Multiparameter models: Probability distributions parameterized by random sets.

Transcription

1 5th International Symposium on Imprecise Probability: Theories Applications, Prague, Czech Republic, 007 Multiparameter models: Probability distributions parameterized by rom sets. Thomas Fetz Institut für Grundlagen der Bauingenieurwissenschaften, Arbeitsbereich Technische Mathematik Universität Innsbruck, Austria Abstract This paper is devoted to the construction of sets of joint probability measures for the case that the marginal sets of probability measures are generated by probability measures with uncertain parameters where the uncertainty of these parameters is modelled by rom sets. Further we show how different conditions on the choice of the weights of the joint focal sets on the probability measures associated to these sets lead to different sets of joint probability measures including the cases of strong independence, rom set independence unknown interaction. Keywords. Rom sets, lower upper probabilities, sets of probability measures, parameterized probability measures, sets of joint probability measures, strong independence, rom set independence, unknown interaction. 1 Introduction Let a mapping g : D R m R : (x 1,..., x m g(x 1,..., x m be given. The variables x k are assumed to be uncertain where the uncertainty is modelled by sets of probability measures for each variable separately. What we want to know is the lower upper probabilities if the value g(x, x (x 1,..., x m, is lower (or greater than a certain value. Therefore we had to propagate the uncertainty of the variables x k through this multivariate model g, c.f. [1]. As a short motivation we want to mention a few applications where this problem of propagating uncertain variables is arising: Reliability analyis: In this case the above mapping g is the so called failure function where g(x 0 means failure g(x > 0 means no failure of buildings like bridges tunnels in civil engineering; or of slopes dams in geotechnical engineering. The aim is to describe the risk of failure, that means we want to have the upper probability P ({g(x 0} of failure. The variables x k are parameters as elastic modulus E, angle of friction φ or flood heights. Construction management: Here the values of g(x are costs or durations which should not exceed a certain bound a where the variables x k are costs, durations or similar parameters as above. Then we want to have the upper probability P ({g(x a}. In most cases all these variables are not precisely known, especially parameters arsing in geotechnical engineering are only very vaguely known. In engineering there are several approaches used to describe the uncertainty of these variables: wellknown ones as probability distributions or intervals more modern ones as fuzzy sets rom sets. The uncertainty of the variables is given separately often modelled by different ways. So a unifying approach is needed to combine propagate the different models of uncertainty trough the function g. This is provided by the concept of sets of probability measures where these sets are generated by rom sets which are including the other three approaches (probability distributions, intervals fuzzy sets. In some applications the type of probability distributions to discribe the uncertainty of a variable is known, e.g. gaussian distributions, exponential distributions in queueing theory, etreme value distributions in flood risk analysis, but the parameters of these probability distributions are often only vaguely known. In these cases we have to model the uncertainty of the parameters of these distributions. So we introduce here the concept of sets of probability measures which are generated by parameterized probability measures where these parameters are uncertain the uncertainty is described by rom sets. All models of uncertainty mentioned before are special cases of this concept.

