancy functions; expected order statistics; lip cancer data; Bayesian model assessment### References:

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