Chapter 2

Question 1

Random sample from a normal distribution in which both the mean μ and the precision τ are unknown, that is, Xi|μ,τ ∼N(μ,1/τ), i = 1,2,…,n (independent), with NGa(b,c,g,h) what posterior do you obtain (define any parameters)/

Answer 1

Get NGa(B,C,G,H)
B=(bc+n*xbar)/c+n
C=c+n
G=g+n/2
H=h+ns^2/2+(xbar-b)^2*cn/[2*(c+n)]
Note NGa is conjugate and E(μ|x) > E(μ) ⇐⇒  ̄x > b.

Question 2

Define the marginal distributions for mu, tau and sigma (both prior and posterior).(6)

Answer 2

The prior
(μ
τ)∼NGa(b,c,g,h) has marginal distributions
•μ ∼t2g(b, h/gc)
•τ ∼Ga(g,h), Also σ = 1/√τ∼Inv-Chi(g,h).
Similarly,
The posterior
(μ
τ)∼NGa(B,C,G,H) has marginal distributions
•μ ∼t2G(B, H/GC)
•τ ∼Ga(G,H), Also σ = 1/√τ∼Inv-Chi(G,H).

Question 3

Is the t distribution symmetric about 0?

Answer 3

Yes, equitailed HDI can therefore be obtained, therefore can obtain for mean but tau and sigma cannot as they are not symmetric.

Question 4

HDI formula for mu in multiparameter.(1)

Answer 4

(B −t2G;α/2√ H/GC, B + t2G;α/2√ H/GC).

Question 5

Predictive distribution for multi-parameter?

Answer 5

Same as univariate case using density or candidates formula (conjugate analysis only).

Question 6

How would you determine the predictive distribution from a Normal with NGa? What about the HDI?

Answer 6

we know Y is a random value from the population Y |µ, τ ∼ N(µ, 1/τ). We also know that the posterior distribution is (µ, τ)^T|x ∼ NGa(B,C, G, H).
Therefore, we can write Y = µ + ε,
where ε|τ ∼ N(0, 1/τ) and µ|x, τ ∼ N(B, 1/Cτ).
Hence Y is the sum of two independent normal random quantities, and so
Y|x, τ ∼ N(B, 1/τ+1/Cτ)≡ N(B, (C + 1)/Cτ ).
Thus using prior results we get:
Y |x ∼t2G{B, H(C + 1)/(GC)} as the final dist
HDI:
(B−t2G;α/2sqrt{H(C + 1)/GC} , B+t2G;α/2sqrt{H(C + 1)/GC)}