Univariate Gaussian Models

Question 1

Inference for µ when σ^2 is known

Conjugate prior, what is the posterior?

Answer 1

π(µ|x1, . . . , xn) ∼ N(a,b^2)

a = [ (n/σ1^2) * xbar + µ0/σ0^2 ] / [n/σ1^2 + 1/σ0^2]

b^2 = 1/ [n/σ1^2 + 1/σ0^2 ]

Question 2

Inference for µ when σ^2 is known

Conjugate prior, what is the posterior mean?

Answer 2

a = [ (n/σ1^2) * xbar + µ0/σ0^2 ] / [n/σ1^2 + 1/σ0^2 ]

Question 3

Inference for µ when σ^2 is known

Conjugate prior, what is the credible interval?

Answer 3

[a − z_α/2 * b, a + z_α/2 * b]

Question 4

What is Jeffrey’s prior?

Answer 4

−E[∂^2/∂^2µ log f (X|µ)]

Question 5

Inference for µ when σ^2 is known

Uninformative prior, what is the posterior distribution and mean?

Answer 5

N(xbar, σ1^2/ n)

Question 6

Inference for µ when σ^2 is known

Uninformative prior, what is the credible interval?

Answer 6

[ xbar ± z(α/2) * (σ1/sqrt(n)) ]

Question 7

Inference for σ^2 when µ is known

Conjugate prior, how do you re-parametrise?

Answer 7

φ = 1 / σ^2

Question 8

Inference for σ^2 when µ is known

Conjugate prior, what is the posterior of φ?

Answer 8

Gamma ( (a+n) / 2 , [ b + (n-1)s^2 + n(xbar - mu)^2] / 2 )

Question 9

Inference for σ^2 when µ is known

Non-informative prior, what is the posterior?

Answer 9

Gamma( n/2, [sum(i=1 to n) (xi - mu1)^2 ] /2 )