Inferences from Multinomial Data: Learning About a Bag of Marbles
多项数据的推断：从一袋弹珠中学习AI 解读 ↓
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多项数据的推断：从一袋弹珠中学习
- 研究问题
- 在缺少先验信息、甚至不知道全部可能结果时，如何从多项观测作出推断，并避免结论依赖样本空间的表示方式？
- 方法与思路
- 作者提出不精确 Dirichlet 模型，以后验概率上下界表达推断，从初始无信息状态随观测逐步收窄，并分析弹珠与医疗试验案例。
- 材料陈述的贡献
- 摘要称模型满足所提出的不变性原则并产生相容推断；案例展示与客观贝叶斯及频率方法的比较和列联表数据分析。
- 阅读关注点
- 应核对不变性与推断相容性的定义、模型参数如何设定、概率界如何更新，以及治疗比较和试验终止判断采用的准则。
- 阅读边界
- 仅据所给 Crossref 摘要；概率界的具体构造及试验决策性能未说明，案例不足以证明该方法在所有决策问题中更优。
原文摘要
SUMMARY A new method is proposed for making inferences from multinomial data in cases where there is no prior information. A paradigm is the problem of predicting the colour of the next marble to be drawn from a bag whose contents are (initially) completely unknown. In such problems we may be unable to formulate a sample space because we do not know what outcomes are possible. This suggests an invariance principle: inferences based on observations should not depend on the sample space in which the observations and future events of interest are represented. Objective Bayesian methods do not satisfy this principle. This paper describes a statistical model, called the imprecise Dirichlet model, for drawing coherent inferences from multinomial data. Inferences are expressed in terms of posterior upper and lower probabilities. The probabilities are initially vacuous, reflecting prior ignorance, but they become more precise as the number of observations increases. This model does satisfy the invariance principle. Two sets of data are analysed in detail. In the first example one red marble is observed in six drawings from a bag. Inferences from the imprecise Dirichlet model are compared with objective Bayesian and frequentist inferences. The second example is an analysis of data from medical trials which compared two treatments for cardiorespiratory failure in newborn babies. There are two problems: to draw conclusions about which treatment is more effective and to decide when the randomized trials should be terminated. This example shows how the imprecise Dirichlet model can be used to analyse data in the form of a contingency table.