Croft and Harper [CH79] based on the formula in Equation 8. This approach ranks documents by a 

function such as idf, assumes the top n documents are relevant, then uses these so called pseudo 

relevance assessments to estimate values for pi and qi in Equation 11. This will be discussed more fully 

in section 3.5. 

This fundamental approach to probabilistic modelling has been extended in many ways, in particular to 

incorporate within document frequency information [RW94]. Pertinent additions or modifications will 

be described, where appropriate, in later sections of this paper. An historical overview of the 

probabilistic model can be found in [SSJ+00a, SSJ+00b]. 

2.2.4 Logical model 

In [Mar64], Maron hinted a potentially useful difference between the Boolean logic exact match 

process and the process of logical implication. This difference distinguishes between the Boolean 

matching of text representations, in which the system is restricted to an exact formula, and the inference 

of information needs, by which process the system can infer more about what may be relevant than is 

stated in the query. 

The advantages of implication or inference as the basis for a retrieval algorithm are demonstrated in the 

logical modelling approach to retrieval. This class of models originates from a proposal by Van 

Rijsbergen [VR86] that relevance can be modelled as a process of uncertain inference. More precisely 

the relevance of a document representation can be measured by the probability that the information in a 

document infers the information in a query

10

, Equation 14. 

P d

(

  q

)

Equation 14: Relevance measured as uncertain inference 

This view was encapsulated in the logical uncertainty principle, [VR86]: 

"Given any two sentences x and y; a measure of the uncertainty of y 

x related to a given 

data set is determined by the minimal extent to which we have to add information to the 

data set, to establish the truth of y 

 x." 

That is if the information in a document, d, does not infer the information in a query q how much would 

d have to be changed to be relevant to q? The degree of necessary change to d allows the calculation of 

the probability of the inference.  

As a simple example, if the query is about animals and a document mentions dogs, ponies, cats, but 

does not explicitly mention animals, then the document would not be retrieved by standard term 

matching retrieval algorithms. By including information that dogs,  ponies, and cats are kinds of 

animals, then it can be asserted that the document may be relevant and should be retrieved. Such an 

approach was taken by Lalmas, [Lal96], who used ontological relationships to express how many 

transformations or substitutions of this type would be necessary before a document s content inferred a 

query. In Lalmas's model, the number of substitutions gave a measure of the uncertainty associated with 

the inference. 

The core logical models are based on non classical logics as the classical notion of inference has 

several undesirable properties for retrieval, e.g. in classical logic the inference, 

d   q

, would hold if d 

did not contain any information, and the majority of logical models of IR are based on a possible 

worlds semantics, in which each possible world represents a possible combination of events. One 

possible representation is one in which a possible world represents a possible combination of terms. For 

example, given a set of indexing terms {t1, t2, t3, ..., t10}, there would be 210 worlds: a world in which 

all terms are true, one in which all terms except t1is true, one in which all terms except t1and t2 are true, 

and so on. In this representation each document and the query is associated with a world. The similarity 

of a document to the query is given by the distance between the document world and the query world

11

. 

10

This is the most common version of the principle. Some authors have tried modelling the inverse; the degree to 

which the information in the query infers the information in the document  P

(

q   d

)

, or a combination of both 

measures, e.g. [Nie89] 

11

 This assumes the Closed World Assumption, i.e. any fact not known to be true is assumed false. 

 13