Bayesian Inference under Ambiguity: Conditional Prior Belief Functions

Transcription

1 PMLR: Proceedings of Machine Learning Research, vol. 6, 7-84, 07 ISIPTA 7 Bayesian Inference under Ambiguity: Conditional Prior Belief Functions Giulianella Coletti Dip. Matematica e Informatica, Università degli Studi di Perugia (Italy) Davide Petturiti Dip. Economia, Università degli Studi di Perugia (Italy) Barbara Vantaggi Dip. S.B.A.I., La Sapienza Università di Roma (Italy) GIULIANELLA.COLETTI@UNIPG.IT DAVIDE.PETTURITI@UNIPG.IT BARBARA.VANTAGGI@SBAI.UNIROMA.IT Abstract Bayesian inference under imprecise prior information is studied: the starting point is a precise strategy σ and a full B-conditional prior belief function Bel B, conveying ambiguity in probabilistic prior information. In finite spaces, we give a closed form expression for the lower envelope P of the class of full conditional probabilities dominating {Bel B, σ} and, in particular, for the related posterior probabilities. The assessment {Bel B, σ} is a coherent lower conditional probability in the sense of Williams and the characterized lower envelope P coincides with its natural extension. Keywords: conditional belief function; Bayesian conditioning rule; inference; ambiguity.. Introduction Bayesian inference is known to naturally fit into de Finetti s theory of coherent (finitely additive) conditional probabilities, where a coherent assessment can be always extended, generally not in a unique way, to any superset of conditional events (de Finetti, 975; Williams, 975). In some application domains (e.g., decision theory, economics, game theory and forensic analysis, to cite some) the prior knowledge could be only partially specified or, even worse, it could refer to a different space of hypotheses. In these circumstances, instead of considering a single prior distribution, one is forced to take into account a set of priors (see, e.g., (Dempster, 967; DeRoberts and Hartigan, 98; Gilboa and Schmeidler, 989)). For instance, suppose that a pension system, based on the social security contributions Λ, is modified by a legal reform so that the new pension scheme takes into account the contribution s years Θ. In order to use the previous information, we need to extract a new prior, starting from the prior distribution P of Λ by taking into account the logical relations between Λ and Θ. There could be possibly infinite probability distributions of (Λ, Θ) compatible with P, determining a lower envelope for the distribution of Θ. In particular, if the initial prior information P is a full conditional probability (Dubins, 975), then for Θ we obtain a full B-conditional belief function (Coletti et al., 06b), i.e., a conditional totally monotone uncertainty measure. Now, considering the profession X and a statistical model connecting X and Θ, the goal could be to draw inferences on Θ belonging to a set of values A (e.g., social pension) under particular values of X (e.g., a person is a clerk). Motivated by the previous discussion, the main aim of this paper is to prove a generalized version of Bayes theorem, working with an ambiguous conditional prior information in the form of a full B-conditional belief function Bel B and a precise statistical model λ, the latter uniquely determining a strategy σ (Dubins, 975). A prior in the form of a full B-conditional belief function 7

