Reasoning with Consistent Probabilities

ABSTRACT

The objective of the present paper is twofold: first, to establish explicit conditions for the elicitation of consistent a priori and conditional probabilities for a set of evenb representing pieces of evidence and hJ1pothue6. Furthermore, an algorithm is proposed which uses these consistent input probabilities to compute lower and upper bound1 for higher order joint probabilities. Secondly, problems concerning the aggregation and propagation of probabilistic estimates are collSidered. It is shown how these could be solved by using the higher order joint probabilities obtained for the elements of the so-<:alled complete 1et of mutuall11 eulU~ive atomic event1.

I INTRODUCTION

A representation of uncertainty in terms of classical Bayesian probabilities requires three basic inputs, namely: (i) a set of n evenb {E. }, each event designating either an evidence, if it is in 1 the premise of an IF - THEN rule, or a hJ1potht~i6, if it appears as a conclusion in such a rule.; (ii) a priori probabilities P(Ei ), 1~i~n for the event1 under consideration; (iii) conditional probabilities P(E. fE. ) and P(E. fE. ). 1 J 1 J

A number of problems are associated with this type of representation of uncertainty. First, the above input probabilities elicited from the expert may not be consistent with respect to the following two conditions (1) and (2) respectively: ·

Second, (1) and (2) are not fulfilled for arbitrary conditional probabiht1es; further requirments for consistency are needed reflectin~ the relationship between the P(Ei )'s and the P(Ei fE. )'s. Third, if the input probabilities are not consistent, Jthen the problem of propagation of uncertainty can not be modeled by the following classical rule:

So, what are the consistency conditions under which we can apply (3) ? Finally, consider questions like: How to determine a consistent P(Ek fE. E. ), 1 J

Dimiter Driankov

Department of Information and Computer Science University of Linkoping S-581 83 Linkoping, Sweden

representing a rule as "IF E. and E. THEN Ek ; How to determine a consistent hi~her ord~r joint probability such as P(E. E. Ek ) etc. These are usually answered under the viry Jmuch restrictive assumption that all E. 's represent independent evidence. Also it is ofteri forgotten that although the input probabilities might have been elicited in a consistent way, the resulting higher order joint probabilities might not be consistent at all.

ll Obtaining consistent input probabilities

Once a set of n event1 {E. } is specified together with thea priori probabilities{P(Ei )} , one is asked to assess the conditional probabilities P(E. fE. ) and P(E. - 1 J 1 /Ej ), thus obtaining the following two matrices: (i) the dependent occurence matrix DO = [ aij ], 1~i~n, 1~<n, a .. = 1, with a .. = P(E. fE. ); -11 lJ Jl

(ii) the dependent non-occurence matrix DNO = [b .. ], - lJ l~i~n, 1~~n, bii = 1, with bij = P(Ej /Ei ).

Now if E. and E. are two independent evidence! then, a .. = i>{E. jt. ) = P(E. ); if they are two lJ J 1 J mutuall11 exclU~ive hJ1pothue6, then a .. = P(E. /Ei ) = 0. Thus, knowledge about indepenJ!nt evitence and mutuall11 eulU~ive hJ1pothue6 is explicitly encoded in the DO-matrix. The problem is that two subjective assessments, say, P(Ej /Ei ) and P(Ei /E. ) might be incons~tent with respect to (2) while P(ti /Ej ) and P(Ei /EJ ) might be inconsistent with respect to (1). An add1tional trouble, concerning implementation, is reflected in the huge number of subjective assessments: in the case of the DO-matrix their number is n.(n-1); n.(n-1) in the case of the DNO-matrix, plus of course the n P(Ei )'s. However, what seems to have been completely ignored is that all P(Ei /Ej )'s for i >j in the DO-matrix, can be computed from the P(E. fE. )'s for i<j by rewriting (2) as: 1 J

Unfortunately, applying (4) produces consistent P (Ei /Ej ) 's only if introducing .some further restrictions on P(E. fE. ) i.e. P(E. fE. ) E [LB .. , UB .. J 1 J 1 Jl Jl J where

The question now is: If P{Ej /Ei ) E [LBji , UBji ] will P{Ei /Ej ) be also within the range of its admissible values ? The answer is, yes, iff P{Ej /Ei ) E [ LBji, UBji] and P{Ei /Ej ) is obtained by {4). Thus, given the P(E. )'s and the P(E. fE. )'s for i<j, the latter within the range of their 1 admfssible values, all remaining P{Ei /Ej ) 's for i>j can be obtained automatically from (4) and vice versa. Also, the number of the subjectively assessed conditional probabilities is reduced to n.(n-1) /2 . As to the DNO-matrix, rewriting (1) as,

we can compute consistent entries using the P(E. )'s and the consistent entries of the DO-matrix, fhus further n.(n-1) subjective assessments being dropped aside.

To conclude we will finally show that the consistent values obtained for P{Ei ), P{Ej ), P(Ei /Ej ) and P(Ei /Ej ) satisfy the condition under which {3) is fullfilled.

