[SOLVED] B555-Written Assignment 3

Description

1. Bayesian Inference for Univariate Normal: Consider an IID sample for a Normal variable

with mean µ and precision λ, and consider a Normal-Gamma prior given by

) Gamma(λ|a₀,b₀).

Calculate the posterior for µ and λ and show that it is also decomposable as a Normal-Gamma distribution, with new parameters a_N,b_N,β_N,µ_N. In particular explicitly calculate and show the parameters of the posterior when given in this form.

2. Calculate one iteration of the Newton-Raphson method for minimizing the function f(x) = where x = (x₁,x₂)^T. Use x = (1,1)^T as the initial value.
3. Solve problem 4.18 (page 223) in the textbook.
4. In this question we develop the properties for the scaled-variants of exponential family distributions that were used in the derivation of GLM.
   • Show that if p₁(x) is a normalized probability density function then) is also a normalized probability density function.
   • Now consider the scaled variant of the one dimensional exponential family distribution. Show that the expectation parameter θ = E[x|η,s] satisfies
   • Show that the variance satisfies
5. Consider a Poisson likelihood function, Poisson(, with prior, Gamma(λ|a,b), on λ. Develop a Gaussian approximation to p(λ|x,a,b) using the Laplace approximation. First develop the formula in general and then apply it to the case a = 1, b = 1, x = 3. For this case, plot the approximation and the true function for 0 ≤ λ ≤ 10.