S. E. R obertson, M . E. M aronand W . S. C ooper7is the probability that the /nth document dm will be judged relevant,independent o f the query term that the patron submits. (Sometimes it is called thea priori probability of relevance.)P (lf |/4 , g?£) is the probability that if dm were to be judged relevant by a patron,then he would be using Ij as his query term.
Given the values of P(d*n\A) and P(lf\A,cC), the system can merely take theirproduct and use that value (after it has been normalized) to rank the outputdocuments. Where would the values of those probabilities come from?
Feedback from users could provide statistics needed to estimate the values ofP(d^,\A). Similarly, user feedback could provide statistics needed to estimate thevalues o f P (lf \A,d^). However, in lieu of actual past statistics, an indexer couldattempt to estimate the values of P(<fm|A ) and P (lf \A,cf„). In fact, according to thismodel, called by Maron and Kuhns ‘Probabilistic Indexing’, the task o f the indexeris precisely defined; his job is to estimate the values of P (lf \A,cfm), for all Ij and alldm and then to assign those index terms Ij to the corresponding documents with thevalues o f those estimates. Thus we end up with a theory o f weighted indexing, wherethe weights are estimates of the probability P (lf \A,d%).

3.2 Model 2: the formal presentationWe now consider the second probability model, called here Model 2. The origins ofthis probabilistic model rest on the work o f Robertson and Sparck Jones (1976). Itsmost complete formulation was presented by van Rijsbergen (1979). In thedescription that now follows we have kept the key ideas o f their model; however,our formulation is somewhat different. What is important for our purpose here isnot the computational details o f how to calculate probability o f relevance, butrather the way probability o f relevance is interpreted.
Let U be the class o f events each o f which consists o f a document being judgedrelevant by the same patron, relative to his information ‘need’. Then not- U,designated ‘U ’, is the complement of U and consists of the class of events each ofwhich consists of a document being judged not relevant by that same patron.
Every document has one or more of s possible properties, which we designate IhI2,1 3 ..........Is- These properties are often the properties of being assigned some termby an indexer. The problem of the document retrieval system, under this model, is tocompute the probability P(U\Ip , , , It) that a randomly selected document will bejudged relevant, given that it possesses the properties Ip , , , /„ where *,’ denoteslogical conjunction.
For simplicity and clarity of presentation consider a document with only twoproperties, Ij and Ik. In this case the problem for the document retrieval system is tocompute the value ofP(U\Ij,Ik)(2)i.e., the probability that a randomly selected document which has the property Ijand also the property Ik will be judged relevant by the inquiring patron.
Probability of relevance in Model 2 may be computed as follows:P(U\IjJk)

P(U)-P(Ij,Ik\U)
PUjJk)(3)

8Probability o f relevanceIf one makes the simplifying assumption that the properties Ij and Ik areindependent of one another relative to U, then we may rewrite (3) as follows:P(U\Ij,h)P(U)-P{Ij\U)-P(Ik \U)PdjJk)(4)(Expansions other than (4), based on other independence assumptions, are possible.)Let us look at what the probabilities on the right hand side o f (4) mean.
We have already said that P(U) is simply- the probability that a randomly selecteddocument will be judged relevant by this inquiring patron. We can imagine that thepatron himself might make some estimate of P(U) and input that estimate to thesystem. (As a matter o f fact, the use of the model for ranking documents does notrequire a value to be assigned to P(U).) P{Ij\U) is the probability that if this patronwere to judge a randomly selected document as being relevant, then that documentwould have the property Ij, and similarly for Ik. In order for the system to computeprobability of relevance, it would need some estimate o f these probabilities for eachterm in the query. We can distinguish at least three different ways of obtainingestimates of P(Ij\U). One way would be to ask the patron himself to estimate thevalue o f P(Ij |(7) or some related quantity. A second would be to establish, from theresults of previous searches, term properties that could be used (by human ormachine) as predictors of P(Ij\U): an example is term frequency. A third way wouldbe for the system to generate some ‘trial retrieval’ and then ask the inquiring patronto examine those output documents and then divide them into two classes — thosethat he judges relevant and those that he judges as not relevant. Given these data,fed back from the patron, the system would compute the values o f P(Ij\U), P{Ik |L0,etc., assuming o f course that the sample generated by trial retrieval is adequate insize, representation, etc.
What about the third probability — the one that appears as the denominator of(4)? P(Ij,Ik) is simply the relative frequency among all the documents of those thatpossess both properties Jj and Ik. The values for this probability are ‘known’ by thesystem, i.e., it can examine all of the records and compute the value o f P(Ij,Ik).
Thus we see that given the values of the above probabilities, the system couldcompute the probability that a randomly selected document which has properties Ijand Ik would be relevant. And, o f course, it could compute the probabilities for theother three cases, namely where a document has either one or else the other o f thoseproperties, or neither.
If values o f P { f \U) are available for all possible Ij, and again given a sufficientlygeneral independence assumption, then it is possible to compute the probability ofrelevance in the case where a randomly selected document has any number ofproperties.
3.3 A unifying notationA unified model must deal with groups of documents and groups o f uses. (The word‘use’ is used here as shorthand for the event consisting of a person with a need forinformation using the document retrieval system.) Thus the event-space (needed fora unified model) must consist o f all possible document-use pairs. The relevancerelationship will hold between some subset o f the above. Thus the notation that weneed to describe this model requires the following symbols: