The Extension of Imprecise Probabilities Based on Generalized Credal - - PowerPoint PPT Presentation

The Extension of Imprecise Probabilities Based on Generalized Credal Sets

Andrey G. Bronevich1, Igor N. Rozenberg2

1 National Research University ”Higher School of Economics”, Moscow, Russia 2 JSC Research, Development and Planning Institute for Railway Information

Technology, Automation and Telecommunication, Moscow, Russia

8th International Conference on Soft Methods in Probability and Statistics, 12-14 September 2016, Rome, Italy

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 1 / 30

The aim of the investigation

To develop theory of imprecise probailities that works with contradictory information when the avoiding sure condition is not fulfilled.This theory can be based on generalized credal sets introduced in paper Bronevich A.G., Rozenberg I.N. The generalization of the conjunctive rule for aggregating contradictory sources of information based on generalized credal sets. Proceedings of ISIPTA-15. Remark Contradiction in information can be caused by inconsistent assessments

• f a decision-maker, after combining information from contradictory

sources.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 2 / 30

Credal sets

Let X be a finite set and 2X be the powerset of its subsets. Definition A family P of probability measures on 2X is called a credal set if it is convex and closed. Credal sets allow us to model randomness (conflict) by of probability measures and inderminacy (non-specificity)in choosing a probability measure in the credal set.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 3 / 30

Lower previsions

Let K be the set of all functions f : X → R and K′ ⊆ K. A functional E : K′ → R is called a lower prevision if values E(f) are conceived as lower estimates of expectations of random variables f ∈ K. Notation Mpr is the set of all probability measures; EP (f) =

x∈X f(x)P({x}) is the expectation of f ∈ K w.r.t.

P ∈ Mpr. The functional E avoids sure loss if the set of probability measures P =

• P ∈ Mpr|∀f ∈ K′ : E(f) EP (f)
• (1)

is not empty. In this case P is a credal set. In the opposite case, when P = ∅, E is contradictory.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 4 / 30

Natural extension

Let a credal set P corresponds to E. Then the natural extension EP : K → R of E is defined by EP(f) = inf {EP (f)|P ∈ P} . E is called a coherent lower prevision if EP(f) = E(f), f ∈ K′.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 5 / 30

Upper previsions

E : K′ → R is called an upper prevision if it gives us upper expections

• f random variables in K′. It is non-contradictory if it defines a

non-empty credal set P =

• P ∈ Mpr|∀f ∈ K′ : E(f) EP (f)
• and its natural extension EP : K → R of E on K is defined by

EP(f) = sup {EP (f)|P ∈ P} . If EP(f) = E(f), f ∈ K′, then E(f) is called a coherent upper prevision. Models based on credal sets, upper and lower coherent previsons are equivalent if X is finite.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 6 / 30

The conjunctive rule in the theory of imprecise probabilities

Definition The conjunctive rule (C-rule) for credal sets P1, ..., Pm is defined as P = P1 ∩ ... ∩ Pm (1) Remark The C-rule is defined only for the case when P is a non-empty set. Let Ei : K′ → R, i = 1, .., m, be lower previsions on K′. Then the result of the C-rule can be expressed as E(f) = max

i=1,...,mEi(f), f ∈ K′.

Analogously for upper previsions the C-rule is defined using the max

• peration.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 7 / 30

The case of fully contradictory sources of information

Sources of information described by P1, P2 ∈ Mpr. They are fully contradictory if there are disjoint sets A, B ∈ 2X such that P1(A) = P2(B) = 1.Fully contradictory information is described by a set function ηd

X(A) =

1, A = ∅, 0, A = ∅, concieved as a lower probability. Notation ηB(A) = 1, B ⊆ A, 0,

• therwise.

ηB is a categorical belief function. let µ be a monotone measure, i.e. µ(∅) = 0, µ(X) = 1, A ⊆ B implies µ(A) µ(B), then the dual of µ is defined as µd(A) = 1 − µ( ¯ A), A ∈ 2X.

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 8 / 30

The conjunctive rule for probability measures

If probability measures P1 and P2 are fully contradictory we define the conjunctive rule as P1 ∧ P2 = ηd

X.

