Extreme probability distributions
of
random/fuzzy sets and p-boxes
Alberto Bernardini; University of Padua, Italy
Fulvio Tonon; University of Texas, USA
Outline
•Review of Imprecise Probability
•Objective
•Set of probability distributions for:
•Choquet capacities
•Random sets
•Fuzzy sets
•P-boxes
Imprecise Probability (Walley, 1991)
Finite probability space (, F, P),
F is the -algebra generated by a finite partition of  into elementary
events (or singletons) S = s1, s2…, sj ,… sn.
=> Probability space is fully specified by the probabilities P(sj)
Consider bounded and point-valued functions (gambles) fi: S
For a specific precise probability distribution P(sj), the prevision is
equivalent to the linear expectation:
EP  f i  
 fi  s j P  s j 
s j S
T  S  P T   EP  T  
 T  s j P  s j 
s j S
T = 1 if sjT, 0 if sj T
Imprecise Probability (Walley, 1991)
Given ELOW[fi] and ELOW[fi], fi  K;
What can we say about probabilities of events in S?
 E  P : ELOW  fi   EP  fi   EUPP  fi  fi K 
Can E be arbitrary? NO: 2 basic conditions:
•E should be non-empty (avoid sure loss)
E is empty if ELOW[fi] > maxj fi(sj) or EUPP[fi] < minj fi(sj)
•Given lower/upper bounds should be the same as obtained with E (coherence)
ELOW ,c  fi   minE EP  fi 
P
EUPP ,c  fi   maxE EP  fi 
P
Objectives
To give expressions for E in special cases:
•Choquet capacities of order 2
•Random sets
•Fuzzy sets
•P-boxes
Choquet capacities of order 2
Order 2
Set function : P(S)  0, 1 :  () = 0, (S)= 1
 (T1  T2 )   (T1 )   (T2 )   (T1  T2 )
Alternate Choquet Capacity of order k = 2: UPP(Ti) = 1- (Tic)
UPP (T1  T2 )  UPP (T1 )  UPP (T2 )  UPP (T1  T2 )
Order k
 (T1  T2 ...  Tk ) 

 K 1,2..k
UPP (T1  T2 ...  Tk ) 

(1)
 K 1,2..k
K 1
(1)
 
 T
  Ti
iK
K 1
UPP
iK
i
Choquet capacities of order 2
Coherent upper and
lower probabilities
Choquet capacities, k2
Coherent upper and
lower previsions
Choquet capacities of order 2
Given (.):
1. Consider a permutation  of the elements of S = s1, s2…, sj ,… sn
2. Construct probability distribution P:
  
P  s       s
P s  j 1   s (j)=1
 (j)=1
 j 2
...

 

 
, s (j)=2    s (j)=1
 

P s  j  k 1   s (j)=1 ,...sk    s (j)=1 ,...sk 1

3. Repeat 1) and 2) for all n! permutations
4. The set, EXT, of P so constructed is the set of extreme points of E
5. E is the convex hull of EXT
Choquet capacities of order 2
Calculation of expectation for f: S Y=[yL, yR]  
yR
yL
E[ f ]  yL 
yR

P
(
 T  s  S | f (s)   )d
yL
Choquet capacities of order 2
Calculation of expectation bounds for f: S Y=[yL, yR]   when  is given
E[ f ]  yL 
yR

P
(
 T  s  S | f (s)   )d
yL

C ( f ,  )  yL 
yR


(
 T ) d
yL
When f attains values: 1= yR> 2 >… > n = yL :
n 1
C ( f ,  )  yL    ( i T )  i   i 1 
i 1
Same as reordering S => f becomes monotonically decreasing
Random sets
Family of n focal elements Ai  S with weights m(Ai) :
m()=0; i m(Ai)=1
n


   P   mi  P i s j

i 1


  | P  s j   0 if
i
sj  A 


T  S : UPP T   Pla(T )   mi | Ai  T  ,
i
 LOW T   Bel (T )   mi | Ai  T
i

i

 COHERENT!


k 


 is the convex hull of EXT
Expectations may be calculated by reordering S
Random sets, Bel & Pla
Coherent upper and
lower probabilities
Choquet capacities, k2
Choquet capacities, k=
Bel, Pla
Coherent upper and
lower previsions
Fuzzy sets
Membership
function
1=1
m1
m2
m3
2
3
4=0
s1
s2
s3
A1
A2
A3
s4
s



 Nested focal sets


F
Probability (P-) boxes
 P : F (s )  F (s )  P s ,..., s   F (s ), j  1 to | S |
LOW
j
j
1
j
UPP
A1
A2
A3
A4
A5
j

 Focal sets


 not nested

F
Probability (P-) boxes
 P : F (s )  F (s )  P s ,..., s   F (s ), j  1 to | S |
LOW
j
j
1
j
UPP
j
All P  E satisfy the constraints
FLOW ( s1 )  P  s1   FUPP ( s1 );
P  s1   0
FLOW ( s2 )  P  s1   P  s2   FUPP ( s2 );
P  s2   0
...................
j 1
FLOW ( s j )  P  s j    P  si   FUPP ( s j ); P  s j   0
i 1
...................
| S |1
P  s j |S |    P  si   1;
i 1
P  s j |S |   0
Probability (P-) boxes
FLOW ( s1 )  P  s1   FUPP ( s1 );
P  s1   0
FLOW ( s2 )  P  s1   P  s2   FUPP ( s2 );
P  s2   0
F
F
Flow(s2)
Fupp(s1)
Fupp(s1)
Flow(s2)
s1
s2
s
s1
s2
s
Conclusions
•The set of probability distributions compatible with a random set is
equal to the convex hull of the extreme distributions obtained by
permuting elements of S
•Exact bounds on expectation of f may be calculated
•By reordering the set S => f is monotonically decreasing
•By using two elements in the set of extreme distributions
•Fuzzy sets and p-boxes are particular indexable random sets
•Random sets can be easily derived
•Extreme distributions can be easily calculated