2 Since the uncertainty of the variables is given separately, we have to model the joint uncertainty, that means to construct the set of joint probability measures. There are certain ways to generate such sets, e.g. according to strong independence [, 11] if we assume stochastically independence of the variables, or according to unknown interaction [] if we do not know how the variables interact, or according to rom set independence [3] since rom sets are involved. These cases are already studied for sets of probability measures generated by rom sets in [5, 6, 7, 8]. Here in this paper we extend this to sets of probability measures generated by parameterized probability measures with uncertain parameters. To propagate the uncertainty through a multivariate model in a computational efficient way it is essential to make use of the structure of the rom sets. We show how different conditions on the parts of this structure (on the choice of the weights of the joint focal sets on the probability measures associated to these sets lead to different sets of joint probability measures. But our goal is not to create new artificial types of sets of joint probability measures, but to get sets according to strong independence or unknown interaction by using the rom set structure. The plan of this paper is as follows: Section is devoted to rom sets the parameterization of probability measures by rom sets in the univariate case. In Section 3 we construct sets of joint probability measures which are generated by probability measures which are parameterized by ordinary sets as preliminary work for Sec. 5. In Section 4 we recall from [5, 7] the general formulation for constructing sets of joint probability measures for the case where rom sets are involved list different conditions on choosing the weights of the joint focal sets the probability measures associated to these sets. In Section 5 we show that some of these cases lead to strong independence, rom set independence unknown interaction. Sets of probability measures generated by probability measures parameterized by rom sets We want to model the uncertainty about the value of a variable x by a convex set K of probability measures in the univariate case. Here in this paper we generate such sets K by a parameterized probability measure p θ where θ (θ 1, θ,... are the parameters of the probability measure. These parameters are assumed to be uncertain. The uncertainty of θ is modelled by rom sets. So we have to recall the concept of rom sets sets of probability measures generated by rom sets. Further we need two different measurable spaces: (Ω, A for the uncertain variable itself (, A for the uncertain parameters of the probability measures..1 Rom sets First we want to model the uncertainty of a variable x by rom sets. Let a measurable space (Ω, A be given. A rom set (F, m [3, 4] consists of a finite class F {F 1, F,..., F n } A of focal sets of a weight function m : F [0, 1] : F m(f with m(f i 1 where is the number of focal sets. Then the plausibility measure Pl or upper probability P of a set A A is defined by P (A Pl(A m(f i F i A the belief measure Bel or lower probability P by P (A Bel(A m(f i. F i A. Sets of probability measures generated by rom sets The focal set F i has the weight m(f i, but we do not know how this weight is distributed on the elements of the focal set which reflects the uncertainty modelled by a rom set. Let K(F i : {P : P (F i 1} (1 be the set of all probability measures on the focal set F i. Then m(f i K(F i is the set of all possible distributions of the weight on the focal set. A convex set of probability measures K is generated by the rom set (F, m as follows [5]: K : K(F, m : m(f i K(F i : ( P : P m(f i P i, P i K(F i. This set K(F, m coincides with the set of probability measures defined by {P : A A : Bel(A P (A Pl(A},

