We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth-type.

中文翻译：

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关于后密度的形式 Edgeworth 展开的有效性

我们考虑参数贝叶斯理论中的一个基本开放问题，即后验密度的形式 Edgeworth 展开的有效性。虽然对后验分布的有效渐近展开的研究构成了丰富的文献，但正式的 Edgeworth 展开的有效性尚未得到严格的确立。几位作者声称各种后验扩展与经典 Edgeworth 扩展之间存在联系，或者只是假设其有效性。我们的主要结果解决了这个悬而未决的问题。我们还证明了一个关于后累积量阶的引理，这在贝叶斯参数理论中是独立的。最相关的文献被综合并与新派生的 Edgeworth 展开进行比较。