After Bayesian inversion and deletion of  P( x)  (which is identical for both the relevance and non 

relevance case), the odds function from Equation 7 turns into Equation 9

a

. 

The probability of relevance,  P

q

(rel) , and the probability of non relevance,  P

q

(rel) , are identical for 

all x's, That is when we use the odds in Equation 7 to rank documents, the ranking is dependent on the 

values of the probabilities  P

q

( x | rel)  and  P

q

( x | rel) , not on the values  P

q

(rel)  and  P

q

(rel) . We 

can therefore eliminate these elements and arrive at the odds in Equation 9

b

. This is then the odds of 

observing x given relevance or non relevance. 

P

q

( x | rel)P

q

(rel)

P

q

( x | rel)

P

q

( x | rel)P

q

(rel)

P

q

( x | rel)

a 

      b 

Equation 9: 

Odds of relevance, or non relevance, having observed document x 

The odds in Equation 9 refer to the probability of relevance, and non relevance, after viewing the actual 

document text rather than the vector representation of the document. That is, it measures the odds of 

relevance to non relevance based on the content of the document and is independent of the document 

representation. This means that the model can be used for many different types of document indexing 

but it also means that Equation 9 must be ultimately be expressed as a retrieval function based on the 

specific document indexing technique used to represent the documents.  

There are many probabilistic models based on the model outlined so far in this section. In the remainder 

of this section we shall describe the transformation from Equation 9 to a function based on the term 

based representation outlined in section 2.1. Specifically the discussion will be based on the 

probabilistic model known as the Binary Independence Model, as this is the most traditional variant of 

the overall probabilistic approach. This model was one of the first probabilistic models of IR, and will 

be used as an example of how the theoretical model is transformed into an actual retrieval model. 

Before converting Equation 9 into an equation that can be estimated based on the probability of 

relevance and non relevance of the terms in document x, it is necessary to consider how the 

probabilities of relevance and non relevance interact. In particular, two aspects of retrieval are 

important: the independence of terms and what information is used to order documents. 

The probabilistic model assumes that terms are distributed independently of other terms, that is the 

probability of seeing term t in a document is not affected by seeing term s in the same document. This is 

a simplifying assumption that reduces the computational complexity of the model. However it is 

necessary to define over what sets the independence holds. Two versions of the independence 

assumption were proposed in [RSJ76]. Both term independence assumptions assume that terms, query 

terms in particular, are distributed independently in the set of relevant documents: the probability of a 

term appearing in the relevant documents is not dependent on the probabilities of other terms appearing 

in the relevant documents. The two assumptions differ in whether the relevant document set should be 

distinguished from the whole document collection or only from the set of non relevant documents. 

 Independence assumption I1: The distribution of terms in relevant documents is independent and their 

distribution in all documents is independent  

 Independence assumption I2: The distribution of terms in relevant documents is independent and their 

distribution in irrelevant

7

 documents is independent  

These two versions of the independence assumption are important in distinguishing whether we should 

measure the difference in the probability of a term's occurrence against the non relevant documents (I2) 

or against its probability of occurrence the collection as a whole (I1). 

The probabilistic model ranks documents according to their probability of being relevant to a query   

the  ordering principle. Two versions of this principle distinguish between the case where this 

7

 The labels irrelevant and non relevant are treated as synonymous in this paper. 

 9