Monte Carlo Markov Chains

Question 1

What is the main idea of MCMC?

Answer 1

generate samples from a MC such that eventually a stationary distribution is reached and that stationary distribution is the target distribution

Question 2

What is a reversible markov chain?

Answer 2

πi * Pr( θt = j | θt-1 = i) = πj * Pr( θt = i | θt-1 = j)

ie, πi * pij = πj * pji

Question 3

Key theoretical result of MCMC chains

Answer 3

if a markov chain is aperiodic, irreducible and has stationary distribution π, then π is unique

Question 4

What is the objective of the Metropolis-Hastings algorithm?

Answer 4

generate a sample from a target distribution

Question 5

Outline the Metropolis-Hastings algorithm

Answer 5

• at iteration i, have value θ
  1. Generate θc from q(θc | θi)
  2. calculate MH-ratio
  = p(θc) * q(θi | θc) / p(θi) * q(θc | θi)
  3. generate u from uniform(0,1)
  4. if u ≤ min{1, MHR}, θi+1 = θc, else θi+1 = θi
  5. set i = i+1 and return to 1

Question 6

Give the BGR statistic

Answer 6

R = width of 80% credible interval of all chains combined / average width of 80% credible interval for individual chains

Question 7

Effective sample size

Answer 7

neff = n / 1 + 2*sum correlations

Want neff close to sample size

Question 8

Outline the Gibbs-Sampler algorithm

Answer 8

1. generate θi(t+1) from p(θi | θ1(t+1), …, θ(i-1)(t+1), θ(i+1)(t), .., θq(t) )