Let X, X2, Xn be i.i.d.~ N(, 02), u, o2 both unknown. Consider the set up...

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Let X₁, X2, Xn be i.i.d.~ N(μ, 02), u, o2 both unknown. Consider the set up of Jeffreys test (see class note). (a) Show that if X → ∞ and S² is bounded, BF01 goes to zero for the Cauchy prior but does not go to zero for normal prior. (b)Consider Cauchy prior for this problem. Using the representation of the Cauchy density 91 (μo) as a scale mixture of normals, express the integrated likelihood under H₁ as a one-dimensional integral (over the mixing variable 7). Let X₁, X2, Xn be i.i.d.~ N(μ, 02), u, o2 both unknown. Consider the set up of Jeffreys test (see class note). (a) Show that if X → ∞ and S² is bounded, BF01 goes to zero for the Cauchy prior but does not go to zero for normal prior. (b)Consider Cauchy prior for this problem. Using the representation of the Cauchy density 91 (μo) as a scale mixture of normals, express the integrated likelihood under H₁ as a one-dimensional integral (over the mixing variable 7).

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