Question

Gelman's (2006) paper on noninformative prior distributions for variance parameters has attracted over 3,100 citations (Google scholar). The Half-t distribution introduced in that paper is now considered the "default" prior to use for any standard deviation parameter. a) Show that if σ 2 is generated from the following hierarchical model. λ= 1/A"), a ~ Inv-Gamma(α=1/2, then the (positive) square root σ has a marginal Half-t(v, A) distribution with density Here, the "scale" A > 0 and "degrees-of-freedom" v > 0 are fixed hyperparameters. b) Simulate N 100,000 1.1.d. samples of σ using the above hierarchical model with A- v- 1. Plot a KDE of the sample. c) After each simulation, compute the running sample mean and plot this against n, forn -1,2,...,100,000. d) Does the sample mean settle down to some fixed value as the sample size n increases? If so, what is that value?

Answer 1

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