If there is no contradiction between probability measures P1 and P2, i.e. P1 = P2 = P. Then P1 ∧ P2 = P. Lemma For any P1, P2 ∈ Mcpr It is always possible to find representations P1 = (1 − a)P (1) + aP (2)

1 , P2 = (1 − a)P (1) + aP (2) 2 ,

such that P (1), P (2)

1

, P (2)

2

∈ Mpr, a ∈ [0, 1], and P (2)

1

, P (2)

2

are fully

• contradictory. These representations are defined uniquely if a ∈ (0, 1).

(HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 9 / 30

The conjunctive rule for probability measures

Thus, we define the C-rule for arbitrary probability measures as P1 ∧ P2 = (1 − a)P (1) + a(P (2)

1

∧ P (2)

2 ).

The last formula can be rewritten as P1 ∧ P2 =

n

• i=1

min {P1({xi}), P2({xi})} η{xi} + aηd

X.

where a = 1 −

n

• i=1

min {P1({xi}), P2({xi})}. The value a is called the amount of contradiction between P1 and P2, denoted by Con(P1, P2).

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 10 / 30

• Example. Conjunction of probability measures.

Let X = {x1, x2, x3} and probability measures P1 and P2 on 2X are given in the following table. {x1} {x2} {x3} P1 0.3 0.2 0.5 P2 0.3 0.3 0.4 min{P1, P2} 0.3 0.2 0.4 a = 0.1 Then P1 ∧ P2 = 0.3η{x1} + 0.2η{x2} + 0.4η{x3} + 0.1ηd

X.

Notation Mcpr is the set of all set functions of the type P = a0ηd

X + n

• i=1

aiη{xi}, where ai 0, i = 0, ..., n,

n

• i=0

ai = 1.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 11 / 30

The definition of the conjunctive rule for general measures in Mcpr

The interpretation of the conjunctive rule through the order Let P1, P2 ∈ Mpr. Then P1 ∧ P2 is the exact lower bound of the set {P ∈ Mcpr|P P1, P P2}. It allows us to define the conjunctive rule for arbitrary P1, ..., Pm ∈ Mcpr as an exact lower bound of the set {P ∈ Mcpr|P P1, ..., P Pm} . If any P = a0ηd

X + n

• i=1

aiη{xi} is identified with a point (a1, ..., an) in

• Rn. Then P = P1 ∧ ... ∧ Pm for P = (b1, ..., bn), Pi = (a(i)

1 , ..., a(i) n ) if

bk = min

i=1,..,ma(i) k .

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 12 / 30

Generalized credal sets

Definition A subset P ⊆ Mcpr is called an upper generalized credal set if

1 P1 ∈ P, P2 ∈ Mcpr, P1 P2 implies that P2 ∈ P.

(The next two properties are essential for the most models of imprecise probabilities (cf. credal sets).)

2 if P1, P2 ∈ P then aP1 + (1 − a)P2 ∈ P for any P1, P2 ∈ P and

a ∈ [0, 1].

3 the set P is closed in a sense that it can be considered as a subset

• f Euclidian space (any P = a0ηd

X + n

• i=1

aiη{xi} is a point (a1, ..., an) in Rn).

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 13 / 30

Generalized credal sets

Notation Md

cpr = {P d|P ∈ Mcpr}

If we consider measures from ¯ Mcpr as (contradictory) lower probabilities, then any P ∈ Md

cpr:

P = a0ηX +

n

• i=1

aiη{xi}, can be concieved as the (contradictory) upper probability. Definition P is the lower generalized credal set if Pd =

• P d|P ∈ P
• is the upper

generalized credal set.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 14 / 30

The profile of a generalized credal set

Let P be a upper generalized credal set in Mcpr. A subset profile(P) consisting of all minimal elements in P is called the profile of P.

• Example. Let X = {x1, x2} and an upper generalized credal set P is

defined by a profile(P) = {aP1 + (1 − a)P2|a ∈ [0, 1]}, where P1 = (0.4, 0.6), P2 = (0.6, 0.4). Then P can be depicted in R2 as

1

a

2

a

1

P

2

P 0.4 1 1 0.6 0.6 0.4

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 15 / 30

The profile of a generalized credal set

Any profile uniquely defines the corresponding credal set. If P describes information without contradiction, then profile(P) is a credal set in usual sense, i.e. profile(P) is a set of probability measures. Analogously, the profile of lower generalized credal set is defined. Let P be a lower generalized credal set in Md

