The Divergence from Randomness (DFR) paradigm is a generalisation
of one of the very first models of Information Retrieval, Harter's 2-Poisson
indexing-model [1]. The 2-Poisson model
is based on the hypothesis that the level of treatment of the informative
words is witnessed by an *elite set* of documents, in which these
words occur to a relatively greater extent than in the rest of the documents.

On the other hand, there are words, which do not possess
elite documents, and thus their frequency follows a random distribution,
which is the *single* Poisson model. Harter's model was first explored
as a retrieval-model by Robertson, Van Rijsbergen and Porter [4]. Successively it was combined with standard
probabilistic model by Robertson and Walker [3] and gave birth to the family of the BMs IR models (among
them there is the well-known BM25 which is at the basis the Okapi system).

DFR models are obtained by instantiating the three components of the framework: selecting a basic randomness model, applying the first normalisation and normalising the term frequencies.

The DFR models are based on this simple idea: "The more
the divergence of the within-document term-frequency from its frequency
within the collection, the more the information carried by the word *t*
in the document *d*". In other words the term-weight is inversely
related to the probability of term-frequency within the document *d*
obtained by a model *M* of randomness:

where the subscript *M* stands for the type of
model of randomness employed to compute the probability. In order to choose
the appropriate model *M* of randomness, we can use different urn
models. IR is thus seen as a probabilistic process, which uses random
drawings from urn models, or equivalently random placement of coloured
balls into urns. Instead of *urns* we have *documents*,
and instead of different *colours* we have different *terms*,
where each term occurs with some multiplicity in the urns as anyone of
a number of related words or phrases which are called *tokens*
of that term. There are many ways to choose *M*, each of these provides
a *basic DFR model*. The basic models are derived in the following
table.

Basic DFR Models | |

D | Divergence approximation of the binomial |

P | Approximation of the binomial |

B_{E} |
Bose-Einstein distribution |

G | Geometric approximation of the Bose-Einstein |

I(n) | Inverse Document Frequency model |

I(F) | Inverse Term Frequency model |

I(n_{e}) |
Inverse Expected Document Frequency model |

If the model *M* is the binomial distribution,
then the basic model is *P* and computes the value^{1}:

*TF*is the term-frequency of the term*t*in the Collection*tf*is the term-frequency of the term*t*in the document*d**N*is the number of documents in the Collection*p*is 1/*N*and*q*=1-*p*

Similarly, if the model *M* is the geometric distribution,
then the basic model is *G* and computes the value:

When a rare term does not occur in a document then it
has almost zero probability of being informative for the document. On
the contrary, if a rare term has many occurrences in a document then it
has a very high probability (almost the certainty) to be informative for
the topic described by the document. Similarly to Ponte and Croft's [2] language model, we include a risk component
in the DFR models. If the term-frequency in the document is high then
the risk for the term of not being informative is minimal. In such a case
Formula (1) gives a high value, but a *minimal
risk * has also the negative effect of providing a *small*
information gain. Therefore, instead of using the full weight provided
by the Formula (1), we *tune* or *smooth*
the weight of Formula (1) by considering only
the portion of it which is the amount of information gained with the term:

The more the term occurs in the elite set, the less term-frequency
is due to randomness, and thus the smaller the probability *P _{risk}*
is, that is:

We use two models for computing the information-gain with
a term within a document: the Laplace *L* model and the ratio of
two Bernoulli's processes *B*:

Before using Formula (4) the
document-length *dl* is normalised to a standard length *sl*.
Consequently, the term-frequencies *tf* are also recomputed with
respect to the standard document-length, that is:

A more flexible formula, referred to as *Normalisation2*, is given below:

*DFR Models are finally obtained
from the generating Formula (4), using a basic
DFR model (such as Formulae (2) or (3)) in combination with a model of information-gain
(such as Formula 6) and normalising the
term-frequency (such as in Formula (7) or Formula
(8)).*

Included with Terrier, are many of the DFR models, including:

Model | Description |

BB2 | Bernoulli-Einstein model with Bernoulli after-effect and normalisation 2. |

IFB2 | Inverse Term Frequency model with Bernoulli after-effect and normalisation 2. |

In_expB2 | Inverse Expected Document Frequency model with Bernoulli after-effect and normalisation 2. The logarithms are base 2. This model can be used for classic ad-hoc tasks. |

In_expC2 | Inverse Expected Document Frequency model with Bernoulli after-effect and normalisation 2. The logarithms are base e. This model can be used for classic ad-hoc tasks. |

InL2 | Inverse Document Frequency model with Laplace after-effect and normalisation 2. This model can be used for tasks that require early precision. |

PL2 | Poisson model with Laplace after-effect and normalisation 2. This model can be used for tasks that require early precision [7, 8] |

Recommended settings for various collection are provided in Example TREC Experiments.

