Lecture 20 | Bayesian analysis

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STATS 200: Introduction to Statistical InferenceAutumn 2016Lecture 20 Bayesian analysisOur treatment of parameter estimation thus far has assumed that is an unknown butnon-random quantity it is some fixed parameter describing the true distribution of data,and our goal was to determine this parameter. This is the called thefrequentistparadigmof statistical inference. In this and the next Lecture , we will describe an alternativeBayesianparadigm, in which itself is modeled as a random variable. The Bayesian paradigm natu-rally incorporates our prior belief about the unknown parameter , and updates this beliefbased on observed Prior and posterior distributionsRecall that ifX,Yare two random variables having joint PDF or PMFfX,Y(x,y), then themarginal distributionofXis given by the PDFfX(x) = fX,Y(x,y)dyin the continuous case and by the PMFfX(x) = yfX,Y(x,y)in the discrete case; this describes the probability distribution ofXalone.

In Bayesian analysis, before data is observed, the unknown parameter is modeled as a random variable having a probability distribution f ( ), called the prior distribution. This distribution represents our prior belief about the value of this parameter. Conditional on = , the observed data Xis assumed to have distribution f Xj (xj ), where f Xj ...

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