2 COLETTI ET AL. is not so uncommon. For instance, in Example, starting from an automatic system S which evolves according to a Markov chain, we show how to generate a full B-conditional prior belief function on the algebra spanned by the states of another unobservable automatic system T, taking into account the logical constraints among the states of S and those of T. Focusing on finite spaces, we provide a characterization of the lower envelope P of the class of full conditional probabilities dominating {Bel B, σ}. The assessment {Bel B, σ} is a Williamscoherent lower conditional probability and P turns out to be its natural extension (Williams, 975). Our results are connected with those proved in (Walley, 98, 99; Wasserman, 990a,b; Wasserman and Kadane, 990): we generalize them in a finite context, since no assumption of positivity for the (lower or upper) probability of the conditioning events is required.. Preliminaries Let A be a Boolean algebra of events E s, and denote with ( ) c, and the usual Boolean operations of negation, disjunction and conjunction, respectively, and with the partial order of implication. The sure event Ω and the impossible event coincide, respectively, with the top and bottom elements of A. If A is finite, we denote with C A the subset of its atoms which form the finer partition of Ω contained in A. Denote A 0 = A\{ }, N is the set of natural numbers, I stands for an arbitrary index set and {E i } i I indicates the Boolean algebra generated by the set of events {E i } i I. A conditional event E H is an ordered pair of events (E, H) with H. In particular, any event E can be identified with the conditional event E Ω. An arbitrary set of conditional events G = {E i H i } i I can always be embedded into a minimal set A A 0, where A = {E i, H i } i I. Recall the definition of coherent conditional probability essentially due to (de Finetti, 975; Holzer, 984; Regazzini, 985; Williams, 975). Definition Let G = {E i H i } i I be a set of conditional events. A function P : G [0, ] is a coherent conditional probability if and only if, for every n N, every E i H i,..., E in H in G and every real numbers s,..., s n, denoting B = {E ij, H ij } j=,...,n with set of atoms C B = {C,..., C m }, the random gain defined on C B as G = n j= s j( Eij P (E ij H ij )) Hij satisfies min G(C r ) 0 max G(C r ), C r H0 0 C r H0 0 where H 0 0 = n j= H i j and, for every E B, E is its indicator defined on C B as E (C r ) = if C r E and 0 otherwise. In particular, if G = A A 0 where A is a Boolean algebra, then P ( ) is a coherent conditional probability if and only if it satisfies the following conditions: (C) P (E H) = P (E H H), for every E A and H A 0 ; (C) P ( H) is a finitely additive probability on A, for every H A 0 ; (C) P (E F H) = P (E H) P (F E H), for every H, E H A 0 and E, F A. In this case P ( ) is simply said a full conditional probability on A according to (Dubins, 975). If G = {E i H i } i I is an arbitrary set, the coherence condition is equivalent to the existence of a full conditional probability on the A = {E i, H i } i I extending the given assessment. This is 74

3 BAYESIAN INFERENCE UNDER AMBIGUITY: CONDITIONAL PRIOR BELIEF FUNCTIONS a consequence of the conditional version of the fundamental theorem for probabilities (de Finetti, 975; Regazzini, 985; Williams, 975): every coherent conditional probability P on an arbitrary G can be extended, generally not in a unique way, to every superset of conditional events G. If A is a finite Boolean algebra, every full conditional probability P ( ) on A is in bijection with a unique linearly ordered class {P 0,..., P k } of probability measures on A whose supports form a partition of Ω (Krauss, 968). The class {P 0,..., P k } is called complete agreeing class and represents the full conditional probability P ( ) in the sense that, for every F K A A 0, there Pα(F K) P α(k). is a minimum index α {0,..., k} such that P α (K) > 0 and P (F K) = If G is an arbitrary set, we can have more complete agreeing classes, each of them obtained by solving a suitable sequence of linear systems (Coletti and Scozzafava, 00). The set P = { P ( )} of all coherent extensions of P to a superset G is a compact subset of the space [0, ] G endowed with the product topology and the projection set on each element of G is a (possibly degenerate) closed interval. The pointwise envelopes P = min P and P = max P, are known as coherent lower and upper conditional probabilities (Coletti and Scozzafava, 00), where coherence here is intended in the sense of (Williams, 975) (namely, Williams-coherence). The envelopes P and P satisfy the duality property, i.e., P (E H) = P (E c H), for every E H, E c H G, so in the following we mainly deal with P. In general, Williams-coherent lower conditional probabilities can be introduced without starting from a precise coherent conditional probability (Williams, 975): Definition A function P ( ) on a set of conditional events G = {E i H i } i I is a Williamscoherent lower conditional probability if there is a class of P = { P ( )} of coherent conditional probabilities on G such that P = inf P. The extendibility of every coherent conditional probability implies the extendibility, generally not in a unique way, of every Williams-coherent lower conditional probability: the pointwise minimal of such extension is referred to as natural extension (Williams, 975). For checking that an assessment is Williams-coherent in a finite setting and to find its natural extension see (Capotorti et al., 00; Coletti and Scozzafava, 00).. Full B-conditional belief and plausibility functions A belief function Bel (Dempster, 967; Shafer, 976) on a finite Boolean algebra A with set of atoms C A is a function such that Bel( ) = 0, Bel(Ω) = and satisfying the n-monotonicity property for every n, i.e., for every E,..., E n A, ( n ) Bel E i i= =I {,...,n} ( ) ( ) I + Bel E i. The associated dual function P l, defined as P l(e) = Bel(E c ), for every E A, is said plausibility function. Both Bel and P l are particular (normalized) capacities (Choquet, 95), i.e., they are monotone with respect to the relation. A belief function Bel (and so its dual P l) on a i I 75

