The intersection probability and its properties

Transcription

1 The intersection probability and its properties Fabio Cuzzolin INRIA Rhône-Alpes 655 avenue de l Europe Montbonnot, France Abstract In this paper we introduce the intersection probability, a Bayesian approximation of belief functions derived from purely geometric arguments, and study its properties. In particular its interpretation in terms of degrees of belief is given, by understanding the way it assigns mass to singletons in proportion to their non-bayesian contribution. Its behavior in terms of convex combination and Dempster s sum is also analyzed, providing a justification of its name and pointing out its affinity with the pignistic transformation. 1 Introduction In the theory of evidence belief functions (b.f.s) [14] generalize finite probabilities, with classical probabilities forming a subclass P of b.f. called Bayesian belief functions. The interplay of belief and Bayesian functions is of course of great interest in the theory of evidence. In particular, many people worked on the problem of finding a probabilistic approximation of an arbitrary belief function. A number of papers [10, 13, 18, 19, 5] have been published on this issue (see [1] for a review). The connection between belief functions and probabilities is as well the basement of a popular approach to the theory of evidence, Smets pignistic model [15]. The study of the interplay between belief functions and probabilities has been recently posed in a geometric setup [2, 12]. Amongst others, F. Cuzzolin has proposed a geometric approach to the theory of evidence in which belief functions are represented as points of a linear space [8]. In this paper we use tools provided by the geometric approach to introduce a new Bayesian approximation called intersection probability, derived from purely geometric considerations. 2 Definition of intersection probability 2.1 RATIONALE Let us consider an interval probability system, i.e. a system of constraints on the probability values of a probability measure p : Θ [0, 1] on a finite domain Θ of the form (l, u). = {l(x) p(x) u(x), x Θ}. (1) The system (1) determines obviously an entire set of probability measures whose values are constrained to belong to a closed interval. We can define the intersection probability associated with the interval probability system (1) as the unique probability such that p[(l, u)](x) = l(x) + α(u(x) l(x)). Figure 1: An illustration of the notion of intersection probability for an upper/lower probability system. the intersection probability can be written as with p[(l, u)](x) = β[(l, u)]u(x) + (1 β[(l, u)])l(x) (2) β[(l, u)] = 1 ( l(x) ). (3) u(x) l(x) 2.2 INTERPRETATIONS The ratio β (3) clearly measures the fraction of the probability interval which we need to add to the lower bound u(x) to obtain a valid probability function which sums to one.

2 Another interpretation of p[b] come from its alternative form where p[(l, u)](x) = l(x) + ( 1 x R[(l, u)](x). = y u(x) l(x) = (x) (u(y) l(y)) l(x) ) R[(l, u)](x) (4) y (y), (5) (x) measuring the size of the probability interval on x. R(x) measures how much the uncertainty on the probability value on a singletons weights on the total uncertainty represented by the interval probability (1). It is the natural to call it relative uncertainty of singletons. 2.3 EXAMPLE Consider as an example an interval probability on a domain of size 3: 0.2 p(x) 0.8, 0.4 p(y) 1, 0 p(x) 0.4. The size of the corresponding intervals is (x) = 0.6, (y) = 0.6, (z) = 0.4 respectively. Computing the intersection probability is very easy. By Equation (3) the fraction of uncertainty to sum to l(x) to get an admissible probability is β = = = 1 4. The intersection probability then has values (4) p[(l, u)](x) = = 0.35, p[(l, u)](y) = = 0.55, p[(l, u)](z) = = 0.1. It is important to notice that, on the other side, the probabilities we obtain by normalizing the lower bound l(x) = l(x)/ y l(y) or the upper bound ũ(x) = u(x)/ y u(y) of the interval are not consistent with the interval itself. Indeed, 2.4 INTERVAL PROBABILITY FOR BELIEF MEASURES Belief measures. A basic probability assignment (b.p.a.) over a finite set or frame of discernment Θ is a function m : 2 Θ [0, 1] on its power set 2 Θ = {A Θ} such that 1. m( ) = 0; 2. A Θ m(a) = 1; 3. m(a) 0 A Θ. Subsets of Θ associated with non-zero values of m are called focal elements. The belief function b : 2 Θ [0, 1] associated with a basic probability assignment m on Θ is defined as: b(a) = B A m(b). (6) Their inverse relation is given by the Moebius inversion formula m b (A) = B A( 1) A B b(b). (7) A finite probability or Bayesian belief function is just a special b.f. assigning non-zero masses to singletons only: m b (A) = 0, A > 1. A dual mathematical representation of the evidence encoded by a belief function b is the plausibility function (pl.f.) pl b : 2 Θ. [0, 1], where pl b (A) = 1 b(a c ) = B A m b(b) b(a). Interval probability of a belief function. When the interval probability system is that associated with a belief function, i.e. u(x) = pl b (x), l(x) = b(x) = m b (x) the intersection probability can be written as with β[b] = p[b](x) = β[b]pl b (x) + (1 β[b])m b (x) (8) 1 m b(x) ( plb (x) m b (x) ) = 1 k b (9) k plb k b where k plb. = pl b(x) is the total plausibility of singletons, while k b. = m b(x) is the total mass of singletons. 2.5 RELATION WITH PIGNISTIC FUNCTION An interesting parallelism between p[b] and BetP [b] emerges on the other side when we note that, if A > B >1 1, β[b A ] = m b(b) B >1 m = 1 b(b) B A so that both p[b](x) and BetP [b](x) assume the form m b (x) + m b (A)β A, A x,a x where β A = const = β[b] for p[b], while β A = β[b A ] in case of the pignistic function. Example: location of BetP. 2.6 RELATION WITH RELATIVE PLAUSIBILITY AND BELIEF Given a b.f. b the relative plausibility of singletons (rel.plaus.) pl b as the unique probability that assigns to each singleton its normalized plausibility: pl b (x) = pl b (x) y Θ pl b(y). (10) Dually, its relative belief of singletons (rel.bel.) assigns to the elements of the frame where b lives their normalized belief values: b(x). = b(x) (11) b(y). y Θ

