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# Theoretical Background

This document aims to provide some theoretical background to Xapian.

## Documents and terms

In Information Retrieval (IR), the items we are trying to retrieve are called documents, and each document is described by a collection of terms. These two words, document and term, are now traditional in the vocabulary of IR, and reflect its Library Science origins. Usually a document is thought of as a piece of text, most likely in a machine readable form, and a term as a word or phrase which helps to describe the document, and which may indeed occur one or more times in the document. So a document might be about dental care, and could be described by corresponding terms tooth, teeth, toothbrush, decay, cavity, plaque, diet and so on.

More generally a document can be anything we want to retrieve, and a term any feature that helps describe the documents. So the documents could be a collection of fossils held in a big museum collection, and the terms could be morphological characteristics of the fossils. Or the documents could be tunes, and the terms could then be phrases of notes that occur in the tunes.

If, in an IR system, a document, D, is described by a term, t, t is said to index D, and we can write,

t → D

In fact an IR system consists of a set of documents, D1, D2, D3 ..., a set of terms t1, t2, t3 ..., and set of relationships,

ti → Dj

i.e. instances of terms indexing documents. A single instance of a particular term indexing a particular document is called a posting.

For a document, D, there is a list of terms which index it. This is called the term list of D.

For a term, t, there is a list of documents which it indexes. This is called the posting list of t. (Document list would be more consistent, but sounds a little too vague for this very important concept.)

At a simple level a computerised IR system puts the terms in an index file. A term can be efficiently looked up and its posting list found. In the posting list, each document is represented by a short identifier. To over-simplify a little, a posting list can be thought of as a list of numbers (document ids), and term list as a list of strings (the terms). Some systems represent each term by a number internally, so the term list is then also a list of numbers. Xapian doesn't - it uses the terms themselves, and uses prefix compression to store them compactly.

The terms needn't be (and often aren't) just the words from the document. Usually they are converted to lower case, and often a stemming algorithm is applied, so a single term connect might derive from a number of words, connect, connects, connection, connected and so on. A single word might also give rise to more than one term, for example you might index both stemmed and unstemmed forms of some or all terms. Or a stemming algorithm could conceivably produce more than one stem in some cases (this isn't the case for any of the stemming algorithms Xapian currently supports, but consider the Double Metaphone phonetic algorithm which can produce two codes from a single input).

## Xapian's context within IR

In the beginning IR was dominated by Boolean retrieval, described in the next section. This could be called the antediluvian period, or generation zero. The first generation of IR research dates from the early sixties, and was dominated by model building, experimentation, and heuristics. The big names were Gerard Salton and Karen Sparck Jones. The second period, which began in the mid-seventies, saw a big shift towards mathematics, and a rise of the IR model based upon probability theory -probabilistic IR. The big name here was, and continues to be, Stephen Robertson. More recently Keith van Rijsbergen has led a group that has developed underlying logical models of IR, but interesting as this new work is, it has not as yet led to results that offer improvements for the IR system builder.

Xapian was built as a system for efficiently implementing the probabilistic IR model (though this doesn't mean it is limited to only implementing this model - other models can be implemented providing they can be expressed in a suitable way). Xapian tries to implement the probabilistic model faithfully, though in some places it can be told to use short-cuts for efficiency.

The model has two striking advantages:

1. It leads to systems that give good retrieval performance. As the model has developed over the last 25 years, this has proved so consistently true that one is led to suspect that the probability theory model is, in some sense, the "correct" model for IR. The IR process would appear to function as the model suggests.
2. As new problems come up in IR, the probabilistic model can usually suggest a solution. This makes it a very practical mental tool for cutting through the jungle of possibilities when designing IR software.

In simple cases the model reduces to simple formulae in general use, so don't be alarmed by the apparent complexity of the equations below. We need them for a full understanding of the general case.

## Boolean retrieval

A Boolean construct of terms retrieves a corresponding set of documents. So, if:

t1 indexes documents 1 2 3 5 8
t1 indexes documents 2 3 6

then

t1 AND t2 retrieves 2 3
t1 OR t2 retrieves 1 2 3 5 6 8
t1 AND_NOT t2 retrieves 1 5 8
t2 AND_NOT t1 retrieves 6

The posting list of a term is a set of documents. IR becomes a matter of constructing other sets by doing unions, intersections and differences on posting lists.

For example, in an IR system of works of literature, a Boolean query :

(lang:en OR lang:fr OR lang:de) AND (type:novel OR type:play) AND century:19

might be used to retrieve all English, French or German novels or plays of the 19th century.

Boolean retrieval is often useful, but is rather inadequate on its own as a general IR tool. Results aren't ordered by any measure of how "good" they might be, and users require training to make effective use of such a system. Despite this, purely boolean IR systems continue to survive.

By default, Xapian uses probabilistic ranking to order retrieved documents while allowing Boolean expressions of arbitrary complexity (some boolean IR systems are restricted to queries in normal form) to limit those documents retrieved, which provides the benefits of both approaches. Pure Boolean retrieval is also supported (select the BoolWeight weighting scheme using enquire.set_weighting_scheme(Xapian::BoolWeight());).

## Relevance and the idea of a query

Relevance is a central concept to the probabilistic model. Whole academic papers have been devoted to discussing the nature of relevance but essentially a document is relevant if it was what the user really wanted! Retrieval is rarely perfect, so among documents retrieved there will be non-relevant ones; among those not retrieved, relevant ones.

Relevance is modelled as a black or white attribute. There are no degrees of relevance, a document either is, or is not, relevant. In the probabilistic model there is however a probability of relevance, and documents of low probability of relevance in the model generally correspond to documents that, in practice, one would describe as having low relevance.

