5_Sampling

Question 1

Rejection Sampling (Algorithm)

Answer 1

(1) Generate z0 ~q(y) [x-axis]
(2) Generate u0 ~ U[0, kq(z0)]
(3) If u0 > \tilde(p)(z0), reject the sample; retain z0 otherwise

Question 2

Markov Chain Monte Carlo Objective

Answer 2

Generate samples from an unknown target distribution.

Target distribution: the distribution we are interested in (e.g., posterior)

Question 3

MCMC detailed balance

Answer 3

Detailed balance, for all x, x’:

p^(x) T(x’ | x) = p^(x’) T(x | x’)

Also ensures that the Markov chain is reversible

Question 4

Metropolis-Hastings Algorithm

Answer 4

(1) Generate proposal x’ ~ q(x’ | x^{(t)})
(2) If

(q(x^{(t)} | x’) * \tilde{p}(x’)) / (q(x’ | x^{(t)}) * \tilde{p}(x^{(t)}) >= u,
where u ~ U[0, 1]

accept the sample x^{(t+1)} = x’; otherwise set x^{(t+1)} = x^{(t)}

Question 5

What can you tell about the acceptance rate in the beginning of the Metropolis-Hastings Algorithm?

Answer 5

In the beginning, a lot of samples get accepted as we move towards the area of our \tilde{p} distribution

Question 6

Is there a relation between step size and acceptance rate of the Metropolis-Hastings Algorithm?

Answer 6

• Big step size => low acceptance rate

- Small step size => high acceptance rate

Question 7

Slice Sampling Algorithm

Answer 7

Repeat the following steps:

(1) Draw u | x^{(t)} ~U[0, \tilde{p}(x)]
(2) Draw x={(t+1)} | u ~ U[{x: \tilde{p}(x) > u }], this is our ‘slice’