4 COLETTI ET AL. finite Boolean algebra is completely characterized by its Möbius inversion m : A [0, ], also known as basic probability assignment (Shafer, 976), defined, for every E A, as m(e) = B E( ) C E\B Bel(B), where C E\B = {D r C A : D r E B c }. In particular, m satisfies m( ) = 0 and E A m(e) =, and, for every E A, it holds Bel(E) = B E m(e) and P l(e) = B E m(e). Denote with F Bel the set of focal elements of Bel, where an event A in A is a focal element of Bel whenever m(a) > 0. Given a finite partition L = {H,..., H n } of Ω, a capacity ϕ on A L = L and a function X : L R, the Choquet integral of X with respect to ϕ is defined as C X(H i )ϕ(dh i ) = n [ X(Hρ(i) ) X(H ρ(i ) ) ] ϕ ( ) H ρ(i)... H ρ(n), i= where ρ is a permutation of {,..., n} such that X(H ρ() )... X(H ρ(n) ) and X(H ρ(0) ) := 0. We write dh i since we are integrating a function defined on the partition L = {H,..., H n } with respect to a capacity defined on A L = L. We recall the definitions of C-class and full B-conditional belief and plausibility functions given in (Coletti et al., 06b). Definition Let A be a finite Boolean algebra. A linearly ordered class {Bel 0,..., Bel k } of belief functions on A with sets of focal elements {F Bel0,..., F Belk } is said a covering class, or C-class for short, if it satisfies the following covering condition E k α=0 F Belα E = Ω. () By duality, a linearly ordered class {P l 0,..., P l k } of plausibility functions on A is said C-class if the corresponding class of dual belief functions {Bel 0,..., Bel k } is. By means of a C-class of belief functions we define the concept of full B-conditional belief function, where B stands for Bayesian. Definition 4 Let A be a finite Boolean algebra. A function Bel B : A A 0 [0, ] is a full B-conditional belief function on A if there exists a C-class {Bel 0,..., Bel k } of belief functions on A whose dual plausibility functions are {P l 0,..., P l k }, such that, for every E H A A 0, if E H = H then Bel B (E H) =, while if E H H Bel B (E H) = Bel αe,h (E H) Bel αe,h (E H) + P l αe,h (E c H), () where α E,H = min{α {0,..., k} : Bel α (E H) + P l α (E c H) > 0}. 76

5 BAYESIAN INFERENCE UNDER AMBIGUITY: CONDITIONAL PRIOR BELIEF FUNCTIONS The previous definition introduces a full B-conditional belief function through a generalized Bayesian conditioning rule corresponding to the one originally given in (Walley, 98) for - monotone capacities. The Bayesian conditioning rule has been discussed for belief functions in (Dempster, 967; Dubois and Denœux, 0; Fagin and Halpern, 99; Jaffray, 99). The difference with the previous ones is that the rule given in Definition 4 covers also the case in which Bel(E H Ω) + P l(e c H Ω) = Bel 0 (E H) + P l 0 (E c H) = 0, since it relies not on a single belief function but on a linearly ordered class of belief functions. For H A 0, the dual conditional function P l B of a full B-conditional belief function Bel B on A is defined, for every E A, as P l B (E H) = Bel B (E c H), and is called full B-conditional plausibility function. By duality we immediately have P l B (E H) = 0 when E H =, while if E H it holds P l B (E H) = Bel B (E c H) = P l αe c,h (E H) P l αe c,h (E H) + Bel αe c,h (E c H). () Notice that a full conditional probability P on A is both a full B-conditional belief function and a full B-conditional plausibility function. In this paper conditional belief functions are always intended in the sense of Definition 4: notice that full conditional probabilities reveal to be both full B-conditional belief and full B-conditional plausibility functions. In (Coletti et al., 06b) it is proved that the conditional measures Bel B and P l B determine the non-empty compact set P B = { π : π is a full conditional probability on A, Bel B π P l B }, (4) for which it holds Bel B = min P B and P l B = max P B. In the same paper we prove that, for every full B-conditional belief function Bel B on A it is always possible to find a different finite Boolean algebra B and a full conditional probability P on B such that P B can be recovered as the set of coherent extensions of P to A A 0 and, thus, Bel B and P l B as the envelopes of P B. In (Coletti et al., 06a) it is also shown that if all the belief functions in a C-class reduce to necessity measures then the corresponding full B-conditional belief function is a full B-conditional necessity measure and its dual is a full B-conditional possibility measure. In particular, interpreting the conditional measures Bel B and P l B as envelopes, a necessary and sufficient condition (involving the finite Boolean algebras B and A and the full conditional probability P ) is given for Bel B and P l B to be full B-conditional necessity and possibility measures. Since a full B-conditional belief function Bel B determines the non-empty compact set P B of full conditional probabilities dominating it, its use in a Bayesian inferential procedure implies an ambiguous specification of a full conditional probability. 4. Bayesian inference with full B-conditional prior belief functions Let L = {H,..., H n } and E = {E,..., E m } be two finite partitions of Ω and consider the Boolean algebras A L = L, A E = E, A = A L A E. The partitions L and E play the roles of the sets of mutually exclusive and exhaustive hypotheses and evidences, respectively. 77