3 c.f. [3, 4, 10]. Remark: There is a second approach of defining rom sets: using multivalued mappings measurable selections [9, 10]. This approach leads to a set M of probability measures which is a subset of K which is not convex in general. The set of probability measures associated to the measurable selctions is P Ω (Γ {P X : X S(Γ}, where Γ : Ω A is a multivalued mapping defined on a probability space (Ω, A, P Ω. S(Γ is the set of measurable selections of Γ, that means the class of rom variables X : Ω Ω with X(ω Γ(ω. Now let P X P (Γ, X S(Γ, be given. Then P X (A P Ω (X 1 (A P Ω ({ω i }χ A (X(ω i Ω m(f i χ A (ω i m(f i δ ω i(a with ω i X(ω i Γ(ω i F i P Ω ({ω i } m(f i where χ A is the indicator function of A. So in our above notation the set M would be generated by M : M(F, m : m(f i M(F i with M(F i {δ ω : δ ω (F i 1} {δ ω : ω F i } K(F i (3 where δ ω is the Dirac measure at ω Ω corresponding to the selections. The connections between M K are discussed in [9, 10]..3 Sets of parameterized probability measures Now we generate the set K of probability measures by a probability measure p θ on (Ω, A which is parameterized by an uncertain θ. For modelling the uncertainty of the parameter θ we need the following: A measurable space (, A where is the universal set for θ, A a σ-algebra K a set of probability measures µ on (, A. The σ-algebra A has to be chosen in a way that for all A A the mapping is A-measurable. θ p θ (A The set K is defined by { } K : K(K, p θ : P p θ ( µ(dθ : µ K. (4 Then the upper lower probabilities for a set A A is computed as follows: P (A sup{p (A : P K} sup p θ (A µ(dθ, µ K P (A inf{p (A : P K} inf p θ (A µ(dθ. µ K In the following the set K is either a set of probability measures generated by ordinary sets or by rom sets. The usage meaning of the symbols K K is summarized in the following table: notation set of probability measures K(F on (Ω, A generated by a set F K(F on (, A generated by a set F K(F, m on (Ω, A gen. by a rom set (F, m K(F, m on (, A gen. by a rom set (F, m K(K, p θ on (Ω, A gen. by K p θ as in (4 where K is either a K(F or K(F, m So K is always a set of probability measures on (Ω, A K a set of probability measures on the parameter space of θ, namely..4 Generation of K by probability measures µ on ordinary sets F, K : K(F We take the set K : K(F of probability measures on F A K : K(K(F, p θ for the set K of probability measures which are generated by K(F the parameterized probability measure p θ. Then the upper lower probability are given by P (A sup µ K(F sup θ 0 F p θ (A µ(dθ (5 p θ (A δ θ0 (dθ sup p θ0 (A θ 0 F P (A inf θ0 F p θ0 (A. Further we have for the special case (, A : (Ω, A p ω : δ ω : K(F K(K(F, δ ω, (6 because { } K(K(F, δ ω P δ ω ( µ(dω : µ K(F Ω {µ K(F } K(F K(F ω p ω (A δ ω (A χ A (ω is A-measurable for all A A. So the set of probability measures generated by an ordinary set is integrated into the new concept.

4 .5 Generation of K by rom sets, K : K(F, m Here we take K : K(F, m K : K(K(F, m, p θ. A probability measure P K is written as follows: P p θ ( µ(dθ p θ ( ( m(f i µ i (dθ m(f i p θ ( µ i (dθ m(f i P i where µ K(F, m. µ m(f i µ i is a decomposition of µ according to the focal sets P i p θ ( µ i (dθ is a probability measure in K(K(F i, p θ. So for the set K(K(F, m, p θ we also can write K(K(F, m, p θ m(f i K(K(F i, p θ (7 which is formula Eq. ( but with K(F i replaced by K(K(F i, p θ. The set K(F i used in Eq. ( is a set of probability measures on F i, but the probability measures in the set K(K(F i, p θ are only associated to F i via the parameter θ. Similar to the section above we have for the upper lower probability: P (A m(f i sup µ i K(F i p θ (Aµ i (dθ m(f i sup p θ0 (A m(f i P i (A θ 0 F i P (A m(f i inf p θ0 (A m(f i P i (A. θ 0 F i.6 An example for p θ, gaussian distribution (Ω, A : (R, B(R, (, A (R R >0, B(R R >0, θ : (µ, σ. The function (µ, σ p (µ,σ 1 (A : χ A (x πσ R (x µ e σ dx is continuous therefore A-measurable. We compute the upper lower probability of the set A [a, using a set K(F, m to describe the uncertainty of µ σ with F i [µ i, µ i ] [σ i, σ i ], i 1,..., n, as follows: Then P i ([a, P i ([a, P ([a, P ([a, { p (µ i,σ i ([a, µ i a, p (µ i,σ i ([a, otherwise, { p (µ i,σ i ([a, µ i a, p (µ i,σ i ([a, otherwise. n m(f i P i ([a, n m(f i P i ([a,. 