一种结合客观贝叶斯和基于信息的推理的客观先验
An objective prior that unifies objective Bayes and information-based inference
摘要：
统计推断有三种主要范式：（i）贝叶斯，（ii）基于信息的和（iii）频率论推断。我们描述了一个客观的先验（加权或$w$先验），它统一了客观贝叶斯和基于信息的推理。选择$w$ -先验，使边际概率成为模型预测性能的无偏估计量。这个定义还有其他几种自然的解释。从先验的信息内容的角度来看，$w$ -先验是均匀的和最大的无信息的。$w$ -先验也可以理解为导致参数空间中可区分模型的均匀密度。最后证明了正则模型的$w$ -先验与赤池信息准则（AIC）在渐近极限上是等价的。$w$ -先验似乎一般适用于统计推断，并且没有{ \it ad hoc}正则化。抑制复杂性的机制类似于AIC：模型复杂性降低了模型的预测性。我们期望这种新的客观贝叶斯推理方法广泛适用于包括奇异模型在内的机器学习问题。
英文摘要：
There are three principle paradigms of statistical inference: (i) Bayesian, (ii) information-based and (iii) frequentist inference. We describe an objective prior (the weighting or $w$-prior) which unifies objective Bayes and information-based inference. The $w$-prior is chosen to make the marginal probability an unbiased estimator of the predictive performance of the model. This definition has several other natural interpretations. From the perspective of the information content of the prior, the $w$-prior is both uniformly and maximally uninformative. The $w$-prior can also be understood to result in a uniform density of distinguishable models in parameter space. Finally we demonstrate the the $w$-prior is equivalent to the Akaike Information Criterion (AIC) for regular models in the asymptotic limit. The $w$-prior appears to be generically applicable to statistical inference and is free of {\it ad hoc} regularization. The mechanism for suppressing complexity is analogous to AIC: model complexity reduces model predictivity. We expect this new objective-Bayes approach to inference to be widely-applicable to machine-learning problems including singular models.
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