6 COLETTI ET AL. In the standard Bayesian setting, a statistical model (see, e.g., (Torgersen, 99)) is given on A E L, where the latter is a function λ : A E L [0, ] such that, for every H i L: (L) λ(b H i ) = 0 if B H i = and λ(b H i ) = if B H i = H i, for every B A E ; (L) λ( H i ) is a probability on A E. Proposition in (Petturiti and Vantaggi, 07) implies that any statistical model λ on A E L uniquely extends to a strategy on A L (see, e.g., (Dubins, 975)) which is a function σ : A L [0, ] such that, for every H i L: (S) σ(h i H i ) = ; (S) σ( H i ) is a probability on A. By Theorem 5 in (Dubins, 975), for every full conditional prior probability π on A L, the assessment {π, σ} (and, in particular, {π, λ}) is a coherent conditional probability, therefore it can be extended, generally not in a unique way, to a full conditional probability on A. This implies that, given a full B-conditional prior belief function Bel B on A, the assessment {Bel B, σ} (and, in particular, {Bel B, λ}) is a Williams-coherent lower conditional probability. Remark 5 The assessment {Bel B, σ} determines a Williams-coherent lower conditional probability P on the set of conditional events G = (A L A 0 L ) (A L) such that P A L A 0 L = Bel B and P A L = σ so, with a little abuse of terminology, {Bel B, σ} is directly referred to be a Williamscoherent lower conditional probability. Remark 6 Recall that, in view of the finite setting adopted in this paper, the notion of conditioning for lower probabilities due to Williams coincides with that due to (Walley, 99) since in this case the conglomerability condition is automatically satisfied. Let Bel B be a full B-conditional belief function on A L and σ a strategy on A L and denote with P B the set of full conditional probabilities on A L dominating Bel B. Consider P = { P : P is a full conditional probability on A extending { π, σ}, π PB }, which is a non-empty compact subset of [0, ] A A0 endowed with the product topology, whose envelopes are P = min P and P = max P. The lower envelope P ( ) turns out to be the natural extension of the Williams-coherent lower conditional probability {Bel B, σ}. The following theorem provides a characterization of the lower envelope P ( ), relying on the functions defined, for every F, K A and A A 0 L such that K A, as { n } n L(F, K; A) = min σ(f K H i ) π(h i A) : σ(f c K H i ) π(h i A) = P (F c K A), π P B U(F, K; A) = max π P B i= { n σ(f K H i ) π(h i A) : i= i= } n σ(f c K H i ) π(h i A) = P (F c K A), where we write F K and F c K in place of F K and F c K to save space. i= 78