3 Sets of joint probability measures generated by ordinary sets 3.1 Preliminaries In this paper we restrict ourselves to the combination of only two sets of probability measures. In the following we always need the measurable spaces (Ω 1, A 1, (Ω, A (Ω, A with Ω Ω 1 Ω A A 1 A for the uncertain variables ( 1, A 1, (, A (, A with 1 for the uncertain parameters of the probabilty measures with σ-algebras such that mappings like θ p θ (A are measurable. A set A will be always in A. The generation of a set of joint probability measures by two marginal sets K 1 K of probability measures will be written as K(K 1, K. First we recall two general ways of combining sets K 1 K of probability measures. Unknown interaction: The set of joint probability measures according to unknown interaction [] is generated by K U : {P : P ( Ω K 1, P (Ω 1 K }. (U Strong independence: The set of joint probability measures according to strong independence [, 11] is generated by K S : {P 1 P : P 1 K 1, P K } K U. (S

5 Notation for the corresponding probabilities: P S (A : sup{p S (A : P S K S }, P S (A : inf{p S (A : P S K S }, P U (A : sup{p U (A : P U K U }, P U (A : inf{p U (A : P U K U }. In the following we are analyzing very special cases of sets of joint probability measures which is a preliminary work for Sec. 4 5 for dealing with the joint focals sets in these sections. 3. K k : K(F k K(K(F k, δ ωk Given subsets F k A k, k 1,, we generate the sets K U K S of joint probability measures by the sets K(F 1 K(F. K U (K(F 1, K(F is the set of joint probability measures generated by the two sets K(F 1 K(F of probability measures according to (U. Since the marginals of all probability measures on F 1 F are in the sets K(F 1 K(F, respectively, we have K U (K(F 1, K(F K(F 1 F. To get the upper lower probability P U (A P U (A it is sufficient to put a Dirac measure at the appropriate place. Since a Dirac measure is a product measure we get P S (A P U (A P U (A P S (A. Now we make a first step towards sets of joint probability measures generated by parameterized probabilities doing the same for K ( K(F k, δ ωk in the more general notation. We already know that K(F k K ( K(F k, δ ωk therefore K U K U ( K(F1, K(F K(F 1 F K ( K(F 1 F, δ ω1 δ ω. Further we have for strong independence K S K S ( K(F1, K(F K S ( K(K(F1, δ ω1, K(K(F, δ ω K ( K S (K(F 1, K(F, δ ω1 δ ω, because P S (A (P 1 P (A P (A ω1 P 1 (dω 1 (8 Ω 1 ( χ (ω Aω1 δ (dω ω µ (dω P 1 (dω 1 Ω Ω 1 Ω ( χ (ω Aω1 δ (dω ω µ (dω Ω 1 Ω 1 Ω Ω δ ω 1 (dω 1 µ 1 (dω 1 ( χ (ω Aω1 δ (dω ω δ ω 1 (dω 1 Ω 1 Ω Ω 1 Ω µ (dω µ 1 (dω 1 ] [(δ ω δ 1 ω (A µ (dω µ 1 (dω 1 Ω 1 Ω Ω 1 Ω ] [(δ ω δ 1 ω (A µ(d(ω 1, ω with µ 1 K(F 1, µ K(F, µ K S (K(F 1, K(F A ω1 {ω Ω : (ω 1, ω A}. 3.3 K k : K(K(F k, p θ k k Now we replace the Diracs by parameterized probability measures p θ k k analyze the cases of strong independence unknown interaction Strong independence Similar to Eq. (8 it holds: ( P S (A So we get 1 Ω 1 1 Ω ( 1 Ω 1 Ω χ Aω1 (ω p θ (dω µ (dθ p θ1 1 (dω 1µ 1 (dθ 1 χ Aω1 (ω p θ (dω p θ1 1 (dω 1 [( p θ1 1 pθ (A µ (dθ µ 1 (dθ 1 ] µ (dθ µ 1 (dθ 1. K S K ( K S (K(F 1, K(F, p θ1 1 pθ Unknown interaction For strong independence the joint probability measure generated by p θ1 1 pθ was pθ1 1 pθ, a single probability measure. In case of unknown interaction we would need the whole set of all possible joint probability measures on (Ω, A. Maybe on the other h we have more information how the joint probability measure, say p θ, is generated by p θ1 1 p θ than how the parameters of the joint probability measure interact. So we introduce the sets K (US K (Upθ of joint probability measures for which the choice of µ is according to (U the choice of the joint parameterized probability measure is according to (S or defined by p θ.

