ch27-lecture_Page_01
ch27-lecture_Page_02
ch27-lecture_Page_02
ch27-lecture_Page_03
ch27-lecture_Page_03
ch27-lecture_Page_04
ch27-lecture_Page_04
ch27-lecture_Page_04
ch27-lecture_Page_05
ch27-lecture_Page_05
ch27-lecture_Page_06
2
ch27-lecture_Page_06
2
ch27-lecture_Page_07
ch27-lecture_Page_08
ch27-lecture_Page_09
ch27-lecture_Page_10
Why? Because Z needs to be computed byintegration which is often a reason why we needsampling.
ch27-lecture_Page_10
ch27-lecture_Page_10
Idea: Draw a random sample (x,y)and keep x if y = 1.
Need to enforce p(x)/q(x) is a prob.
ch27-lecture_Page_11
ch27-lecture_Page_12
ch27-lecture_Page_14
ch27-lecture_Page_14
How do we handleevidence?
ch27-lecture_Page_16
ch27-lecture_Page_16
ch27-lecture_Page_16
ch27-lecture_Page_17
ch27-lecture_Page_18
ch27-lecture_Page_20
Screen Shot 2013-12-08 at 4.30.24 PM.png
ch27-lecture_Page_21
Need to enforce the right side to be a pdf!
Same as Rejection Sampling:
Draw x’ with probability q(x’|x) =
a)Draw x’ with probability q(x’|x) AND
b)Accept x’ when a unif(0,1) u < f(x’,x)
~
ch27-lecture_Page_21
Enforcing the stationary distribution requirement:
ch27-lecture_Page_21
Enforcing the stationary distribution requirement:
ch27-lecture_Page_22
ch27-lecture_Page_23
ch27-lecture_Page_23
ch27-lecture_Page_24
ch27-lecture_Page_24
ch27-lecture_Page_25
q(x) is proposaldistribution
ch27-lecture_Page_25
q(x) is proposaldistribution
ch27-lecture_Page_26
Particle Filtering
A particle is a pair of random variables : state x and its weight  0
A particle set for a PDF f is an algorithm to generate (xii) such that for anyfunction g:
limn ig(xi) = E(g(x))   “Converge by distribution”
Probability
State
xi = centroid of ellipse
= area of ellipse
Operations on particles
Idea: use particles {xii}i=1,..N to represent P(xt|y1,...,yt)
Recall
time update : P(xt+1 | y1,...,yt)=∫ P(xt+1|xt) P(x| y1,...,yt) dxt   “Convolution”
prediction : P(yt+1| y1,...,yt)= P(yt+1= z |xt+1) P(xt+1 | y1,...,yt) dxt+1 “Convolution”
measurement: P(xt+1 | y1,...,yt+1 P(yt+1| xt+1) P(xt+1| y1,...,yt“Multiplication”
Assume we know how to evaluate and generate randomsamples from all functions in red.
How to “convolve” and “multiple” sets of particles with otherfunctions?
Multiplication and Convolution ofparticles
Multiply by q(x)
           xi xi
               i q(xi i
Convolution with q(y|x)
1.Resampling {xii}i=1..N to a new set of particles {xi’, i’} i=1..N
{xii}i=1..N
       {xi’,i’}i=1..N
2.Reweighting                xi’  Sample based on q(x|xi’)
               i i
Why resampling?
=
Without resampling:
0
With resampling:
there are a lot more ...
 better resampling : fewer xi have the same values
 how many particles?  Effective sampling size