7 BAYESIAN INFERENCE UNDER AMBIGUITY: CONDITIONAL PRIOR BELIEF FUNCTIONS Theorem 7 The lower envelope P ( ) is such that, for every F K A A 0, if F K = K, then P (F K) =, otherwise: (i) if K A 0 L, then P (F K) = C σ(f H i )Bel B (dh i K); (ii) if K A 0 \ A 0 L, then if there exists A A0 L such that K A and P (K A) > 0 we have that { } P (F K A) P (F K) = min P (F K A) + U(F c, K; A), L(F, K; A), L(F, K; A) + P (F c K A) otherwise P (F K) = 0. Proof The statement is trivial if F K = K since in this case P (F K) = for every P P, thus suppose F K K. Condition (i) follows since, if K A 0 L then P (F K) = min P P = min π C BelB ( K) P (F K) = min π P B n i= n σ(f H i ) π(h i K) i= σ(f H i ) π(h i K) = C σ(f H i )Bel B (dh i K), where C BelB ( K) = { π( K)} is the core of probability measures on A L induced by Bel B ( K) and the last equality follows by Proposition in (Schmeidler, 986). To prove condition (ii) let us consider K A 0 \ A 0 L. If there exists A A0 L such that K A and P (K A) > 0 we have that P (K A) > 0 for every P P (F K A) P, thus P (F K) = min P (F K A)+ P. So, the conclusion (F c K A) follows since x x+y is increasing in x and decreasing in y, and, for every P P, P ( A) is the convex combination of probabilities P ( A) and P ( A), P, P P, attaining the lower and the upper envelopes, respectively, on F K (or on F c K) and P (F K) min{p (F K), P (F K)}. The remaining case, realizing when for all A A 0 L with K A it holds P (K A) = 0, is proven in analogy to the proof of Lemma in (Petturiti and Vantaggi, 07). P P Restricting to a finite setting, the previous theorem generalizes some results proved in (Coletti et al., 04) in which an ambiguous unconditional prior is considered, either in the form of a belief function or a -monotone capacity. A simplification of condition (ii) of Theorem 7 is obtained when the functions on L, defined as X( ) = σ(f H ) and ( Y ( )) = ( σ(f c H )), are comonotonic (see, e.g., (Denneberg, 994)), i.e., for every H h, H k L, [X(H h ) X(H k )] [( Y (H h )) ( Y (H k ))] 0, as shown by the following Proposition 8. In particular, this happens for all conditional events in A L A 0 E related to posterior probabilities, obtaining, for a finite setting, a generalization of results in (Wasserman, 990a). Proposition 8 For every F K A A 0 such that F K K, K A 0 \A 0 L and there exists A A 0 L such that K A and P (K A) > 0, if X( ) = σ(f H ) and ( Y ( )) = ( σ(f c H )) are comonotonic then P (F K A) P (F K) = P (F K A) + P (F c K A). 79

8 COLETTI ET AL. Proof For every A A 0 L, Bel B( A) is a totally monotone capacity on A L inducing a core C BelB ( A) = { π( A)} of probability measures on A L, moreover, the functions X( ) and ( Y ( )) are comonotonic. By Proposition 6.6 in (Troffaes and de Cooman, 04) there exists π( A) C BelB ( A) such that n i= X(H i) π(h i A) = C X(Hi )Bel B (dh i A) and n i= ( Y (H i)) π(h i A) = C ( Y (H i ))Bel B (dh i A). Since n i= ( Y (H i)) π(h i A) = n i= Y (H i) π(h i A) and C ( Y (Hi ))Bel B (dh i A) = C Y (Hi )P l B (dh i A), it follows C Y (Hi )P l B (dh i A) = n i= Y (H i) π(h i A) and this implies P (F K A) = C X(Hi )Bel B (dh i A) = n i= X(H i) π(h i A) = L(F, K; A) and P (F c K A) = C Y (Hi )P l B (dh i A) = n i= Y (H i) π(h i A) = U(F c, K; A). The following example shows that, though Bel B ( K) is a belief function on A L, for every K A 0 L, and σ( H i) is a probability measure on A, for every H i L, the function P ( K) can fail -monotonicity, for some K A 0. Example Let L = {H, H, H } and E = {E, E, E, E 4 } be logically independent partitions of Ω, and take A L = L, A E = E and A = A L A E. Let Bel B be the full B-conditional belief function on A L determined by the C-class of belief functions {Bel 0, Bel } displayed below A L H H H H H H H H H Ω Bel Bel where F Bel0 = {H, H H } and F Bel = {H, H H }, thus condition () of Definition is satisfied. Let λ be the statistical model on A E L such that λ(e j H ) = λ(e j H ) = 6, for j =,,, and λ(e 4 H ) = λ(e 4 H ) =, λ(e H ) = λ(e H ) =, and λ(e H ) = λ(e 4 H ) = 0. which uniquely extends to a strategy σ on A L by Proposition in (Petturiti and Vantaggi, 07). Let K = H H, A = E E and B = E E. Simple computations show that Bel B ( K) is a belief function vacuous at K, so, we have P (A B K) = C σ(a B H i )Bel B (dh i K) = inf σ(a B K) = H i K, P (A K) = C σ(a H i )Bel B (dh i K) = inf σ(a K) = H i K, P (B K) = C σ(b H i )Bel B (dh i K) = inf σ(b K) = H i K, P (A B K) = C σ(a B H i )Bel B (dh i K) = inf σ(a B K) = 0. H i K Since P (A B K) < P (A K) + P (B K) P (A B K), P ( K) is not -monotone. 80

