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

The Extension of Imprecise Probabilities Based on Generalized Credal Sets Andrey G. Bronevich 1 , Igor N. Rozenberg 2 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 of 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 2 X be the powerset of its subsets. Definition A family P of probability measures on 2 X 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 M pr is the set of all probability measures; E P ( f ) = � x ∈ X f ( x ) P ( { x } ) is the expectation of f ∈ K w.r.t. P ∈ M pr . The functional E avoids sure loss if the set of probability measures P ∈ M pr |∀ f ∈ K ′ : E ( f ) � E P ( f ) � � P = (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 E P : K → R of E is defined by E P ( f ) = inf { E P ( f ) | P ∈ P } . E is called a coherent lower prevision if E P ( 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 of random variables in K ′ . It is non-contradictory if it defines a non-empty credal set P ∈ M pr |∀ f ∈ K ′ : E ( f ) � E P ( f ) � � P = and its natural extension E P : K → R of E on K is defined by E P ( f ) = sup { E P ( f ) | P ∈ P } . If E P ( 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 P 1 , ..., P m is defined as P = P 1 ∩ ... ∩ P m (1) Remark The C-rule is defined only for the case when P is a non-empty set. Let E i : K ′ → R , i = 1 , .., m , be lower previsions on K ′ . Then the result of the C-rule can be expressed as i =1 ,...,m E i ( f ), f ∈ K ′ . E ( f ) = max Analogously for upper previsions the C-rule is defined using the max operation. (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 P 1 , P 2 ∈ M pr . They are fully contradictory if there are disjoint sets A, B ∈ 2 X such that P 1 ( A ) = P 2 ( B ) = 1.Fully contradictory information is described by a set function � 1 , A � = ∅ , η d � X � ( A ) = 0 , A = ∅ , concieved as a lower probability. Notation � 1 , B ⊆ A, η � B � ( A ) = η � B � is a categorical belief function. 0 , otherwise. 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 µ d ( A ) = 1 − µ ( ¯ A ), A ∈ 2 X . (HSE, Moscow, Russia) The Extension of Imprecise Probabilities SMPS’2016 8 / 30

The conjunctive rule for probability measures If probability measures P 1 and P 2 are fully contradictory we define the conjunctive rule as P 1 ∧ P 2 = η d � X � . If there is no contradiction between probability measures P 1 and P 2 , i.e. P 1 = P 2 = P . Then P 1 ∧ P 2 = P. Lemma For any P 1 , P 2 ∈ M cpr It is always possible to find representations P 1 = (1 − a ) P (1) + aP (2) 1 , P 2 = (1 − a ) P (1) + aP (2) 2 , such that P (1) , P (2) , P (2) ∈ M pr , a ∈ [0 , 1], and P (2) , P (2) are fully 1 2 1 2 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 P 1 ∧ P 2 = (1 − a ) P (1) + a ( P (2) ∧ P (2) 2 ) . 1 The last formula can be rewritten as n � min { P 1 ( { x i } ) , P 2 ( { x i } ) } η �{ x i }� + aη d P 1 ∧ P 2 = � X � . i =1 n � where a = 1 − min { P 1 ( { x i } ) , P 2 ( { x i } ) } . i =1 The value a is called the amount of contradiction between P 1 and P 2 , denoted by Con ( P 1 , P 2 ). (HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 10 / 30

Example. Conjunction of probability measures. Let X = { x 1 , x 2 , x 3 } and probability measures P 1 and P 2 on 2 X are given in the following table. { x 1 } { x 2 } { x 3 } P 1 0.3 0.2 0.5 P 2 0.3 0.3 0.4 min { P 1 , P 2 } 0.3 0.2 0.4 a = 0 . 1 Then P 1 ∧ P 2 = 0 . 3 η �{ x 1 }� + 0 . 2 η �{ x 2 }� + 0 . 4 η �{ x 3 }� + 0 . 1 η d � X � . Notation M cpr is the set of all set functions of the type n P = a 0 η d � � X � + a i η �{ x i }� , i =1 n where a i � 0, i = 0 , ..., n , � a i = 1. i =0 (HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 11 / 30

The definition of the conjunctive rule for general measures in M cpr The interpretation of the conjunctive rule through the order � Let P 1 , P 2 ∈ M pr . Then P 1 ∧ P 2 is the exact lower bound of the set { P ∈ M cpr | P � P 1 , P � P 2 } . It allows us to define the conjunctive rule for arbitrary P 1 , ..., P m ∈ M cpr as an exact lower bound of the set { P ∈ M cpr | P � P 1 , ..., P � P m } . n If any P = a 0 η d � � X � + a i η �{ x i }� is identified with a point ( a 1 , ..., a n ) in i =1 R n . Then P = P 1 ∧ ... ∧ P m for P = ( b 1 , ..., b n ), P i = ( a ( i ) 1 , ..., a ( i ) n ) if i =1 ,..,m a ( i ) b k = min k . (HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 12 / 30

Generalized credal sets Definition A subset P ⊆ M cpr is called an upper generalized credal set if 1 P 1 ∈ P , P 2 ∈ M cpr , P 1 � P 2 implies that P 2 ∈ P . (The next two properties are essential for the most models of imprecise probabilities (cf. credal sets).) 2 if P 1 , P 2 ∈ P then aP 1 + (1 − a ) P 2 ∈ P for any P 1 , P 2 ∈ P and a ∈ [0 , 1]. 3 the set P is closed in a sense that it can be considered as a subset n of Euclidian space (any P = a 0 η d � X � + � a i η �{ x i }� is a point i =1 ( a 1 , ..., a n ) in R n ). (HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 13 / 30

Generalized credal sets Notation M d cpr = { P d | P ∈ M cpr } If we consider measures from ¯ M cpr as (contradictory) lower probabilities, then any P ∈ M d cpr : n � P = a 0 η � X � + a i η �{ x i }� , i =1 can be concieved as the (contradictory) upper probability. Definition P is the lower generalized credal set if P d = � 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 M cpr . A subset profile ( P ) consisting of all minimal elements in P is called the profile of P . Example. Let X = { x 1 , x 2 } and an upper generalized credal set P is defined by a profile ( P ) = { aP 1 + (1 − a ) P 2 | a ∈ [0 , 1] } , where P 1 = (0 . 4 , 0 . 6), P 2 = (0 . 6 , 0 . 4). Then P can be depicted in R 2 as a 2 1 P 1 0.6 P 2 0.4 a 1 0 0.4 0.6 1 (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 M d cpr . A subset consisting of all maximal elements in P is called the profile of P and it is denoted by profile ( P ). M cpr , then P d is Obviously, if P be an upper generalized credal set in ¯ the lower generalized credal set in ¯ M d cpr and profile ( P d ) = profile ( P ) d . (HSE, Moscow, Russia) The Extension of Imprecise ProbabilitiesSMPS’2016 16 / 30