Priors

Question 1

What are hyperparameters?

Answer 1

Hyperparameters are parameters of a prior distribution, e,g the hyperparameters of Gamma(α,β) are α and β

Question 2

Give two uninformative priors

Answer 2

Beta(1,1)

Uniform(0,1)

Question 3

What is an improper prior?

Answer 3

Prior where θ is continuous

Question 4

When are improper priors okay to use and why?

Answer 4

Used in an attempt to construct non-informative prior, okay as long as posterior is proper

Question 5

Define a conjugate prior

Answer 5

If p(θ|y) is in the same family of distributions as π(θ)

Question 6

For mixture distributions, give the marginal distribution

Answer 6

∫[qπ1(θ) + (1-q)π2(θ)] * f(y | θ) dθ

Question 7

For mixture distribution prior, give expectation

Answer 7

qμ1 + (1-q)μ2

Question 8

Give posterior for normal with unknown μ but known σ^2

Answer 8

N( (mμ0 + nybar) / m+n, σ^2/n)

Question 9

Give posterior for normal with unknown σ^2 but known μ

Answer 9

Gamma ( α + n/2, β + z^2/ 2 )

Question 10

Give the formula for Jeffreys’ prior

Answer 10

π(θ) ~ sqrt( - E ( second derivative of log-likelihood) )