• cpr. A subset consisting of

all maximal elements in P is called the profile of P and it is denoted by profile(P). Obviously, if P be an upper generalized credal set in ¯ Mcpr, then Pd is the lower generalized credal set in ¯ Md

cpr and

profile(Pd) = profile(P)d.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 16 / 30

The C-rule for generalized credal sets

Definition. Let P1, ..., Pm be non-empty upper generalized credal sets in Mcpr. Then the credal set P produced by the C-rule is defined as P = P1 ∩ ... ∩ Pm. Let us observe that this definition generalizes the introduced C-rule for probability measures. Actually, let we have two credal sets P1, P2 in ¯ Mcpr with profile(Pi) ∈ Mpr, where Mpr is the set of all probability measures on 2X. Then profile(P1 ∩ P2) = profile(P1) ∧ profile(P2). In the same way the C-rule for lower generalized credal sets are defined.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 17 / 30

The C-rule for generalized credal sets (Example)

Let X = {x1, x2} and upper generalized credal sets P1 and P2 are given by their profiles: profile(P1) = {aP3 + (1 − a)P4|a ∈ [0, 1]}, profile(P2) = {aP1 + (1 − a)P5|a ∈ [0, 1]}, where P1 = (0, 0.4), P3 = (0.4, 0), P4 = (0.1, 0.9), P5 = (0.9, 0.1).

1

a

2

a P

1

P

2

P

3

P 1 1 0.6 0.6

4

P

5

P

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 18 / 30

The C-rule for generalized credal sets (Example)

Then their intersection (conjunction) is the credal set P3 with profile(P3) = {aP1 + (1 − a)P2|a ∈ [0, 1]} ∪ {aP2 + (1 − a)P3|a ∈ [0, 1]}. Remark profile(P3) is not a convex set.

1

a

2

a P

1

P

2

P

3

P 1 1 0.6 0.6

4

P

5

P

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 19 / 30

The amount of contradiction of a generalized credal set

Let P is a generalized credal set in Mcpr. Then the amount of contradiction is defined as Con(P) = inf {Con(P)|P ∈ P}. From previous example: Con(P1) = Con(P4) = 0, Con(P2) = Con(P5) = 0, Con(P3) = Con(P2) = 0.4,P2 = (0.3, 0.3), Con(P2) = 1 − 0.3 − 0.3 = 0.4.

1

a

2

a P

1

P

2

P

3

P 1 1 0.6 0.6

4

P

5

P

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 20 / 30

The ways of defining generalized credal sets

Let lower probability P ∈ Mcpr, i.e. P = a0ηd

X + n

• i=1

aiη{xi} and f : X → R. Then the lower expectation EP(f) of f w.r.t. P can be computed by the Choquet integral: EP (f) = (C)

• fdP = a0 max

x∈X f(x) + n

• i=1

aif(xi). Let E : K′ → R be a lower prevision. Then it defines the upper generalized credal set P(E) =

• P ∈ ¯

Mcpr|∀f ∈ K′ : EP (f) E(f)

• .

This set is not empty if E(f) max

x∈X f(x) for all f ∈ K′.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 21 / 30

The ways of defining generalized credal sets

The same constructions can be defined for lower generalized credal sets. If P ∈ Md

cpr, i.e. P = a0ηX + n

• i=1

aiη{xi} and f : X → R. Then the upper expectation EP (f) of f w.r.t. P can be computed by the Choquet integral: EP(f) = (C)

• fdP = a0 min

x∈X f(x) + n

• i=1

aif(xi). Let E : K′ → R be an upper prevision. Then it defines the lower generalized credal set P(E) =

• P ∈ Md

cpr|∀f ∈ K′ : ¯

EP(f) ¯ E(f)

• .

This set is not empty if E(f) min

x∈X f(x) for all f ∈ K′.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 22 / 30

Upper and lower previsions based on generalized credal sets

We can define the lower prevision based on non-empty upper generalized credal set P as EP(f) = inf

P ∈P EP (f), f ∈ K.

