Large Deviation Principles for Posterior Distributions of the Normal Parameters

概要

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Suppose that X1,X2,... are conditionally i.i.d. random variables withdistribution Pθ given ＝θ，where is an unknown parameter. If Pθis a normal distribution with mean θ and known variance σ2, and if theprior of is chosen from the conjugate family N(μ,v2) or proportionalto the Lebesgue measure, then it follows that the posterior distributionsgiven X1,...,Xn obey a large deviation principle with a rate function. IfPθ is a normal distribution with known mean and unknown precision θ,and if as a prior we choose the gamma distribution with parameters αand β or the improper distribution (1/θ)dθ，the Jeffereys' prior, thenthe posterior distributions of given X1,...,Xn are shown to satisfy alarge deviation principle. The G artner-Ellis theorem plays the key roleto prove these large deviation principles for the posterior distributions.
2012-03-25