Robertson / Sparck Jones Formalization

Robertson and Sparck Jones describe a probabilistic system as a system that determines the weights of terms in document and query vectors probabilisticly [24]. Using log likelihoods, they derive the following expression for the ranking of a document based on sum of probabilisticly-derived weights:
 
where is the probability that term is in a relevant document and is the probability that is in a non-relevant document. m is the number of terms in the collection. The question now comes to determining how to calculate and .

For a given term and a query Q, Robertson and Sparck Jones define:

N

The number of documents in the collection, i.e. |D|

R

The number of relevant documents, i.e.

The number of documents containing term , i.e.

The number of relevant documents containing , i.e.

and create the following contingency table used to determine the number of relevant and irrelevant documents when a given term is present or absent:

Using this table, and , the probabilities of a document being relevant when a term is or is not present, are defined by Robertson and Sparck Jones as

and by substituting back into Equation (4) they arrive at

also referred to as the f4 formula. In order to avoid undefined weights when is 0, a common case in operational systems, a factor of 0.5 is added to each term, resulting in the f4 point 5 formula:

N and are easily obtainable. However, since is unknown to the system, it is necessary to estimate R and for the initial query. There are a variety of estimation methods available [21, ] for initial queries, although the common technique is to use the f4 point 5 or similar [36] formula and set R and to 0 resulting in an initial uniform weight for all terms in a query.

Relevance Feedback is implemented in this system by successively updating R and after each iteration. The theory is that after enough iterations, the values for and will converge upon the true weights, and thus the ranking formula defined by Equation (4) will produce the full set of relevant documents .