Monte Carlo Methods

Question 1

what are monte carlo methods

Answer 1

methods for generating random variables –
i.e. samples of numbers which
behave as if they are drawn
from some particular pdf (e.g.
uniform, Gaussian, Poisson etc).

Question 2

algorithms only generate pseudo-random numbers which are

Answer 2

very long (deterministic) sequences of numbers which are
approximately random (i.e. no discernible pattern).

Question 3

the better the random number generator, the

Answer 3

better it approximates U[0,1]

Question 4

phase portraits

Answer 4

scatterplots of the ith value against against the (i+1)th value

Question 5

we can compute the auto-correlation function where j is known as

Answer 5

the lag

Question 6

if the sequence is uniformly random, we expect

Answer 6

p(j)=1 for j=0
p(j)=0 otherwise

Question 7

generating random numbers from other pdfs can be done by

Answer 7

transforming random numbers drawn from simpler pdfs

Question 8

probability integral transform steps

Answer 8

1. sample a random variable y~U[0,1]
2. compute x such that y=p(x)
3. then x~p(x) ie x is drawn from the pdf corresponding to the cdf

Question 9

MCMC provides a

Answer 9

simple metropolis algorithm for generating random samples of points from L(a,b)

Question 10

MCMC steps

Answer 10

1. sample random initial point
2. create a new pdf Q called the proposal density on p1
3. sample tentative new point P’ from Q
4. compute R
5. if R>1 we accept next point
6. If R<1 we reject

Question 11

markov chain

Answer 11

acceptance probability depends only on the previous point

Question 12

If the variance of the proposal density is too small (σ=0.01) then

Answer 12

not all of the distribution is sampled

Question 13

If the variance of the proposal density is too large (σ=50) then

Answer 13

values jump around significantly

Question 14

if the variance of the proposal density is correct (σ=1) then

Answer 14

the entire distribution is sampled