Importance Sampling, Sequential Importance Sampling and more.

CS-295Spring 2008

Estimation

Function : An assignment of weights to a domain

Summation Problems

Ratio over sums

Xx

xf )(

Yy

Xx

yg

xf

)(

)(

Example: Function over 3 variables

A B C Weight

0 0 0 0.3

0 0 1 0.4

0 1 0 0.6

0 0 1 0.5

1 0 0 0.1

1 1 1 0.9

1 1 0 0.4

1 1 1 0.5

Specification: Function is a Product of smaller Functions

X: A set over n variables fi(y), i=1 to m is a function over a

subset of X f(X) is the product of all functions

XyyfxxXfm

iin

1

1 )(}),..,{(

Distribution: A function whose weights sum to 1.

))(|()(

))(|( :CPTs

1ii

i

XpaXPXP

XpaXPn

i

i

lung Cancer

Smoking

X-ray

Bronchitis

Dyspnoea

Xx

xxh )()(

USE REJECTION SAMPLING

g(x)f(X)

Rejection Control

Given a constant c>0. For j=1 to m

Accept x(j) with probability r(j) = min {1,w(j)/c)

If x(j) is accepted update its weight as w(*j)=w(j)/r(j)

Let g*(x) be the sampling distribution, then:

)(

)(

)(*

)(

xg

xVar

xg

xVar