6 Then it holds: K S : K ( K S (K(F 1, K(F, p θ1 1 pθ K ( K U (K(F 1, K(F, p θ1 1 pθ K ( K(F 1 F, p θ1 1 pθ : K(US. For the upper lower probabilities we have P S (A P (US (A und P S (A P (US (A, because we can obtain the upper lower probabilities from K (US by means of Dirac measures in K(F 1 F which are also in K S (K(F 1, K(F. 4 General formulation of the generation of sets of joint probability measures by rom sets Let rom sets (F k, m k, k 1,, be given for modelling the uncertainty of the variables x 1 x. As a consequence of Dempster s rule of combination [3, 4] the joint rom set (F, m is defined by where F {F ij : i 1,..., n 1 ; j 1,..., n } F ij : F i 1 F j m(f i 1 F j : m 1(F i 1m (F j (9 which is the case of rom set independence (RSindependence. For our more general approach we start with the multivariate analogon of Eq. (: F 1 F K? m(f1 i F j K?(K i 1, K j j1 where the question mark in K? (K i 1, K j indicates the possibility of different choices in combining the sets of probability measures K i 1 K j associated with the marginal focal sets F1 i F j. Further we have to define how the joint weights m(f1 i F j are computed (perhaps not in the way of Eq. (9 to think about possible interactions between probability measures in the different sets K? (K i 1, K j. The consequences of these different choices are different sets of joint probability measures K? the goal is to generate sets according to strong independence, unknown interaction RS-independence. In the following we describe the different choices we have for the above formula discuss their consequences for the set of joint probability measures. 4.1 The choice of the joint weights m(f i 1 F j The weights m 1 m are discrete probability measures on the sets of focal sets {F1 1,..., F n1 1 }, {F 1,..., F n } respectively. So if we want to choose the joint focal sets in a stochastically independent way, then m m 1 m which means m(f1 i F j m 1 (F1m i (F j for all i, j. If we do not know how m 1 m interact, we allow all possible combinations, that means unknown interaction. Case (U : Unknown interaction, m must satisfy the following conditions: F m 1 (F1 i m(f1 i F j, i 1,..., F 1, j1 F 1 m (F j m(f1 i F j, j 1,..., F. In this case m is not uniquely defined is determined later on by solving an optimization problem for the lower or upper probabilities. Case (S : Stochastic independence: m(f i 1 F j : m 1(F i 1m (F j. 4. The choice of P ij, K ij, respectively P ij K ij is a probability measure associated to the joint focal set F i 1 F j. How a P ij looks like depends on how K ij is constructed from K i 1 K j. Case ( U : K ij U : K U(K i 1, K j which is the set of all joint probability measures generated by the sets K i 1 K j according to condition (U. Case ( S : K ij S : K S(K i 1, K j which is the set generated according to strong independence (S. 4.3 The choice of interactions between the P ij Case ( 1: Row- columnwise equality conditions on the marginals of the probability measures on the joint focal sets: where P i 1 : P i,i1 1 P i,in i, i 1,..., n 1, P j : P j,1j P j,n1j i, j 1,..., n P i,ik 1 P ik 1 ( Ω P j,kj P kj (Ω 1. This condition seems to be very artificial, but we need this to get results according to strong independence later on.