3 Even though relative belief and plausibility () are not guaranteed to be consistent with a probability interval (1), the intersection probability has nevertheless a direct relation with them. By Equation (8) p[b] = m b (x)b x + β[b] (pl b (x) m b (x))b x = (1 β[b]) m b (x)b x + β[b] pl b (x)b x and if we call the quantities. pl b = pl b (x)b x b = m b (x)b x (12) plausibility of singletons and belief of singletons respectively, we have that p[b] = β[b] pl b + (1 β[b]) b (13) i.e. p[b] lies on the line joining pl b and b, in the same relative position of ς[b] on the segment Cl(b, pl b ). 3 GEOMETRY IN THE PROBABILITY SIMPLEX The intersection probability has originally been proposed in the framework of the geometric approach to belief measures. Given a frame of discernment Θ, a b.f. b : 2 Θ [0, 1] is completely specified by its N 1 belief values {b(a), A Θ, A }, N =. 2 Θ, and can then be seen as a point of R N 1. The geometry of the intersection probability it corresponds indeed to the intersection of the line a(b, pl b ) with the Bayesian simplex P. In the binary case it coincides with the pignistic function p[b] = BetP [b]. It is more interesting though to study the geometry of the intersection probability in the probability simplex P = {p : p : Θ [0, 1]}. It is well known, indeed, that the pignistic function [16, 17] BetP [b] = b x A x m b (A) A (14) is, analogously to what we appreciated in Figure??, the center of mass of the set of probabilities consistent with b [3, 11, 7]: P[b]. = {p P : p(a) b(a) A Θ}. (15) The polytope (15) can be naturally decomposed as the intersection of the regions P[b] = n 1 i=1 T i (16) T i. = {p P : p(a) b(a) A : A = i} formed by all probability meeting the lower probability constraint for size i events. 3.1 UPPER AND LOWER SIMPLICES In particular we focus on the polytope of the lower constraint on singletons T 1 or lower simplex, T 1. = {p P : p(x) b(x) x Θ} and the polytope of the upper constraint on singletons T 1n or upper simplex. It is in fact easy to see that T n 1. = {p P : p(a) b(a) A : A = n i} = {p P : p({x} c ) b({x} c ) x Θ} = {p P : p(x) pl b (x) x Θ} expresses the upper probability constraint on singletons. Clearly, then, the pair formed by the upper and lower simplices is the geometric counterpart of an interval probability in the probability simplex, exactly as the polytope of consistent probabilities P[b] there represents a belief function. Theorem 1. The set T 1 of all probabilities meeting the lower probability constraint on singletons is the simplex T 1 = Cl(t 1 x, x Θ), with vertices t 1 x = y x Proof. We need to show that m b (y)b y + ( 1 y x m b (y) ) b x. (17) 1. all points which belong to Cl(t 1 x, x Θ) satisfy p(x) m b (x); 2. all points which do not belong to the above polytope do not meet the constraint. Concerning item 1 p Cl(t 1 x, x Θ) p(x) = y Θ α y t 1 y(x) = = m b (x) y x α y + (1 k b)α x + m b (x)α x where y α y = 1 and α y 0 y, as t 1 y(x) = m b (x) if x y, t 1 y(x) = m b (x) + 1 k b if x = y. Therefore p(x) = m b (x)(1 α x ) + (1 k b)α x + m b (x)α x = m b (x) + (1 k b)α x m b (x) as 1 k b and α x are both non-negative quantities. Point 2: if p Cl(t 1 x, x Θ) then p = y α yt 1 y where z Θ such that α z < 0. But then p(z) = m b (z) + (1 k b)α z < m b (z) as (1 k b)α z < 0, unless k b = 1 in which case b is already a probability.

