Part III. U se of Conditional and Relational Events in Data Fusion

Transcription

1 Part III U se of Conditional and Relational Events in Data Fusion

2 341 Scope of Work In Part III here, we consider a number of topics associated with conditional and relational event algebra, focusing on potential applications to data fusion issues. The work is divided into eight basic chapters as follows: Chapter 9 presents introductory material, with Section 9.1 providing the philosophy of approach: the presentation revolves around a number of motivating examples and conditional and relational event algebra are shown to address these issues in a more meaningful manner than previously developed approaches. Section 9.2 presents an overview of the problems involved and how an algebraic approach - in conjunction with appropriate numerical implementations - can provide more coherency and a stronger logical foundation instead of consider ing numerical techniques alone. Section 9.3 describes further details of the algebraic approach. Section considers partitioning of information. Section establishes the basic analytic framework used throughout Pad III - boolean and sigma algebras and probability spaces - and introduces the concepts of deduction and "enduction" ( backward deduction). Section introduces the idea of algebraic combining or averaging of information, prior to taking probability evaluations. Section also contains an informal introduction to the concept of a "constant-probability" event, a critical part of the development of relational event algebra (which properly begins in Chapter 15), and itself a particular type of "conditional event" (the main introduction being in Chapter 10). Section 9.4 presents the basic problem of determin ing similarity of events and introduces two measures of similarity (or probability distance functions), the absolute value of the difference of probability of events dp,l, which we call the "naive probability distance" function, and the probability of the boolean symmetric difference (or sum) of events dp,2, which we call the "natural absolute probability distance" function (due to its well-known role in the standard development of metrics spaces from probability spaces). This section also briefly discusses the actual role conditional and relational event algebra play in implementing such measures. Chapter 10 introduces in a more formal manner conditional event algebra. In Section 10.1, Example 1 is presented to illustrate how conditional event algebra can be used to address the modeling of implicational statements and rules of inference in Expert Systems. In addition, a fundamental inequality (eq.(10.7)) is presented showing the discrepancy between using the material conditional operation and conditional probability in evaluating conditional statements. Section 10.2 introduces the basic properties all conditional event algebras should satisfy. The Frechet-Hailperin bounds on the probability of logical combinations of events - app!ied to conditiona! events when in a boolean setting - are also discussed and applied to deriving inequalities for some measures of similarity of conditiona! events. Section 10.3 continues the discussion on constant-probability events begun in Section 9.3. Chapter 11 introduces briefly the three best-investigated conditiona! event a!gebras for interpreting conditiona! statements. Section 11.1 provides general remarks; Section 11.2 introduces the DeFinetti-Goodman-Nguyen-Walker

3 342 (DGNW) conditional event algebra (cea); Section 11.3 introduces the Adams Calabrese (AC) cea; Section 11.4 provides some comparisons between AC and DGNW cea's; Section 11.5 presents in some detail Lewis' negative result on forcing boolean conditional events to lie in the same boolean algebra as unconditional events (Theorem O); and Section 11.6 presents an introduction to the only well-analyzed boolean cea, Product Space (PS). Chapter 12 details further important development of PS cea. Section 12.1 is devoted to the determination of equivalence, ordering relations and the basic calculus oflogical operations (all detailed in Theorems 1-9). Section 12.2 treats additional properties of PS, including: comparison of two natural types of events that occur in PS (Section ), showing that though they are similar, they are non-isomorphic distinct entities; Lewis' Theorem and its relation to PS (Section ); higher order conditionals in PS (Section ); and other properties (Section ), including emphasis on the fact that PS obeys alllaws of probability (while DGNW and AC do not) as well as natural independence properties (which again the other two cea's do not have). Section 12.3 considers the compatibility of PS conditional events with conditioning of random variables (Theorem 10). Section 12.4 illustrates the fact that boolean cea's other than PS are possible to construct - but necessarily, in light of the characterization theorem (Theorem 11) of Section 12.5, lack certain fundamental properties PS enjoys. Chapter 13 provides details for PS as a means for both analyzing cea issues and serving as an umbrella where ali three leading cea's can be related. Section 13.1 shows direct ordering relations (Theorem 12), and a key result showing ali six notions of deduction and enduction, relative to the nontrivial simple conditiona! events of PS can be completely characterized - quite simply and entirely! - by the imbedded DGNW and AC operations. (See eqs.(13.28)-(13.31) and Theorems 13, and 14.) Section 13.2 returns to Example 1 introduced in Section 10.1 and shows specifically how PS can address it. Section 13.3 provides a full calculus of operations for constant probability events together with an important result on combining constant-probability events with ordinary events or other conditional events (Theorem 15). Chapter 14 considers particular details in the testing of hypotheses for the distinctness of events when they are indirectly given via numerical functions of probabilities. Section with Example 2 - simply illustrates classical statistical test ing of hypotheses for the same event relative to different probability measures, while Section with Example 3 - considers the main problem to be addressed here: iesting of hypotheses of possibly different events, which, as staied above, may noi be directly known a priori, relative to a common probabiliiy measure. Section 14.3 presents a basic higher order probability assumption (Q), undei' which ali hypotheses testing here is to be implemented and an outline of the actual test procedure is provided. Essentially, Q states that ali relative atomic probabilities involved in the formulation of the problem are themselves assumed to have a joint uniform distribution over their natural dom ain of values as ali possible probabilities and the events are varied. Also, in Section 14.3 the cumulative distribution functions associated with the two

