Journal ArticleDOI
Inferences from Multinomial Data: Learning About a Bag of Marbles
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In this article, the imprecise Dirichlet model is proposed for multinomial data in cases where there is no prior information and the probabilities are expressed in terms of posterior upper and lower probabilities.Abstract:
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.read more
Citations
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TL;DR: An application of the measure of total uncertainty on convex sets of probability distributions, also called credal sets, to the construction of classification trees, using a total uncertainty measure (entropy + nonspecificity) as branching criterion.
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Inferring a possibility distribution from empirical data
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On Nonparametric Predictive Inference and Objective Bayesianism
TL;DR: An overview of recently developed theory and methods for nonparametric predictive inference (NPI), which is based on A(n) and uses interval probability to quantify uncertainty, and a discussion of NPI and objective Bayesianism.
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Credal-C4.5: Decision tree based on imprecise probabilities to classify noisy data
TL;DR: A modification of C4.5, called Credal-C4.
Book ChapterDOI
Imprecise Reliability: An Introductory Overview
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TL;DR: A lot of methods and models in classical reliability theory assume that all probabilities are precise, that is, that every probability involved is perfectly determinable.
References
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TL;DR: In this article, the effect of non-normality on inference about a population mean with generalizations was investigated. But the authors focused on the effect on the mean with information from more than one source.