4 A dual proof can be provided for the simplex associated with the lower probability constraint on size n 1 events. We just need to replace belief with plausibility values on singletons.. Theorem 2. The set T n 1 = {p : p(a) mb (A) A : A = n 1} of all probabilities meeting the lower probability constraint on size n 1 events is the simplex T n 1 = Cl(t n 1 x, x Θ) with vertices t n 1 x = pl b (y)b y + ( 1 pl b (y) ) b x. (18) y x y x 3.2 INTERSECTION PROBABILITY AS FOCUS OF UPPER AND LOWER SIMPLICES Consider as an example the case of a belief function m b (x) =, m b (y) =, m b (z) =, m b ({x, y}) =, m b ({x, z}) =, m b (Θ) = defined on a ternary frame Θ = {x, y, z}. Figure 2 illustrates the geometry of its consistent simplex P[b]. We can notice that by Equation (16) P[b] (shaded area) is in this case the intersection of two triangles (2-dimensional simplices) T 1 and T 2. The intersection probability p[b] is the unique intersection of the lines joining the corresponding vertices of the upper T 2 and lower T 1 simplices. This fact is true in the general case, and can be formalized by the notion of focus of a pair of simplex. The pignistic function Intersection probability and consistent polytope Figure 2: The intersection probability is the focus of the two simplices T 1 and T n 1. In the ternary case the two simplices reduce to triangles. Their focus is geometrically the intersection of the lines joining the corresponding vertices of the two triangles. can also be similarly interpreted Definition 1. Consider a pair of simplices S = Cl(s 1,..., s n ), T = Cl(t 1,..., t n ) in R n. We call focus of the pair (S, T ) the unique point f(s, T ) of S T which has the same simplicial coordinates in both simplices: f = n α i s i = i=1 n β j t j, j=1 n n α i = 1, β j = 1. i=1 j=1 (19) It is easy to see that the focus of a pair of simplices is indeed the unique point which lies in the intersection of the lines a(s i, t i ) joining corresponding vertices of S and T : f = n i=1 a(s i, t i ) (see Figure 3-left). The barycenter itself of a simplex is a special case of focus. Indeed, the center of mass of a d-dimensional simplex S is the intersection of the medians of S, i.e. the lines joining each vertex with the barycenter of the opposite (d 1 dimensional) face (see Figure 3-right). But those barycenters for all d 1 dimensional faces form themselves a simplex. Figure 3: The focus of a pair of simplices is the unique intersection of the lines joining corresponding vertices of the two simplices (left). The barycenter of a simplex is a special case of focus (right). Theorem 3. The intersection probability is the focus of the pair of simplices (T n 1, T 1 ). Proof. We need to show that p[b] has the same simplicial coordinates in T 1 and T n 1. Remembering the expression () of p[b](x) p[b](x) = m b (x) + β[b](pl b (x)) m b (x) = (1 β[b])m b (x) + β[b]pl b (x) Theorem 3 completes the picture of the dual behavior of pignistic function and intersection probability. The both respect rationality principles for belief functions and interval probabilities respectively. Geometrically, this translates into a similar behavior in the probability simplex, in which they are the center of mass of the consistent polytope and the focus of the pair of lower and upper probability simplices. 4 INTERSECTION PROBABILITY AND CONVEX COMBINATION We conclude by discussing the behavior of the intersection probability with respect to the operators acting on belief measures. In particular, it is well known

