论文标题
进一步看贝叶斯盲点
A Further Look at the Bayes Blind Spot
论文作者
论文摘要
Gyenis和Redei已经证明，有限代数上的任何先前的P（无论选择）严重限制了通过杰弗里（Jeffrey）调节在非平凡分区上可以从p访问的后期。他们的演示涉及表明，相对于三种常见的大小衡量标准，即具有基数c，（归一化的）lebesgue Measure 1，而Baire第二类对于自然拓扑而言。在本文中，我们为在任何无限无限集的亚集的无限sigma代数上定义的概率措施建立了类似的结果。但是，我们需要采用明显不同的方法来确定基数，尤其是在无限情况下贝叶斯盲点的拓扑和测量方法。有趣的是，我们为一个先前的P所建立的所有结果继续在许多先生的贝叶斯盲点的交叉点上保持。这使我们猜想贝叶斯学习本身可能与先验在使大贝叶斯盲点存在时所施加的限制一样。
Gyenis and Redei have demonstrated that any prior p on a finite algebra, however chosen, severely restricts the set of posteriors accessible from p by Jeffrey conditioning on a nontrivial partition. Their demonstration involves showing that the set of posteriors not accessible from p in this way (which they call the Bayes blind spot of p) is large with respect to three common measures of size, namely, having cardinality c, (normalized) Lebesgue measure 1, and Baire second category with respect to a natural topology. In the present paper, we establish analogous results for probability measures defined on any infinite sigma algebra of subsets of a denumerably infinite set. However, we have needed to employ distinctly different approaches to determine the cardinality, and especially, the topological and measure-theoretic sizes of the Bayes blind spot in the infinite case. Interestingly, all of the results that we establish for a single prior p continue to hold for the intersection of the Bayes blind spots of countably many priors. This leads us to conjecture that Bayesian learning itself might be just as culpable as the limitations imposed by priors in enabling the existence of large Bayes blind spots.
