Lecture 9

Question 1

Markov Chain Monte Carlo Method

Answer 1

1. Sample random initial point P1 = (a1,b1)
2. Create a new pdf, Q, called the proposal density, on P1
3. Sample tentative new point P’ = (a’,b’) from Q
4. Compute R = L(a’,b’)/L(a1,b1)

Question 2

MCMC If R>1

Answer 2

this means P’ is uphill from P1. We accept P’ as the next point

Question 3

MCMC If R<1

Answer 3

this means P’ is downhill from P1. In this case we MAY reject P;.

how do we do this…?

a) generate a random number x ~ U[0,1]
b) If x < R then accept P’ and set P2 = P’
c) If x > R then reject P’ and set P2 = P1

Question 4

The sequence of points represents

Answer 4

a sample from the LF L(a,b)

Question 5

The sequence for each coordinate samples the

Answer 5

likelihood of a

Question 6

Fourier Series

Answer 6

f(x) = 1/2 ao + ∞ Σ n = 1 an cosnx + ∞ Σ n = 1 bn sinnx

Question 7

Markov Chain Monte Carlo - definition

Answer 7

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

Question 8

d(t) = δ(t-1) + δ(t+1)

Answer 8

D(f) = (inf ∫ -inf) δ(-1) e^(-2πift)dt + (inf ∫ -inf) δ(1) e^(-2πift)dt

Question 9

Modulus squared

Answer 9

H(f) H*(f)

multiplied by the conjugate

Question 10

rms

Answer 10

read off time or variance

Question 11

upper frequency

Answer 11

nyquist sampling theorem f measured = 2f source

Question 12

lower frequency

Answer 12

total data length

Question 13

parsavel’s theorem

Answer 13

the power in the time domain signal should be equal to the power in the frequency domain