这是一篇来自英国的关于SEMESTER 1 EXAMINATION 2022/23似然推理和贝叶斯推理的作业代写

1. [25 marks.] Suppose that we have a random sample of normal data Yi ∼ N(µ, σ2 ), i = 1, . . . , n  where σ 2 is known but µ is unknown. Thus for µ the likelihood function comes from the distribution Y¯ ∼ N(µ, σ2 /n), where Y¯ =1n P n i=1 Yi . Assume the prior distribution for µ is given by N(γ, σ2/n0) where γ and n0 are known constants.

(a) [10 marks] Show that the posterior distribution for µ is normal with mean = E(µ|y¯) =n0γ + ny¯n0 + n , variance = var(µ|y¯) = σ 2 n0 + n .

(b) [4 marks] Provide an interpretation for each of E(µ|y¯) and var(µ|y¯) in terms of the prior and data means and prior and data sample sizes.

(c) [4 marks] By writing a future observation Y˜ = µ + ϵ where ϵ ∼ N(0, σ2 ) independently of the posterior distribution π(µ|y¯) explain why the posterior predictive distribution of Y˜ given y¯ is normally distributed. Obtain the mean and variance of this posterior predictive distribution.

(d) [7 marks] Suppose that in an experiment n = 2, y¯ = 130, n0 = 0.25, γ = 120 and σ 2 = 25. Obtain:

(i) the posterior mean, E(µ|y¯) and variance, var(µ|y¯),

(ii) a 95% credible interval for µ given y¯,

(iii) the mean and variance of the posterior predictive distribution of a future observation Y˜ ,

(iv) a 95% prediction interval for a future observation Y˜ .

2. [25 marks.]

Suppose that y1, . . . , yn are i.i.d. observations from a Poisson distribution with mean θ where θ > 0 is unknown. Consider the following three models, that differ in the specification of the prior distribution of θ:

M1 : θ = 1,

M2 : π(θ) =b a Γ(

1. a) θ a−1 e −bθ, a > 0, b > 0,M3 : π(θ) ∝ p I(θ),

where I(θ) is the Fisher information number (see the formula sheet for its definition).

(a) [8 marks] Write down the likelihood function. Hence obtain the Jeffreys prior for θ given by π(θ) = p I(θ).

(b) [10 marks] Derive an expression for the Bayes factor for comparing models M1 and M2. If a = b = 1, n = 2 and y1 + y2 = 4, find the values of the Bayes factor. Which model is preferred?

(c) [7 marks] Explain why it is problematic to use the Bayes factor to compare M3 with any of the other two models. Describe an alternative approach for comparing M3 with any other model and discuss how it can be implemented using Monte Carlo methods.

3. Assume that we want to use a Gibbs sampler to estimate P(X1 ≥ 0, X2 ≥ 0) for a normal distribution N(µ, Σ), where µ =  µ 1µ2 and Σ =  σ 21σ12σ12σ 22 . The pdf of this distribution is f(x1, x2) ∝ exp ( − 12 x1 − µ1x2− µ2 T  σ 21σ12σ12σ 22 −1  x1 − µ1x2− µ2 ) .

(a) Show that f(x1|x2) ∝ exp  

−  x1−  µ1+σ12σ 22(x2−µ2)  22σ 21− σ 212σ 22, i.e.

X1|(X2 = x2) ∼ N  µ1 + σ12σ 22(x2 − µ2), σ21− σ 212σ 22 .

(b) Write down the conditional distribution of X2|(X1 = x1)

(c) Write down what the Gibbs sampler step is for some t = 1, 2, . . .

(d) Consider the special case µ1 = µ2 = 0, σ 2 1 = σ2 2 = 1 and σ12 = 0.4. Write the code in R to implement Gibbs sampler for this case (consider t = 1, 2, . . . , 4000 and x (0) 2 ∼ N(µ2, σ2 2 )).

(e) Plot the resulting chains. Estimate P(X1 ≥ 0, X2 ≥ 0). From the resulting chains, calculate and plot the evolution of P(X1 ≥ 0, X2 ≥ 0) over time t = 1, 2, . . . , 4000. Run the sampler 100 times and plot the replications of P(X1 ≥ 0, X2 ≥ 0) over time in grey (all in one plot). Add the original estimate of P(X1 ≥ 0, X2 ≥ 0) as a black line over the plotted range.