STAT 462-Computational DataAnalysis
Chapter 8- Part 1
Nasser Sadeghkhani
October 2017
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Outline
Background and Concepts
I Bayes vs. Classical Statistics
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The basic tool of Bayesian statistics is Bayes theorem. It is namedafter Reverend Thomas Bayes, a nonconformist minister who livedin England in the first half of the eighteenth century. The theoremwas published posthumously in 1763 in ”An essay towards solvinga problem in the doctrine of chances”.
Figure: T. Bayes, 1702–1761
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Bayesian vs. Frequentist
The basic philosophical difference between the frequentist andBayesian paradigms is that Bayesians treat unknown parameters asrandom and use probability to quantify their uncertainty about it.In contrast, frequentists treat unknown parameters fixed.
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Bayes’ Theorem
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Continuous version in terms of a parameter.
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Historical Example
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More generally
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Normal model– Uniform Prior
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Normal model–Normal prior
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Conjugate PriorsDefinition: Let π(θ) be a class of prior distributions. Then it isconjugate for the model P (y|θ) whenever π(θ|y) belongs to thesame class (of prior).
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How to choose the prior distribution?
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Jeffrey’s Prior
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Example of Jeffrey’s prior
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Nuisance parameters
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