Show that the normalized likelihood distribution is a

Question

Q1. A doctor gives a patient a test for a disease. The test is pretty accurate: if given to a person with the disease, it will give a positive result 99% of the time, and if given to a person without the disease, it will give a negative result 99% of the time. This particular patient tests positive, and the doctor claims that now the probability this patient has the disease is 99%. Show that this is not necessarily true by constructing an example where the correct probability is less than 99%.

Q2. Let X₁, ..., X_n be a random sample from a geometric distribution with parameter θ, i.e, p(x_i|θ) = θ(1 - θ)^{x_i -1}. Let the prior for θ be a beta distribution with parameters α and β:

p(θ|α, β) = (Γ(α+β)/Γ(α)Γ(β))θ^α-1(1-θ)^β-1

(for θ ∈ [0, 1]). Recall that E(θ)= α/(α+β) and E(X_i|θ) = 1/θ.

(a) Find the maximum likelihood estimate for θ.

(b) Show that the beta distribution is conjugate for θ, i.e., find the posterior distribution for θ and show that it is a beta distribution.

(c) Show that the posterior mean is a weighted average of the prior mean and the maximum likelihood estimate.

(d) Make up three observations and appropriate values for α and β. Compute the standard 95% posterior interval estimate and the 95% HPD interval estimate for θ. Compare your intervals.

Q3. Suppose we are going to observe data that are from a uniform distribution with unknown upper endpoint, i.e.:

          y_i ∼^iid Unif[0, θ] i = 1, . . . , n

with prior θ ∼ Pareto(α, β)

                    where p(θ|α, β) = αβ^αθ^-(α+1)I_{{θ ≥ β}}

Assuming that α > 2, the Pareto has mean  αβ/α-1 and variance αβ²/(α-1)(α-2).

(a) Show that the normalized likelihood distribution is a Pareto(n - 1, m,) where n is the sample size and 9n is the maximum of the observations.

(b) Compare the expressions for the posterior mean, prior mean, and the mean of the likelihood.

(c) Write out the posterior variance. When considered as a function of the sample size n, what is unusual about this expression to most standard cases?

Q4. Show (derive) that under the normal model with a non-informative prior distribution:

y₁, . . . , y_n ∼^iid N(μ, σ²)

p(μ, σ²) ∝ (σ²)^-1

that the posterior predictive distribution for a new observation y^∼ is a t distribution with location y^- and scale (1 + 1/n)^½ s, and degrees of freedom n-1. You do not need to derive p(μ|σ², data) or p(σ²|data), you can use the results for these distributions found in class.

Q5. In March 2011, a 9.0 magnitude earthquake occurred off the coast of Japan, triggering an aggressive tsunami. The effects of the earthquake and tsunami were of the most devastating in recorded history. The nuclear power plant in Fukashima was essentially obliterated, from nuclear meltdowns to chemical explosions. Fukashima was the largest nuclear accident since Chernobyl in 1986. Radioactive material spilled into the surrounding land and water mass exposing people, plants, and animals to radiation. Local fisheries were particularly concerned with the level of radioactive cesium in consumable fish (seafood). The local government regulations state that the maximum consumable of level of radioactive cesium to be 100 becquerels per kilogram. The dataset radioces.R describe the radioactive cesium levels of 20 fish from a particular location off the coast of Japan in 2012. Assume the following normal conjugate model for these data:

y₁, . . . , y_n ∼^iid N(μ, σ²)

               μ ∼ N(m_μ, s²_μ)

              σ² ∼ Γ^-1(2, b_σ(scale))

(a) Suppose we know that the levels of radioactive cesium in a nearby location ranged from 40 - 120 becquerels per kilogram. Use this prior information to obtain values for your prior parameters. Check you values by obtaining a plot of your prior predictive distribution.

(b) Derive the posterior for p(σ²|data) and p(μ|σ², data). Use Monte Carlo to obtain posterior samples from the joint posterior distribution, and provide a plot of your posterior sample pairs.

(c) Derive the posterior for p(μ|data). Provide plots overlaying the empirical MC and actual marginal distribution of σ² and μ.

(d) Simulate from the posterior predictive distribution, and plot your results. What is the probability that a new observed specimen will surpass the maximum consumable of level of radioactive cesium?

(e) Derive the full conditional posterior for p(σ²|μ data) and p(μ|σ², data) (although the later has been done). Use Gibbs sampling to obtain posterior samples from the joint posterior distribution, and provide a plot of your posterior sample pairs. Compare your results to (b).

Attachment:- Assignment File.rar