Proposition · Let P{Ei /Ej ) defines the point {1, P{Ei /Ej )) in the positive quadrant of a coordinate system, while P(E. ) and P(E. ) defme the point {P(E. 1 - J _J ), P(E. )) and P(E. fE. ) defines a point {0, P(E. fE. 1 1 J 1 J )). Then, if P{Ei /Ej ) is obtained by (5) the point (0, P{Ei /Ej )) belongs to the straight line that passes through the points {P(E. ), P(E. )) and {1, P(E. fE. )). J 1 1 J

Ill Constructing joint probabilities

The entries of the DO and the DNO matrices can not be used for representing rules of the form, IF E. and E. THEN Ek . Thus, there is a need fo~ determining a consistent higher order conditional probability such as P(Ek fE. E. ) which is not directly assessed by the expert, but c~ be computed as,

Here, we have to determine a consistent P{Ei Ej Ek ) , given the already consistent a priori and conditional probabilities related to Ei ,Ej and Ek . This in turn, can be used to obtain bounds for P{Ek /Ei Ej ) by use of (6).

In (E. Kounias 1968) it has been proved that the best upper (UB) and lower (LB) bounds for a union of events i.e. P{E 1 U E 2 U ... U En ) are given as:

However, we are interested in bounds on higher order joint probabilities while the above. result is valid for bounds on union of events. However, this is not a serious obstacle since one has that:

Hence, we wish to obtain bounds on P{D 1 D 2 ... D ) where, each D. can be either E. or E .. To be able n 1 1 J _ to achieve this, we require bounds on P(D 1 U D 2 U ... U Dn ) using as input the following already consistent probabilities: {i) the P(E. )'s; (ii) the P(E. fE. )'s, and 1 1 J (iii) the P(Ei /Ej )'s. Then the algorithm proposed is as follows:

Step 1. Compute a matrix C as follows:

Step 3 . The upper bound UB is to be obtained as:

1. Compute a set of lower bounds SLB as follows:

1. Compute the best lower bound LB as the maximum ot all SLB's

Finally, once we have obtained both LB and its corresponding UB we can use (11) and thus obtain:

It is to be stressed here that the bounds on the higher order joint probabilities might be consistent for some of them and inconsistent for others. To resolve this problem will require to determine explicit conditions for the input probabilities so, that the bounds on all higher order joint probabilities be consistent. This in turn will produce consistent higher order conditional probabilities. The consistency condition for each particular higher order joint probability is that, LB$ UB i.e. an inequality which two parts are linear functions of certain particular a priori probabilities and certain entries, c.. of C. IJ, Furthermore, each cij can be represented as a linear function of particular input a priori and conditional probabilities. Thus, if we want all higher order joint probabilities to have consistent bounds, we have to find such values for certain of the input probabilities, which are solutions to a system of simultaneous inequalities each inequality representing the consistency condition for the bounds of each particular higher order joint probability.

IV Reasoning with consistent probabilities

The possibility to obtain consistent bounds for an arbitrary set of higher order joint probabilities· helps in determining the probability of conjunctions of any number of E. 's; the probability of disjunctions of any number of E. 's, and also higher order conditional probabilities lor IF - THEN rules which premises and/or conclusions are conjunctions of an arbitrary number of Ei 's.

However, when determining the probability of an arbitrary compound logical proposition or a conditional probability consisting of such propositions, an arbitrary set of higher order joint probabilities is simply not enough. In this special case one needs higher order joint probabilities for the following set of N = 2n mutual/11 exclusive events:

Here, 1 expresses that an went occurs and 0 that it does not occur. The rows represent a complete set of mutually exclusive atomic events where P. is the I probability of the i-th of those events and Ei=t'N P. = 1. Now applying the technique proposed in he previous section we can obtain consistent bounds for each atomic event Pi that confine its actual value. Obtaining consistent probabilities for 2n complete and mutually exclusive events, guarantees that we can always find a concrete single value for each of the Pi 's, so that they sum up to 1 though, one can as well use the intervals confining it. Then, having assigned such a single value (or an interval) to each atomic event we can, as shown in (Konolige 1982), express any logical formula as a disjunction of some subset of (13). Furthermore, the probability of the proposition of interest can be determined by simply summing the · probabilities of the corresponding mutually exclusive atomic events, members of the disjunction. One advantage that comes from this representation in terms of higher order joint probabilities is that loops in the inference net does not matter - the algorithm through which these joint probabilities are obtained does not make any use of the concept of directionality.

V Conclusion

The paper presents a new method for computing consistent probabilities for arbitrary logical propositions the main advantages being that: (i) it allows the expert to assess in a consistent way a ·minimal amount of input-data in terms of a priori and conditional probabilities; (ii) the amount of computational effort for determining higher order joint probabilities is much less when compared· to methods based on the minimum-information assumption (Konolige 1982) and (Cheeseman 1983), since in our case onlr systems of linear inequalities are considered, and (iii) inconsistencies in the intermediate and/or final results, are traced back to the input data and, thus can be resolved by introducing direct changes in some of the input probabilities.