Transcript No Slide Title

2nd INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS, ISIPTA’01
LATTICE STRUCTURE OF THE FAMILIES OF
COMPATIBLE FRAMES OF DISCERNMENT


Fabio Cuzzolin and Ruggero Frezza, Dipartimento di Elettronica e Informatica, Università di Padova, Italy
FAMILIES OF FRAMES (F,R)
.0
• collections F of finite domains of belief functions (different
representations of a same phenomenon) and refinings R
between them, satisfying several axioms
MEASUREMENT CONFLICT
DEMPSTER’S RULE
.1
REFINING .0
• the combination is guaranteed only for trivially interacting
.1
.2
.3
belief functions
.4
.00 .01 .10 .11
• Axiom 1: composition of refinings
0.00
• every collection of belief functions over Q1 ,..., Q n
is combinable

0.09
• Axiom 2: identity of coarsenings
0.49
• Axiom 3: identity of refinings
0.90
0.99
• Axiom 4: existence of coarsenings
• Axiom 5: existence of refinings
• Axiom 6: existence of common refinements
COMMON
REFINEMENT
• Q1 ,..., Q n are independent

FINITE PARTITIONS OF A SET
0
0.25
0.5
0.75
1
Ai
AiBj=A
 Example: a function y  [0,1] is estimated resting on three
different quantized measurements with:
Bj
whenever 
• F and R are both commutative monoids with no annihilator
LATTICE STRUCTURE
• F is a locally Birkhoff (semimodular with finite length)
• Q1  ...  Q n  Q1  ...  Q n
w1(A1)  …  wn(An)   (1)
•A collection of frames is independent if
- 1 binary digit, 2 binary digits, 1 base-5 digit
MONOIDAL STRUCTURE
ABBmlAeB)()(
 A i  Qi " i
INDEPENDENCE RELATIONS AND MODULARITY
1F
•Several candidate independence relations are related one to the others in a way that
depends on the class the underlying lattice belongs to
lattice bounded below
maximal coarsening
QW
LI3
LI1
LI1
Q
LI2
W
minimal refinement
QW
• F is a complemented semimodular lattice
INTERNAL AND EXTERNAL INDEPENDENCE
L semimodular
L modular
PSEUDO GRAM-SCHMIDT ALGORITHM
•A Gram-Schmidt type technique is useful to detect the largest set of compatible data
of commutative monoids like the collection
of vector subspaces of a linear space
• the internal independence of frames as Boolean
subalgebras of their common refinement is equivalent to
their external independence LI3 as elements of a lattice
LI2=LI3
Q1
Q2
Qn
APPROXIMATE
FEATURE SPACES
Q
with Q'1, … , Q'm independent.
Q
APPROXIMATE
PARAMETER SPACE
PARAMETER SPACE