7 Case ( 0: No interactions, this means that we can choose a P ij K ij on F1 i F j irrespective of the probability measures chosen on other joint focal sets. Remark: It is clear that it should hold that the convex sum k 1 i m 1 (F1 im(f 1 F j i,ik P1 is in K i 1. This is always true for convex sets K i 1 of probability measures, but for sets which are generated by measurable selections (see Eq. (3 it is not true in general. In this case one should introduce a more restrictive condition than ( The choice of the joint marginals We emphasize that the choice of the Cartesian products F1 i F j as joint focals is no restriction of generality. Joint focal sets V F1 i F j of arbitrary shape can be subsumed in our approach by restricting sets of joint probability measures on F1 i F j to those whose support lies in V. Such subsets would describe specific types of dependence or interaction between the marginal focal sets F1 i F j. But such interactions are not investigated in this paper. 5 The different cases Now we will discuss combinations of the above cases which lead to rom set independence, unknown interaction, strong independence. The cases are indicated by indices of the form (ABC where for example (SU0 means m according (S, P ij according to ( U no interaction between the P ij. We want to stress that again, that it is not our goal to introduce a number of eight (all possible combinations new types of joint probability measures, but to identify the combinations which leads to the desired types of sets joint probability measures. We do this for RS-independence, unknown interaction strong independence. In this very technical part we first recall for each of these types the case where pure rom sets are used, that means the case where no parameterized probabilities are involved. Then we generalize the results to the case of parameterized probabilities. So the sets K i k are first sets of probability measures K(Fk i then in a second part replaced by sets K(K(Fk i, p θ associated with the marginal focal set Fk i. 5.1 (SU0, (SS0 RS-independence General formulation The sets K SU0 K SS0 of joint probability measures are generated by F 1 F K SU0 m 1 (F1m i (F j K U(K i 1, K j j1 F 1 F K SS0 m (F j m (F j K S(K i 1, K j. j K i 1 : K(F i 1, K j : K(F j We obtain the upper probability P SU0 (A for a set A A by where F 1 F P SU0 (A m 1 (F1m i (F j P ij U(A j1 P ij U(A sup { P ij ij U (A : PU K U(K(F1, i K(F j } K U (K(F i 1, K(F j K(F i 1 F j. So P ij U(A is computed very easily by P ij U(A sup{δ ω (A : ω F1 i F j { } 1 ω A F1 i F j, 0 else which leads to the formula for the joint plausibility measure P R (A : P SU0 (A Pl(A m 1 (F1m i (F j i,j: F i 1 F j A which is the joint upper probability in the case of RSindependence indicated by the index R. Further we have P SU0 P SS0 because δ ω δ (ω1,ω δ ω1 δ ω. is a product measure (case ( S. Similar to the upper probability we get for the lower probability P R : Bel P SU0 P SS0. Contrary to the above equalities we have for the corresponding sets of joint probability measures only K R : K SU0 K SS0.