4 343 probability distance functions dp,l and dp,2 are derived under assumption Q for use in testing of the hypotheses. Two additional measures of association or probability distance functions are introduced in Section 14.4: dp,3, the "relative probability distance" function and dp,4, the "negative form relative probability distance" function. Section 14.5 presents a brief review (Theorems 16 and 17) - with some new results apropos to our interest (Theorem 18) concerning classical (numerically-valued) metrics or distance measures and numerical subadditivity for generating additional metrics. Section 14.6 introduces algebraic metrics and algebraic subadditivity for generat ing additional algebraic metrics (Theorems 19 and 20). Section 14.7 connects probability with algebraic metrics by showing all probability evaluations of algebraic metrics are legitimate metrics and conversely, any binary operations on a boolean algebra which when evaluated by arbitrary probabilities yields numerical metrics is itself an algebraic metric (Theorem 21). It is also shown that three of the four measures of association previously introduced, namely: dp,2, dp,3, dp,4, are actual metrics arising from corresponding algebraic metrics (Theorem 22), justifying the previous use of the term "probability distance " functions. The fourth, dp,l, while obviously a legitimate metric, does not appear to arise as the probability evaluation of an algebraic metric. However, in a related direction, dp,1 is shown to arise as the probability evaluation of an event which in general is probability-dependent itself. See Section 16.4, eq.(16.58) and especially eqs.(16.80)-(16.91), for the construction of the associated relational events 0:(/17) and 0:(118)' AIso, Theorem 23 shows that the only possible unconditional and conditional boolean algebraic metrics - and associated metrics via probability evaluations - are the same three as above. Section 14.8 provides additional relations among the four probability distance functions previously introduced, derives the cumulative distribution functions (under assumption Q) associated with dp,3 and dp,4, and provides in Table 1 a summary of basic properties that the four candidate probability distance functions possess or lack and, as a result, shows in Table 2 a natural preference ordering among the four for implementation. The most desirable, among the four turns out to be dp,3 - not dp,2 - though the latter is much more well-known in the standard development of probability. In any case, the interactive metrics, i.e., the metrics requiring knowledge of the probability of logical conjunctions (01' disjunctions) of the relevant events in addition to the separate probabilities of the events, dp,2, dp,3, dp,4, are ali seen to be superior to the non-interactive metric dp,l' These inieresiing results, iogeiher wiih Theorems 19-23, show nonirivial use of PS cea in deriving meirics for probabiliiy spaces. Chapter 15 turns the focus on natural language and testing of hypotheses for similarities of linguistic narratives 01' descriptions. The basic issue is illustrated by Example 4 in Section 15.1, where it is seen how fuzzy logic can be used as a natural way to model the problem initially. Then, except for Section 15.3, the remainder of the Chapter shows how probability theory can be used to re-interpret the fuzzy logic model - including the case of exponentialform modifiers - via full homomorphic-like one-point coverage representations of appropriately constructed random subsets of the domains of the fuzzy sets involved in the modeling. In turn, the theory of hypotheses testing developed

5 344 previously in Chapter 14 can be applied to this situation - see Section To carry this out, the first tool for dealing with the problem, the dual concepts of "copula" and "cocopula" are briefly treated in Section Before proceeding further, Section 15.3 illustrates, alternatively, a few typical standard ad hoc numerical approaches to the problem, via ordinary real analysis distance functions and fuzzy logic. But, it is seen that such approaches do not go to the heart of the issue: first, determining the natural underlying events represented by the numerical functions of probabilities, and then finding their interactive probability distances - as opposed to the non-interactive numerically-based ones for hypotheses testing. This can lead to a significantly improved specification of values over a possibly wide range of values relative to the potential use of interactive distances. The latter are seen to have a sounder theoretical basis than the non-interactive ones when only the numerical approaches are used. (See again the comparisons in Tables 1 and 2 in Section 14.8.) More details on this are provided at the end of Section Section 15.4 provides a brief review of one-point coverage representations. Section 15.6 shows how the modeling of Example 4 can be further refined by taking into account in a nontrivial way fuzzy logic modifiers in the form of exponentials, and in turn, how to apply relational event algebra in combination with probability distance functions. Section 15.7 gives additional analysis ofthe problem begun in Section 15.6, where a homomorphic-like representation of the full fuzzy logic model of Example 4 can be obtained in relation to the one-point coverage random set representations. Chapter 16, is concerned how relational event algebra - other than solely conditional event algebra - can used to treat data fusion and hypotheses testing issues from an algebraic viewpoint before utilizing numerical implementations. Section 16.1 introduces and addresses, via Example 5, a problem involving the comparison of two weighted averages of probabilities of several possibly overlapping events by different experts. Section 16.2, likewise, considers Example 6, a problem of comparing or contrasting certain types of polynomial or power series (i.e., analytic functions) of a common probability value, while Section 16.3, beginning with Example 7, treats the comparison of quadratic functions in two common probability values. Section 16.4 considers first a full mathematical formulation of the general relational event algebra problem - as treated throughout ali of Part III up to this point - and proposes a weakening of the form of relational (or conditional, as a particular case) events whereby the constantprobability terms acting as coefficients are now allowed to depend on the particular probability measure used to evaluate them. (This is justified theoretically from the development of constant-probability events in general - see Section 13.3.) Such an extension of the concept of relational events allows for the representation of certain functions of probabilities, which up to this point, did not seem to be representable by relational events, including the naive probability distance function. (Again, see the last part of Section 16.4.). Finally, Section 16.5 presents concluding remarks on both conditional and relational event algebras and poses a number of general open questions which, hopefully, will be addressed by future research.