Intro, Beta-Binomial, Dirichlet-Multinomial

Question 1

When is a set of data exchangable?

Answer 1

x1, . . . , xn are exchangeable,
meaning that the distribution p(x1, . . . , xn) is invariant to permutation of the
indices

Question 2

Give the posterior formula

Answer 2

Posterior ∝ Likelihood * Prior

π(θ|x) ∝ product(i=1 to n) fθ(xi) x π(θ)

Question 3

What is a conjugate prior?

Answer 3

prior and posterior are in the same family of distributions

Question 4

Give the posterior of a Beta-Binomial model

Answer 4

π(p|k) ∝ p^(k+α−1) * (1 − p)^(n−k+β−1) ∼ Be(k + α, n − k + β)

Question 5

Give the posterior mean of a Beta-Binomial model

Answer 5

(k + α) / (n + α + β)

Question 6

Give the posterior variance of the Beta-Binomial model

Answer 6

(k + α)*(β + n − k) / (n + α + β)^2 * (α + β + n + 1)

Question 7

Give the formula for a predictive posterior distribution

Answer 7

p(xn+1|x1, . . . , xn) = intgeral fθ(xn+1) * π(θ|x1, . . . , xn)dθ

Question 8

In large samples, what does the mean of the Beta-Binomial model approximately equal?

Answer 8

k / n

Question 9

In large samples, what does the variance of the Beta-Binomial model approximately equal?

Answer 9

k/(n^2) * (1- k/n)

Question 10

What is the posterior distribution of the Dirichlet-Multinomial model?

Answer 10

π(θ1, . . . , θm|y1,…,ym) ∼ Dirichlet (y1 + α1, . . . , ym + αm)

Question 11

Posterior mean of θi DM-model

Answer 11

(yi + αi) / sum(i=1 to m) (yi + αi)