P

q

(x

i

| rel)

r

(

R

)

 w

x

= log

= log

i

P

q

( x

i

)

n

(

N

)

F1 

P

r

(

R

)

w

q

(x

i

| rel)P

q

(rel)

x

= log

= log

i

P

q

(x

i

| rel)P

q

(rel)

(

n

(

- r

) (

N - R

)

)

F2 

P

r R

(

- r

)

w

q

(x

i

| rel) / P

q

(x

i

| rel))

x

= log

= log

i

P( x

i

) /(P(x

i

)

n

(

N - n

)

F3 

P

r R

(

- r

)

w

q

(x

i

| rel) / P

q

(x

i

| rel)

x

= log

= log

i

P

q

(x

i

| rel) / P

q

(x

i

| rel)

n

(

- r

) (

N - n - R + r

)

F4

Figure 5: Term weighting functions F

1 

  F

4 

In [RSJ76], Robertson and Sparck Jones used the four term weighting schemes to carry out two sets of 

experiments. The first set was based on retrospective weighting. This involves deriving optimal weights 

to retrieve the relevant documents already found   the known relevant set. The second group of 

experiments were based on predictive weighting. Predictive weighting uses the weights from the 

retrospective stage to retrieve new documents. If the known relevant set is a representative sample of all 

relevant documents, then predictive weighting should be better at retrieving unseen relevant documents 

than the original term weights. Naturally, it is the latter, predictive, case that is mainly of interest as RF 

is intended to retrieve relevant documents that the user has not yet seen. 

All functions outperformed no relevance weighting, and the idf function. F

1

 and F

2

, and F

3 

and F

4 

perform within the same range with F

3

 and F

4 

outperforming F

1 

and F

2, 

and F

4 

slightly outperforming 

F

3

. This confirms Robertson and Sparck Jones  intuition that ordering principles O2 is correct and that 

it is necessary to consider both presence and absence of query terms. No conclusive evidence was 

provided to distinguish between the two versions of the independence assumption, however Robertson 

and Sparck Jones favoured the second, I2, assumption as the more realistic assumption. 

Given that the preferred weighting scheme is F

4

, the odds function in Figure 6 (Equation 10

a

) can be 

converted to that of Equation 10

b

 by eliminating the division operators. By noting that  P

q

( x

i

| rel)  = 1 

   P

q

( x

i

| rel) , and  P

q

( x

i

| rel)  = 1    P

q

( x

i

| rel)  it is possible to convert the representation of F

4

 in 

Figure 6 to that in Equation 10

c

.  

P

P

P

w

q

(x

i

| rel) / P

q

(x

i

| rel)

q

( x

i

| rel)P

q

(x

i

| rel)

q

(x

i

| rel)(1- P

q

( x

i

| rel))

x

= log

= log

= log

i

P

q

(x

i

| rel) / P

q

(x

i

| rel)

P

q

( x

i

| rel)P

q

(x

i

| rel)

P

q

(x

i

| rel)(1- P

q

( x

i

| rel))

a 

b 

c 

Equation 10: 

Term weighting function based on term s distribution 

in relevant and non relevant documents 

where w

xi

= the weight of term  x

i

This equation (Equation 10

c

), which expresses the F

4

 function solely as a factor of the presence of a 

term in the relevant and non relevant documents, can alternatively be represented as in Equation 11. 

The probability of relevance of a document, then, is measured as the sum of the term weights of the 

query terms in the document, i.e. the sum of the F

4

 weights of each query term in the document. 

 11