Another provided weighting model is a derivation of the BM25 formula from the Divergence From Randomness framework. Finally, Terrier also provides a generic DFR weighting model, which allows any DFR model to be generated and evaluated.

The query expansion mechanism extracts the most informative terms from the top-returned documents as the expanded query terms. In this expansion process, terms in the top-returned documents are weighted using a particular DFR term weighting model. Currently, Terrier deploys the Bo1 (Bose-Einstein 1), Bo2 (Bose-Einstein 2) and KL (Kullback-Leibler) term weighting models. The DFR term weighting models follow a parameter-free approach in default.

An alternative approach is Rocchio's query expansion mechanism.
A user can switch to the latter approach by setting `parameter.free.expansion`
to `false` in the `terrier.properties` file. The default value
of the parameter beta of Rocchio's approach is `0.4`. To change this
parameter, the user needs to specify the property rocchio_beta in the `terrier.properties`
file.

DFR can encapsulate the importance of term occurrences occurring in different fields in a variety of different ways:

- Per-field normalisation: The frequencies from the different fields in the documents are normalised with respect to the statistics of lengths typical for that field. This is as performed by the PL2F weighting model. Other per-field normalisation models can be generated using the generic PerFieldNormWeightingModel model.
- Multinomial: The frequencies from the different fields are modelled in their divergence from the randomness expected by the term's occurrences in that field. The ML2 and MDL2 models implement this weighting.

Proximity can be handled within DFR, by considering the number of occurrences of a pair of query terms within a window of pre-defined size. In particular, the DFRDependenceScoreModifier DSM implements the pBiL and pBiL2 models, which measure the randomness compared to the document's length, rather than the statistics of the pair in the corpus.

A different interpretation of the gain-risk generating
Formula (4) can be explained by the notion of
cross-entropy. Shannon's mathematical theory of communication in the 1940s [5] established that the minimal average code
word length is about the value of the entropy of the probabilities of
the source words. This result is known under the name of the *Noiseless
Coding Theorem*. The term *noiseless* refers at the assumption
of the theorem that there is no possibility of errors in transmitting
words. Nevertheless, it may happen that different sources about the same
information are available. In general each source produces a different
coding. In such cases, we can make a comparison of the two sources of
evidence using the cross-entropy. The cross entropy is minimised when
the two pairs of observations return the same probability density function,
and in such a case cross-entropy coincides with the Shannon's entropy.

We possess two tests of randomness: the first test is
*P _{risk}* and is relative to the term distribution within
its elite set, while the second

Relation (9) is indeed Relation (4) of the DFR framework. DFR models can be equivalently defined as the divergence of two probabilities measuring the amount of randomness of two different sources of evidence.

For more details about the Divergence from Randomness
framework, you may refer to the PhD thesis of Gianni Amati, or to Amati
and Van Rijsbergen's paper *Probabilistic models of information retrieval
based on measuring divergence from randomness*, TOIS 20(4):357-389,
2002.

[2] J. Ponte and B. Croft. A Language Modeling Approach in Information Retrieval. In The 21st ACM SIGIR Conference on Research and Development in Information Retrieval (Melbourne, Australia, 1998), B. Croft, A.Moffat, and C.J. van Rijsbergen, Eds., pp.275-281.

[3] S.E. Robertson and S. Walker. Some simple approximations to the 2-Poisson Model for Probabilistic Weighted Retrieval. In Proceedings of the Seventeenth Annual International ACM-SIGIR Conference on Research and Development in Information Retrieval (Dublin, Ireland, June 1994), Springer-Verlag, pp. 232-241.

[4] S.E. Robertson, C.J. van Risjbergen and M. Porter. Probabilistic models of indexing and searching. In Information retrieval Research, S.E. Robertson, C.J. van Risjbergen and P. Williams, Eds. Butterworths, 1981, ch. 4, pp. 35-56.

[5] C. Shannon and W. Weaver. The Mathematical Theory of Communication. University of Illinois Press, Urbana, Illinois, 1949.

[6] B. He and I. Ounis. A study of parameter tuning for term frequency normalization, in Proceedings of the twelfth international conference on Information and knowledge management, New Orleans, LA, USA, 2003.

[7] B. He and I. Ounis. Term Frequency Normalisation Tuning for BM25 and DFR Model, in Proceedings of the 27th European Conference on Information Retrieval (ECIR'05), 2005.

[8] V. Plachouras and I. Ounis. Usefulness of Hyperlink Structure for Web Information Retrieval. In Proceedings of ACM SIGIR 2004.

[9] V. Plachouras, B. He and I. Ounis. University of Glasgow in TREC 2004: experiments in Web, Robust and Terabyte tracks with Terrier. In Proceedings of the 13th Text REtrieval Conference (TREC 2004), 2004.

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