9 BAYESIAN INFERENCE UNDER AMBIGUITY: CONDITIONAL PRIOR BELIEF FUNCTIONS Proposition 8 is a generalization of the ɛ-contamination model presented in Example. in (Huber, 98), where the author provides a characterization of the lower envelope P on A L E, starting from a statistical model λ and an unconditional prior belief function Bel obtained as the ɛ- contamination of a reference prior probability. In such case, in (Huber, 98) it is stated that P ( E j ) is a -monotone capacity on A L, for every E j E, nevertheless, as shown in our Example the envelope P ( K) can fail -monotonicity on the whole A, for some K A 0. The following example shows that a full B-conditional prior belief function can be obtained starting from a full conditional prior probability defined on a different algebra. Example We consider an automatic system S that can assume three possible states s, s and s. Let S = {S, S, S } be the partition of Ω, where S i = S is in state s i, for i =,,, and denote A S = S. The evolution of S is determined by the Markov chain whose transition matrix and graph are reported in Figure. A = 0 0 Figure : Transition matrix and graph of the Markov chain related to S Suppose that the initial state of S is selected at random and that we observe the system evolve indefinitely in time, so, we can take the limit probabilistic behaviour as our prior information on S. The starting probability distribution on the states of S is π (0) = (,, ), while after n > 0 steps ( we have π (n) = π (n ) A = ( ) n+, ( ) n, ( ) n ). It is easily seen that the probability distribution π (n) is positive for every n 0, so, it uniquely extends to a full conditional probability (still denoted with π (n) ) on A S setting, for every A B A S A 0 S, π(n) (A B) = π(n) (A B). Thus, we have a sequence {π (n) : n = 0,,,...} of full π (n) (B) conditional probabilities on A S converging pointwise to the full conditional probability π ( ) on A S defined below A S S S S S S S S S S Ω π ( ) ( S ) π ( ) ( S ) π ( ) ( S ) π ( ) ( S S ) π ( ) ( S S ) π ( ) ( S S ) 0 0 π ( ) ( Ω)

10 COLETTI ET AL. The full conditional probability π ( ) is determined by the complete agreeing class {P 0, P } of probability measures on A S such that P 0 ( ) = π ( ) ( Ω) and P ( ) = π ( ) ( S S ). Consider now a second automatic system T that is not directly observable: the only information we have is that T can assume three possible states t, t and t, and that if S is in state s i then T is not in state t i, for i =,,. Let T = {T, T, T } be the partition of Ω, where T i = T is in state t i, for i =,,, and denote A T = T with T i S i =, for i =,,. As proven in (Coletti et al., 06b) setting, for every B A T, (B) = {S i S : S i B}, Bel 0 (B) = P 0 ((B) ) and Bel (B) = P ((B) ), we obtain a C-class of belief functions {Bel 0, Bel } on A T B-conditional belief function on A T reported below which, in turn, determines the full A T T T T T T T T T T Ω Bel B ( T ) Bel B ( T ) Bel B ( T ) Bel B ( T T ) Bel B ( T T ) Bel B ( T T ) Bel B ( Ω) Suppose that the state of the unobservable system T can be verified through a detector D that returns one of three possible values d, d and d, with d i corresponding to the state t i, for i =,,, with a reliability of 90% and equal chances on failures. Let D = {D, D, D } be the partition of Ω, where D i = D returns d i, for i =,,, and denote A D = D. Let A = A T A D and consider the statistical model on A D T singled out by λ(d i T i ) = 90%, λ(d j T i ) = λ(d k T i ) = 5%, for different i, j, k {,, }, that uniquely extends to a strategy σ on A T by Proposition in (Petturiti and Vantaggi, 07). The full B-conditional belief function Bel B encodes all our prior information on T and can be used together with σ to draw Bayesian inferences. At this aim, suppose that the detector D shows the value d j, for j =,,, then the lower posterior distribution on the states of T can be easily determined using Proposition 8. For instance, since P (D j Ω) > 0, P (T D j Ω) = 0 and P (T c D j Ω) > 0, for j =,,, we get P (T D j ) = P (T D j Ω) P (T D j Ω) + P (T c D j Ω) = 0, and, analogously, we can compute P (T c D j) =, so, P (T D j ) = P (T D j ) = 0, i.e., the observation of the detector D does not change our degree of belief on T since it is Bel B (T Ω) = P l B (T Ω) = Conclusions We show that, as long as we consider a precise strategy σ, the introduction of ambiguity in the prior information through a full B-conditional belief function Bel B has straightforward treatment: 8

11 BAYESIAN INFERENCE UNDER AMBIGUITY: CONDITIONAL PRIOR BELIEF FUNCTIONS a characterization for the envelopes of the class of full conditional probabilities dominating the assessment {Bel B, σ} is provided. The entire procedure lives inside Williams framework and the characterized lower envelope reveals to be the natural extension of {Bel B, σ}. Our aim for future research is to introduce ambiguity also in the strategy by considering an imprecise strategy β such that β( H i ) is a belief function, for every H i L, possibly removing the finiteness assumption. This would lead to a theory to compare with that of (Walley, 99). Acknowledgments This work was partially supported by INdAM-GNAMPA through the Project 06 U06/0009. References A. Capotorti, L. Galli, and B. Vantaggi. Locally strong coherence and inference with lower upper probabilities. Soft Computing, 7(5):80 87, 00. G. Choquet. Theory of capacities. Annales de l Institut Fourier, 5: 95, 95. G. Coletti and R. Scozzafava. Probabilistic Logic in a Coherent Setting, volume 5 of Trends in Logic. Kluwer Academic Publisher, Dordrecht/Boston/London, 00. G. Coletti, D. Petturiti, and B. Vantaggi. Bayesian inference: the role of coherence to deal with a prior belief function. Statistical Methods and Applications, (4):59 545, 04. G. Coletti, D. Petturiti, and B. Vantaggi. When upper conditional probabilities are conditional possibility measures. Fuzzy Sets and Systems, 04:45 64, 06a. G. Coletti, D. Petturiti, and B. Vantaggi. Conditional belief functions as lower envelopes of conditional probabilities in a finite setting. Information Sciences, 9:64 84, 06b. B. de Finetti. Theory of Probability -. Wiley, 975. A. Dempster. Upper and Lower Probabilities Induced by a Multivalued Mapping. Annals of Mathematical Statistics, 8():5 9, 967. D. Denneberg. Non-Additive Measure and Integral, volume 7 of Series B: Mathematical and Statistical Methods. Kluwer Academic Publishers, Dordrecht/Boston/London, 994. L. DeRoberts and J. Hartigan. Bayesian inference using intervals of measures. Annals of Statistics, 9():5 464, 98. L. Dubins. Finitely additive conditional probabilities, conglomerability and disintegrations. The Annals of Probability, ():89 99, 975. D. Dubois and T. Denœux. Conditioning in Dempster-Shafer Theory: Prediction vs. Revision, pages Springer Berlin Heidelberg, 0. R. Fagin and J. Halpern. Uncertainty in Artificial Intelligence, chapter A New Approach to Updating Beliefs, pages Elsevier Science Publishers, 99. 8

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