geometry of dempsters rule navlab - autonomous navigation and computer vision lab department of...
TRANSCRIPT
Geometry of Dempster’s rule
NAVLAB - Autonomous Navigation and Computer Vision Lab
Department of Information Engineering
University of Padova, Italy
Fabio Cuzzolin
FSKD’02, Singapore, November 19 2002
2
1The talk
introducing the theory of evidence
2presenting the geometric approach: the belief space
3analyzing the local geometry of Dempster’s rule
4perspectives of geometric approach
1The theory of evidence
4
generalize classical finite probabilities
)(pAP
AB
BmAs )(
A
Belief functions
1)( B
Bmnormalization
B2B1
focal elements
5
Dempster’s rule
are combined by means of Dempster’s rule '', ssss
ABBmABel)()(
Ai
Bj
AiBj=A
intersection of focal elements
ji
ji
BAji
ABAji
BmAm
BmAm
Am)()(1
)()(
)(21
21
2Geometry of belief functions
7
it has the shape of a simplex
),( APClS A
Belief space
the space of all the belief functions on a frame
1,02: sS
each subset A A-th coordinate s(A)
8
),(, sA CAPsClSttss
Global geometry of
Dempster’s rule and convex closure commute
conditional subspace: “future” of s
),...,(),...,( 11 nn ssssClssCls
example: binary frame ={x,y}
xx PPs
yy PPs
SP
P
ss
3Local geometry of Dempster’s rule
10
Convex form of
Dempster’s sum of convex combinations
i
iii
ii ssss
jjj
iii
decomposition in terms of Bayes’ rule
AA
A
Bs
s PsBPBm
APAms
,*
*
)()(
)()(
11
Local geometry in S2
12
Constant mass loci
})(:{ kAmsH skA
set of belief functions with equal mass k assigned to a subset A
expression as convex closure
),,)1(( ABBPkPkClH BAkA
13
)1,0[
)(
k
kAA HsvF
intersection of all the subspaces )( kAHsv
Foci of conditional subspaces
),,( ABvF BA
it is an affine subspace
Ak
A Pks )1(lim
generators: focal points
14
4…conclusions
a new approach to the theory of evidence: the belief space
geometric behavior of Dempster’s rule
applications: approximation, decomposition, fuzzy measures