8 5.1.3 K i 1 : K(K(F1, i p θ1 1, Kj : K(K(F j, pθ An idea would be to define K R by K SU0 as before [6], but then we have the same problem as in Sec So another possibility would be to define K R : K S(US0 or K R : K S(Upθ 0. We start with the case of (SS0 get F 1 F K SS0 j1 F 1 F j1 F 1 F m 1 (F i 1m (F j Kij S m 1 (F i 1m (F j Kij (US m 1 (F1m i (F j j1 K ( K(F1 i F j θ1, p1 p θ K ( K(F, m, p θ1 1 p θ : KS(US0 : K R, with K ij S : K ( S K(K(F i 1, p θ1 1, K(K(F j θ, p K ij (US : K( K(F1 F i j, pθ1 1 pθ Eq. (7. (F, m is the joint rom set according to RS-independence. K S(SU0 is the set of probability measures where the parameterized probability measure is the product measure, but the uncertainty of the parameters of this poduct measure is discribed by the set K(F, m of joint probability measures which are generated by the rom set describing the uncertainty of θ 1 θ. For the upper lower probabilities we have P SS0 P S(US0 P SS0 P S(US0 by the same arguments as in Sec (UU0, (US0 unknown interaction 5..1 K i 1 : K(F i 1, K j : K(F j Let K UU0 be the set of probability measures generated according to case (UU0. A computational method for P UU0 (A is obtained in the following way: F 1 F P UU0 (A m (F1 i F j P ij U(A j1 i,j: F i 1 F j A m (F i 1 F j, where P ij U(A is computed by the same Dirac measures as for P R the weights m by solving the following linear optimization problem: m(f1 i F j max! i,j: F i 1 F j A subject to condition (U. Minimization instead of maximization leads to lower probability P UU0 (A. The set K UU0 is just the set of probability measures which is generated by the least restrictive conditions on m P ij. It is proven in [5, 6] that K U K UU0. By the same arguments as in the previous cases we get P U P UU0 P US0 P U P UU0 P US K i 1 : K(K(F1, i p θ1 1, Kj : K(K(F j, pθ Similar to Sec we can define sets F 1 F K UU0 m(f1 i F j j1 F 1 F K US0 m(f1 i F j j1 F 1 F K U(US0 m(f1 i F j j1 ( K U K(K(F i 1, p θ1 1, K(K(F j θ, p K ( K S (K(F1, i K(F j θ1, p1 p θ K ( K(F1 i F j θ1, p1 p θ where in addition the joint weights can be choosen according to (U ; it also holds P US0 (A P U(US0, P US0 (A P U(US0 K UU0 K U(US0 K US0. But unfortunately we do not have K U K UU0 in general what we show in the following example. Example: The sets K 1 K of probability measures are given by K 1 K(F 1, m 1 K(K(F 1, m 1, δ ω K K(K(F, m, p θ where p θ is defined by pθ ({0} θ p θ ({1} 1 θ where Ω 1 Ω {0, 1}, Ω Ω 1 Ω F 1 {{0}, {1}}, m 1 ({0} m 1 ({1} 1, F {{ 1 }}, m ({ 1 } 1. In this very special example both marginal sets of probability measures have only one element, namely the discrete uniform distribution on {0, 1}: K 1 {P 1 1 }, K {P 1 }, P 1 P P 1 ({0} P 1 ({1} 1. But this uniform distribution is generated by two different ways:,

9 1. As a degenerated rom set where the two focal sets are singletons.. As a realization of the parameterized probability measure p θ with a parameterization by a rom set with only one focal set. The sets of probability measures associated with the marginal focal sets are given by K 1 1 {P 1 1 }, P 1 1 ({0} 1, K 1 {P 1 }, P 1 ({1} 1, K 1 {P 1 } {P }. Now we determine the joint focal sets weights: F {F 11, F 1 } with F 11 {(0, 1 }, F 1 {(1, 1 }, m(f 11 m(f 1 1. Since F 1 the joint weights are uniquely determined independent of (S or (U. The sets of probability measures associated with the joint focal sets: K 11 U K U(K 1 1, K 1 K U (P 1 1, P 1 {P 11 U } with PU ({(0, 0} PU ({(0, 1} 1 K 1 U K U(K 1, K 1 K U (P 1, P 1 {P 1 U } with PU 1 1 ({(1, 0} PU ({(1, 1} 1. Let A {(0, 0, (1, 1}. Then P UU0 (A m(f 11 P 11 U (A + m(f 1 P 1 U (A 1 P U 11({(0, 0} + 1 P U 1 ({(1, 1} But it is clear that P U (A sup{p U (A : P U K U (K 1, K } 1 for P U defined by P U ({(0, 0} P U ({(1, 1} The case (SS1, strong independence K i 1 : K(F i 1, K j : K(F j We write a probability measure P SS1 K SS1 in the following way: F 1 F P SS1 (A m 1 (F1m i (F j (P 1 i P j (A ( F1 j1 m 1 (F1 i P1 i (P 1 P (A P S (A ( F m (F j P j (A j1 with P 1 K(F 1, m 1 P K(F, m. This leads to K SS1 K S { P 1 P : P 1 K(F 1, m 1, P K(F, m } which is the case of strong independence. Computational method: Theorem 1. The upper probability P S (A is the solution of the following global optimization problem: subject to F 1 F m(f1 i F j χ A(ω1, i ω j max! j1 ω i 1 F i 1, i 1,..., F 1, ω j F j, j 1,..., F, where χ A is the indicator function of the set A. The lower probability P S (A is obtained by minimization. Proof: see [5, 8]. In general it is very hard to solve the above optimization problem because there may be many local maxima (or minima because the objective function is not continuous. Criteria when we have P S P R are given in [6]. In this case we automatically get P S by using the computationally cheaper P R K i 1 : K(K(F1, i p θ1 1, Kj : K(K(F j, pθ It holds P S (A P SS1 (A ( P 1 P (A ( F1 ( F m 1 (F1P i 1 i m (F j P j (A F 1 F j1 m 1 (F1m i (F j (P 1 i P j j1 F 1 F m 1 (F1m i (F j j1 ( 1 Ω 1 Ω (A χ Aω1 (ω p θ (dω p θ1 1 (dω 1 µ j (dθ µ i 1(dθ 1

10 F 1 F m 1 (F1m i (F j j [( ] p θ1 1 p θ (A µ j (dθ µ i 1(dθ 1 [( p θ1 1 p θ ] (A µ (dθ µ 1 (dθ 1 [( ] p θ1 1 p θ (A µ(d(θ 1, θ with µ K S (K(F 1, m 1, K(F, m, µ i 1 K(F i 1, µ j K(F j, µ 1 K(F 1, m 1 µ K(F, m. Computational method: We get the following optimization problem for the computation of P S (A P S (A, respectively: F 1 F j1 m 1 (F1m i (F j (p θi 1 1 pθj (A sup! subject to θ1 i F1 i θ j F j. Proof: see [6]. 6 Summary (inf! We summarize the results where parameterized probabilities are involved: Fig. 1 depicts the relations between the sets of joint probability measures. For the upper probabilities see Fig.. There are three differences to the results for pure rom sets in [5]. 1. K UU0 is only a subset of K U in general.. New cases induced by ( (US which coincide with ( U for pure rom sets because of the Dirac measures. 3. Generalization of the computational method for P S P S. K U K UU0 K SU0 K U(US0 K R : K S(US0 K US0 K SS0 K S K SS1 Figure 1: Relations between the sets of probability measures. P U P UU0 P U(US0 P US0 References [1] C. Baudrit D. Dubois. Comparing methods for joint objective subjective uncertainty propagation with an example in risk assessment. In Proceedings of the 4th ISIPTA Conference, Pittsburgh, 005. [] I. Couso, S. Moral, P. Walley. Examples of independence for imprecise probabilities. In Proceedings of the 1st ISIPTA Conference, pages , Ghent, [3] A. P. Dempster. Upper lower probabilities induced by a multivalued mapping. Ann. Math. Stat., 38:35 339, [4] A. P. Dempster. Upper lower probabilities generated by a rom closed interval. Ann. Math. Statistics, 39: , [5] Th. Fetz. Sets of joint probability measures generated by weighted marginal focal sets. In Proceedings of the nd ISIPTA Conference, pages , Ithaca (NY, 001. [6] Th. Fetz. Mengen von gemeinsamen Wahrscheinlichkeitsmaßen erzeugt von zufälligen Mengen. PhD thesis, Universität Innsbruck, 003. [7] Th. Fetz. Multi-parameter models: rules computational methods for combing uncertainty. In Oberguggenberger,Vieider, Fellin, Lessmann, editors, Analyzing Uncertainty in Civil Engineering, Berlin, 005. Springer. [8] Th. Fetz M. Oberguggenberger. Propagation of uncertainty through multivariate functions in the framework of sets of probability measures. Reliability Engineering System Safety, 85(1-3:73 87, 004. [9] E. Mira, I. Couso, P. Gil. Rom intervals as a model for imprecise information. Fuzzy Sets Systems, 154(3:386 41, 005. [10] E. Mira, I. Couso, P. Gil. Rom sets as imprecise rom variables. Journal of Mathematical Analysis Applications, 307:3 47, 005. [11] P. Walley. Statistical reasoning with imprecise probabilities. Chapman Hall, London, P SU0 P R : P S(US0 P SS0 P S P SS1 Figure : Relations between the upper probabilities.