Q1. Exponential family expectations: Let p(y|Φ) = c(Φ)h(y) exp{Φt(y)} be an exponential family model.

a) Take derivatives with respect to Φ of both sides of the equation ∫p(y|Φ) dy = 1 to show that E[t(Y)|Φ] = -c'(Φ)/c(Φ).

b) Let p(Φ) ∝ c(Φ)^n_0e^n_0t_0Φ be the prior distribution for Φ. Calculate dp(Φ)/dΦ and, using the fundamental theorem of calculus, discuss what must be true so that E[-c(Φ)/c(Φ)] = t₀.

Q2. Posterior prediction: Consider a pilot study in which n₁ = 15 children enrolled in special education classes were randomly selected and tested for a certain type of learning disability. In the pilot study, y₁ = 2 children tested positive for the disability.

a) Using a uniform prior distribution, find the posterior distribution of θ, the fraction of students in special education classes who have the disability. Find the posterior mean, mode and standard deviation of θ, and plot the posterior density.

Researchers would like to recruit students with the disability to participate in a long-term study, but first they need to make sure they can recruit enough students. Let n₂ = 278 be the number of children in special education classes in this particular school district, and let Y₂ be the number of students with the disability.

b) Find Pr(Y₂ = y₂|Y₁ = 2), the posterior predictive distribution of Y₂, as follows:

i. Discuss what assumptions are needed about the joint distribution of (Y₁, Y₂) such that the following is true:

Pr(Y₂ = y₂|Y₁ = 2) = ₀∫¹ Pr(Y₂ = y₂|θ)p(θ|Y₁ = 2)dθ.

ii. Now plug in the forms for Pr(Y₂ = y₂|θ) and p(θ|Y₁ = 2) in the above integral.

iii. Figure out what the above integral must be by using the calculus result discussed in Section 3.1.

c) Plot the function Pr(Y₂ = y₂|Y₁ = 2) as a function of y₂. Obtain the mean and standard deviation of Y₂, given Y₁ = 2.

d) The posterior mode and the MLE of θ, based on data from the pilot study, are both θ^{^} = 2/15. Plot the distribution Pr(Y₂ = y₂|θ = θ^{^}), and find the mean and standard deviation of Y₂ given θ = θ^{^}. Compare these results to the plots and calculations in c) and discuss any differences. Which distribution for Y₂ would you use to make predictions, and why?

Q3. Tumor counts: A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, A and B. They have tumor count data for 10 mice in strain A and 13 mice in strain B. Type A mice have been well studied, and information from other laboratories suggests that type A mice have tumor counts that are approximately Poisson-distributed with a mean of 12. Tumor count rates for type B mice are unknown, but type B mice are related to type A mice. The observed tumor counts for the two populations are

y_A = (12, 9, 12, 14, 13, 13, 15, 8, 15, 6);

y_B = (11, 11, 10, 9, 9, 8, 7, 10, 6, 8, 8, 9, 7).

a) Find the posterior distributions, means, variances and 95% quantile-based confidence intervals for θ_A and θ_B, assuming a Poisson sampling distribution for each group and the following prior distribution:

θ_A ∼ gamma(120,10), θ_B ∼ gamma(12,1), p(θ_A, θ_B) = p(θ_A) x p(θ_B).

b) Compute and plot the posterior expectation of θ_B under the prior distribution θ_B ∼ gamma(12 x n₀, n₀) for each value of n₀ ∈ {1, 2, ... , 50} . Describe what sort of prior beliefs about θ_B would be necessary in order for the posterior expectation of θ_B to be close to that of θ_A.

c) Should knowledge about population A tell us anything about population B? Discuss whether or not it makes sense to have p(θ_A, θ_B) = P(θ_A) x P(θ_B).

Attachment:- Assignment.rar