Modular Belief Updates and Confusion about Measures of Certainty in Artificial Intelligence Research

Over the last decade, there has been growing Interest. In the use or measures or thangc In belief ror reasoning with uncertainty In artlrlclal Intelligence researt'h. An Important characteristic or several methodologies that reason wlt.b changes In belief or belief updates, Is a property that we term modularit11. We call updates that !atisfy thl! property modular update.. Whereas probabilistic measures or belld update • wblcb satisfy the modularity property were first discovered In tb"e nlneternth t'f'nt.ury, knowledge and discussion or t. hese quantities remains obsrure In artlflc.-lal lnlelll&enre research. \Ve define modular updates and dl�russ their Inappropriate use In two Influential expert systems. lntroduc:tloD • .Most work In reasoning about uncertainty bas c:eutered on the manipulation or measurrs or. d•olutc belief. That Is, most methods for managing uncertainty c:oncern themselves wltb quenlons of the form. Given current evidence, bow certain Is some b�·potbesls! Although more obscure, there has been lnvtstli�ntlon Into the U!e or me3.Surrs or change In belief. Here. qutstlons are or the form. Given a piece or evidence, bow bas the certainty or some hypothesis changed! The distinction between rea.sotllng with quantities representing changes In belief and reasoning wltb quantities representltlg absolute measures of belief 15 significant In the design and characterization of methods for handling uncertainty In reasoning systems. In this paper, we rorus on measures or change In belld �bat satisfy a special proper�y tha� we call modulorit11. We will define modularl�y and describe quanmles c�lled morlulor update•. Modular updates are central t.o t.be Intent or several methodologies for reasoning under uncertainty In expert systems. We will discuss �wo medical expert systems, MYCIN and INTERNIST-I and show that both sYStems bave Implicitly assumed modularity In reasoning with uncertainty. Finally, we argue that MYCIN and INTERNIST-I have used Inappropriate measures or belld update. DeltDltlon of a modular update In this senlon, we rormall:r.e tbe concept or a modular update. Before we examine tbe notion or modularity, le\ us define what we mean by au uprlGte. Informally, a belle! update represents the change of belief In some hypothesis clven a piece or evidence. Thb must be contrasted with the notion or absolute belief In a b)'pothesis given evidence. We shall ass ume In �hi! paper, t. bat belief Is a continuous, ordered quantity. That Is, It *Thil •ark wa• 1uppon•cl lo part b7 d1• l01iall M�. lr. FoUDda&loo aud tb• Htaf7 I. J,:a;,., Family Fonda&iao . .. Orcin of .. lbonbip b-el oo a colo nip. mak�s sen.!e to talk about a tlcfrce or bellet and to say that one belief Is strongtr than another. We Insist that beller updates are al!o continuous and ordered. In addition. the combination or two updates which correspond to two different pieces or evid••nre should Itself be an update. Before tbls basic upd3.te pr�perty Is stated formally, It should be realized thai updating a beller In a hypothtsls H ginn some evidence E may depend oo p:-ior e\"ldenre for the bypot hesis. Therefore. update! logically mu!t haw thrre ar'uments: a hypothesis H. new evidence E. and prior evldenrt e. \\'ltb thl!, we formally capture the notion or an updatr by ln!istlng that there Is some runctlou • such that (!) where t'(H.E1.e) denotes the update on H given E1 and prior evldtnre e. l'(H,E:.E1e) denotes the update on H given E: and prior evidence E1 and e, and U{H.E1E2.e) denotes the combined upd:�te 'Inn prior e\"ldeuce e. Now we lntroduC'e the formal concept or modularity. ·we say that an upd:�te Is modular. If the update Is Independent or prior eYident'e. Form:�lly, U(H,E,e) � U(H,E.t) &i U(H,E} (2) \'l.'e not.e that the notion or modularity Is not new. For �x:�mple. Informal notions of modularity bave been a central theme In the MYCIN expert system project1• As!umlng mod•Jiarity or belief updates may or may not be reasonable depending on the nature or dependencies within the domain or lntert!t. However. the property Is often assumed in expert system researC'h to eliminate the need to consider possible complex Interactions among e\'ldence. The modularity property thereby eases the task or knowledge acquisition and explanation. facilitates the construction and maintenance or knowledge bases and allows the use of relatively simple runetlons for belltr updating. The desirability or these features makes It tempting to use the modularity assumption even wben It Is Inappropriate. History and derlvatloD of a modular update The first modular update was proposed In 1878 by Pelrce2• Tbls quantity, which be called "weight of evidence," Is equal to the logarithm or a rr.tlo or conditional probabilities commonly called the likelihood ratio. Several other researchers Independently disC'overed this simple but powerful me�ure of change In belief. Tbey Include Turlng1, Good4 and Minsky and Seltrldges. To see that thlll quantity Is a modular upd:ll e, first consider Ba�·es· Theorem for evidence E and hypothesis H: p(E He)p(Hie) : ----------........ P (E i e) p(HIEe) Tbr rorrr!pondlng formula Cor tbe negation or the hypothesis. -H. Is P(WHe)p(-Hje) p(-H]Ee) : -----·-······-­ p(Eie) Di\'ldlng tbe two. Wf crt P(H[Ee) P(E!He} p(Hje) ·-----·= ---··--··----p{"HiEe) p(Ej·He) p(-Hje) The quantity P(EIHe)