Ancestral Sampling

31 Pages·2013·0.4 MB·English

Robert Collins
CSE586, PSU
Intro to Sampling Methods
CSE586 Computer Vision II
Penn State Univ
Robert Collins
Topics to be Covered
CSE586, PSU
Monte Carlo Integration
Sampling and Expected Values
Inverse Transform Sampling (CDF)
Ancestral Sampling
Rejection Sampling
Importance Sampling
Markov Chain Monte Carlo
Robert Collins
Integration and Area
CSE586, PSU
http://videolectures.net/mlss08au_freitas_asm/
Robert Collins
Integration and Area
CSE586, PSU
http://videolectures.net/mlss08au_freitas_asm/
Robert Collins
Integration and Area
CSE586, PSU
total # boxes
http://videolectures.net/mlss08au_freitas_asm/
Robert Collins
Integration and Area
CSE586, PSU
http://videolectures.net/mlss08au_freitas_asm/
Robert Collins
Integration and Area
CSE586, PSU
• As we use more samples, our answer should
get more and more accurate
• Doesn’t matter what the shape looks like
(1,1) (1,1)
(0,0) (0,0)
arbitrary region area under curve
(even disconnected) aka integration!
Robert Collins
Monte Carlo Integration
CSE586, PSU
Goal: compute definite integral of function f(x) from a to b
Generate N uniform random
upper bound samples in upper bound volume
c
N
f(x)
a b

K
Answer= * Area of upper bound volume
N
K
K
= * (b-a)(c-0)
N
count the K samples that

fall below the f(x) curve
Robert Collins
Motivation: Normalizing Constant
CSE586, PSU
Sampling-based integration is useful for computing the
normalizing constant that turns an arbitrary non-negative
function f(x) into a probability density function p(x).
Z
Compute this via sampling (Monte Carlo Integration). Then:
1
Z
Note: for complicated, multidimensional functions, this is the
ONLY way we can compute this normalizing constant.
Robert Collins
Motivation : Expected Values
CSE586, PSU
If we can generate random samples x from a given
i
distribution P(x), then we can estimate expected values of
functions under this distribution by summation, rather than
integration.
That is, we can approximate:
by first generating N i.i.d. samples from P(x) and then forming
the empirical estimate: