2.2.3 Probabilistic model 

In the probabilistic model, suggested by Maron and Kuhns [MK60], and developed by amongst others, 

Robertson and Sparck Jones [RSJ76], and Van Rijsbergen [VR79], documents and queries are also 

viewed as vectors but the vector space similarity measure is replaced by a probabilistic matching 

function. The probabilistic model is based on estimating the probability that a document will be 

relevant to a user, given a particular query. The higher this estimated probability, the more likely the 

document is to be relevant to the user

4

. This is instantiated in the probabilistic ranking principle, 

[Rob77]. 

 If a reference retrieval system s response to each request is a ranking of the 

documents in the collection in order of decreasing probability of relevance to the 

user who submitted the request, where the probabilities are estimated as 

accurately as possible on the basis of whatever data have been made available to 

the system for this purpose, the overall effectiveness of the system to its user will 

be the best that is obtainable on the basis of those data.  

The estimated probability of relevance can be expressed as 

P

q

(rel | x)

, the probability of relevance 

given a document x and a query q. This probability can be used to decide whether or not to retrieve a 

document: if 

P

q

(rel | x)

 = 0 then the probability of relevance given x is 0, and x should not be 

retrieved

5

.  

This can be refined by also considering the probability of non relevance given x and q, 

P

q

(rel | x)

. If 

P

q

(rel | x)

 > 

P

q

(rel | x)

 then it can be asserted that the probability of relevance is greater than the 

probability of non relevance and hence x should be retrieved

6

. Thresholds may also be used, i.e. the 

difference between the probability of relevance and the probability of non relevance must be greater 

than some threshold value before x is retrieved, ((

P

q

(rel | x)

P

q

(rel | x)

) > threshold). In this case 

threshold is a value set by the user or system, in order to further restrict the retrieval function.  

Having decided which documents to retrieve, the odds of relevance to non relevance, Equation 7, can 

be used as a document ranking function: the higher the ratio of the probability of relevance to non 

relevance, given x, then the more likely document x is to be relevant to a user. 

P

q

(rel | x)

P

q

(rel | x)

Equation 7: Odds of relevance to non relevance for document x and query q 

Bayes, [Bay63], theorem can be used to calculate 

P

q

(rel | x)

 and 

P

q

(rel | x)

. Equation 8 demonstrates 

this for the relevance case. 

P

P

q

(x | rel)P

q

(rel)

q

(rel | x) =

P(x)

Equation 8: Calculation of 

P

q

(rel | x)

 through Bayesian inversion 

where  

P

q

(rel)

 is the prior probability that any document in the collection is relevant to q 

P

q

( x | rel)

 is the probability of observing document x given relevance information 

P( x)

 is the probability of observing document x irrespective of relevance 

4

The probabilistic model measures the probability of relevance, i.e. the probability that a document will be 

relevant, not the degree of relevance as is sometimes suggested. A good discussion of the difference between these 

two notions is found in [RB78]. 

5

In an operational system 

P

q

(rel| x)

will generally only equal 0 if x does not contain any query terms. This rule 

then decides only to retrieve those documents that contain at least one query term. 

6

In the case where the two probabilities are equal, it is arbitrarily decided that x is non relevant [VR79]. 

 8