5 that pignistic function and orthogonal projection commute with the convex combination of belief functions: BetP [α 1 b 1 + α 2 b 2 ] = α 1 BetP [b 1 ] + α 2 BetP [b 2 ]. It is then worth to study the behavior of p[b] with respect to and Cl, to understand how to classify it in the context of all Bayesian approximations. 4.1 Convex combination The condition under which the intersection probability commutes with convex combination is also quite interesting. Theorem 4. p[b] and convex closure commute, p[α 1 b 1 + α 2 b 2 ] = α 1 p[b 1 ] + α 2 p[b 2 ], α 1 + α 2 = 1, iff R[b 1 ] = R[b 2 ]. Proof. By definition (8) p[α 1 b 1 + α 2 b 2 ] = α 1 m 1 (x) + α 2 m 2 (x)+ α 1 1 (x) + α 2 2 (x) +(1 k α1b 1+α 2b 2 ) y Θ (α 1 1 (y) + α 2 2 (y)) α 1 1 (x)+α 2 2 (x) y Θ (α 1 1 (y)+α 2 2 (y)) be- that after defining R(x) =. comes α 1 m 1 (x) + α 2 m 2 (x) + [1 (α 1 k b1 + α 2 k b2 )]R(x) = = α 1 ( m1 (x) + (1 k b1 )R(x) ) + +α 2 ( m2 (x) + (1 k b2 )R(x) ) The intersection probability does not then have the nice relation with convex combination which characterizes pignistic function and orthogonal projection. However, Theorem 4 states that they commute exactly when each uncertainty interval l(x) p(x) u(x) has the same weight in the two interval probabilities. 5 Conclusions In this paper we have studied the properties of the intersection probability, a Bayesian approximation of belief functions derived from purely geometric arguments. In particular, its behavior in terms of convex combination and Dempster s sum have been analyzed, providing a justification of its name and pointing out its affinity with the pignistic transformation. More extensive studies of its relation with other Bayesian approximations are due in the near future: particular attention to objective distance metrics between all those probabilities is worth to be given. References [1] Mathias Bauer, Approximation algorithms and decision making in the Dempster-Shafer theory of evidence an empirical study, International Journal of Approximate Reasoning 17 (1997), [2] P. Black, An examination of belief functions and other monotone capacities, PhD dissertation, Department of Statistics, Carnegie Mellon University, 1996, Pgh. PA [3] A. Chateauneuf and J. Y. Jaffray, Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical Social Sciences 17 (1989), [4] B. R. Cobb and P. P. Shenoy, A comparison of bayesian and belief function reasoning, Information Systems Frontiers 5(4) (2003), [5] B.R. Cobb and P.P. Shenoy, On the plausibility transformation method for translating belief function models to probability models, International Journal of Approximate Reasoning 41(3) (April 2006), which is equal to α 1 p[b 1 ] + α 2 p[b 2 ] iff [6] F. Cuzzolin, Geometry of Dempster s rule of combination, IEEE Transactions on Systems, Man α 1 (1 k b1 )(R(x) R[b 1 ](x))+α 2 (1 k b2 )(R(x) R[b 2 ](x)) = 0 and Cybernetics part B 34:2 (April 2004), 961 which happens if and only if R(x) = R[b 1 ](x) = 977. R[b 2 ](x) x, as 1 k bi 0 unless b i is a probability, and the thesis is trivially true for α i = 1, α j = 0. [7] Fabio Cuzzolin, Geometry of upper probabilities, Proceedings of the 3 rd Internation Symposium on Imprecise Probabilities and Their Applications (ISIPTA 03), July [8] Fabio Cuzzolin and Ruggero Frezza, Geometric analysis of belief space and conditional subspaces, Proceedings of ISIPTA 01, Cornell University, Ithaca, NY, June [9] A.P. Dempster, Upper and lower probabilities generated by a random closed interval, Annals of Mathematical Statistics 39 (1968), [10] T. Denoeux and A. Ben Yaghlane, Approximating the combination of belief functions using the fast moebius transform in a coarsened frame, International Journal of Approximate Reasoning 31(1-2) (October 2002), [11] Didier Dubois, Henri Prade, and Philippe Smets, New semantics for quantitative possibility theory., ISIPTA, 2001, pp

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