Analogously, we can define the upper prevision based on non-empty lower generalized credal set P as EP(f) = sup

P ∈P

¯ EP(f), f ∈ K. Functionals EP and EP can be considered as counterparts of coherent lower und upper previsions in the theory of imprecise probabilities without contradiction.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 23 / 30

Necessary and sufficient conditions for a fucntional to be EP

A function f : X → R is called normalized if minx∈X f(x) = 0. Theorem A functional Φ : K → R coincides with EP on K for some credal set P in Md

cpr iff it has the following properties:

1) Φ(f + a1X) = Φ(f) + a for any f ∈ K and a ∈ R; 2) Φ(af) = aΦ(f) for any f ∈ K and a 0; 3) Φ(f1) Φ(f2) for f1, f2 ∈ K if f1 f2; 4) Φ (f1) + Φ (f2) Φ (f3) for any normalized functions f1, f2, f3 in K such that f1 + f2 = f3. Remark If 4) in Theorem will be fulfilled for every possible functions f1, f2, f3 in K (not necessarally normalized), then conditions 1)-4) are necessary and sufficient for the functional Φ to be a coherent lower prevision on K.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 24 / 30

Does the functional EP define the underlying generalized credal set P uniquely?

Let X = {x1, x2} and credal sets P1 and P2 in Md

cpr are given by

profile(P1) = {P1}, profile(P2) = {aP2 + (1 − a)P3|a ∈ [0, 1]}, where P1 = (0.5, 0.5), P2 = (0.5, 0), P3 = (0, 0.5). Then, EP1(f) = 0.5f(x1) + 0.5f(x2), EP2(f) = 0.5 max{f(x1), f(x2)} + 0.5 min{f(x1), f(x2)} = EP1(f).

1

a

2

a

1

P

2

P

3

P 1 1

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 25 / 30

When functionals EP1 and EP2 coincide.

Let a lower generalized credal set P is identified with a convex set Rn. The j-th projection of P is Prj P = {Prj P|P ∈ P} , where Prj P = (a1, ..., aj−1, 0, aj+1, ..., an) if P = (a1, ..., an). Clearly, Prj P is also a credal set in Md

cpr.

Theorem Let P1 and P2 credal sets in Md

• cpr. Then EP1(f) = EP2(f) for all

f ∈ K iff Prj P1 = Prj P2, j = 1, ..., n.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 26 / 30

Maximal generalized credal sets.

Definition The LG-credal set P in Md

cpr is called maximal if

P =

• P ∈ Md

cpr|∀f ∈ K : ¯

EP (f) ¯ EP(f)

• .

Theorem Let P be a generalized credal set in Md

cpr whose profile is an usual

credal set in Mpr. Then the credal set P is maximal. Remark The result from the theorem means that in the theory based on generalized credal sets models based on coherent upper previsions and usual credal sets are equal.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 27 / 30

The natural extension based on generalized credal sets

Let E : K′ → R be an upper prevision, then it defines the credal set P =

• P ∈ Md

cpr|∀f ∈ K : EP (f) E(f)

• and ¯

EP can be considered as the natural extension of E on K. The functional E is called the generalized coherent upper prevision if EP(f) = ¯ E(f), f ∈ K′. Theorem Let ¯ E : K′ → R be an upper prevision functional. Then its natural extension ¯ E′ : K → R based on generalized credal sets is E

′

f

• = inf
• k

akE

• fk
• + a|
• k

akfk + a f, fk ∈ K′, ak, a 0

• ,

where f, fk are normalized functions and EP

• f
• = ¯

EP (f) − b, E

• fk
• = E (fk) − bk, b = min

x∈X f(x), bk = min x∈X fk(x).

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 28 / 30

Comparison of reasoning with usual and generalized credal sets

Example Let we have two sources of information that describe possible diseases X = {x1, x2, x3} of a patient. Source 1: E(1{x1,x2}) = E(1{x2,x3}) = 0.5, i.e. probabilities of events {x1, x2} and {x2, x3} are lower or equal to 0.5; Source 2: E(1{x1,x3}) = 0, i.e. this is definitely disease x2. If we describe sources of information by usual credal sets, then sources 1 and 2, described by probability measures P1 = (0.5, 0, 0.5) and P2 = (0, 1, 0) are fully contradictory. If we described them by generalized credal sets P1 and P2, then profile(P1) = {aP1 + (1 − a)P3|a ∈ [0, 1]} and profile(P2) = {P2}, where P2 = (0, 0.5, 0). Thus, profile(P1 ∩ P2) = {P3} and we come to the conclusion that it is disease x2 but with Con(P3) = 0.5.

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 29 / 30

Thanks for you attention

brone@mail.ru I.Rozenberg.gismps.ru

(HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 30 / 30