What the user actually wants has to be expressed in some form, and the expression of the user's need is the query. In the probabilistic model the query is, usually, a list of terms, but that is the end process of a chain of events. The user has a need; this is expressed in ordinary language; this is then turned into a written form that the user judges will yield good results in an IR system, and the IR system then turns this form into a set, Q, of terms for processing the query. Relevance must be judged against the user's original need, not against a later interpretation of what Q, the set of terms, ought to mean.

Below, a query is taken to be just a set of terms, but it is important to realise that this is a simplification. Each link in the chain that takes us from the information need ("what the user is looking for") to the abstraction in Q is liable to error, and these errors compound to affect IR performance. In fact the performance of IR systems as a whole is much worse than most people generally imagine.

## Evaluating IR performance

It is possible to set up a test to evaluate an IR system. Suppose Q is a query, and out of the complete collection of documents in the IR system, a set of documents R of size R are relevant to the query. So if a document is in R it is relevant, and if not in R it is non-relevant. Suppose the IR system is able to give us back K documents, among which r are relevant. Precision and recall are defined as being,

$$\text{precision = }\frac{r}{K}\text{ , recall = }\frac{r}{R}$$

Precision is the density of relevant documents among those retrieved. Recall is the proportion of relevant documents retrieved. In most IR systems K is a parameter that can be varied, and what you find is that when K is low you get high precision at the expense of low recall, and when K is high you get high recall at the expense of low precision.

The ideal value of K will depend on the use of the system. For example, if a user wants the answer to a simple question and the system contains many documents which would answer it, a low value of K will be best to give a small number of relevant results. But in a system indexing legal cases, users will often wish to make sure no potentially relevant case is missed even if that requires they check more non-relevant cases, so a high value of K will be best.

Retrieval effectiveness is often shown as a graph of precision against recall average over a number of queries, and plotted for different values of K. Such curves typically have a shape similar to a hyperbola (y=1/x).

A collection like this, consisting of a set of documents, a set of queries, and for each query, a complete set of relevance assessments, is called a test collection. With a test collection you can test out different IR ideas, and see how well one performs against another. The controversial part of establishing any test collection is the procedure employed for determining the sets Ri, of relevance assessments. Subjectivity of judgement comes in here, and people will differ about whether a particular document is relevant to a particular query. Even so, the averaging across queries reduces the errors that may occasionally arise through faulty relevance judgements, and averaging important tests across a number of test collections reduces the effects caused by accidental features of individual collections, and the results obtained by these tests in modern research are generally accepted as trustworthy. Nowadays such research with test collections is organised from TREC.

## Probabilistic term weights

In this section we will try to present some of the thinking behind the formulae. This is really to give a feel for where the probabilistic model comes from. You may want to skim through this section if you're not too interested.

Suppose we have an IR system with a total of N documents. And suppose Q is a query in this IR system, made up of terms t1, t2 ... tQ. There is a set, R, of documents relevant to the query.

In 1976, Stephen Robertson derived a formula which gives an ideal numeric weight to a term t of Q. Just how this weight gets used we will see below, but essentially a high weight means an important term and a low weight means an unimportant term. The formula is,

$$w(t) =\log\:\:\frac{p(1-q)}{(1-p)q}$$

(The base of the logarithm doesn't matter, but we can suppose it is e.) p is the probability that t indexes a relevant document, and q the probability that t indexes a non-relevant document. And of course, 1 - p is the probability that t does not index a relevant document, and 1 - q the probability that t does not index a non-relevant document. More mathematically,

p = P(t → D|DinR)q = P(t → D|DnotinR)

1 − p = P(tnot → D|DinR)1 − q = P(tnot → D|DnotinR)

Suppose that t indexes n of the N documents in the IR system. As before, we suppose also that there are R documents in R, and that there are r documents in R which are indexed by t.

p is easily estimated by r/R, the ratio of the number of relevant documents indexed by t to the total number of relevant documents.

The total number of non-relevant documents is N - R, and the number of those indexed by t is n - r, so we can estimate q as (n - r)/(N - R). This gives us the estimates,

$$p =\:\frac{r}{R}\:\:,\:\:1-q =\:\frac{N-R-n+r}{N-R}$$

$$1-p=\:\frac{R-r}{R}\:\:,\:\:q=\:\frac{n-r}{N-R}$$

and so substituting in the formula for w(t) we get the estimate,

$$w(t) = \log\:\:\frac{r(N - R - n + r)}{(R - r)(n - r)}$$

Unfortunately, this formula is subject to violent behaviour when, say, n = r (infinity) or r = 0 (minus infinity), and so Robertson suggests the modified form

$$w(t) = \log\:\:\frac{(r+\frac{1}{2})(N-R-n+r+\frac{1}{2})}{(R-r+\frac{1}{2})(n-r+\frac{1}{2})}$$

with the reassurance that this has "some theoretical justification". This is the form of the term weighting formula used in Xapian's BM25Weight.

Note that n is dependent on the term, t, and R on the query, Q, while r depends both on t and Q. N is constant, at least until the IR system changes.

At first sight this formula may appear to be quite useless. After all, R is what we are trying to find. We can't evaluate w(t) until we have R, and if we have R the retrieval process is over, and term weights are no longer of any interest to us.

But the point is we can estimate p and q from a subset of R. As soon as some records are found relevant by the user they can be used as a working set for R from which the weights w(t) can be derived, and these new weights can be used to improve the processing of the query.

In fact in the Xapian software R tends to mean not the complete set of relevant documents, which indeed can rarely be discovered, but a small set of documents which have been judged as relevant.

Suppose we have no documents marked as relevant. Then R = r = 0, and w(t) becomes,

$$\log\:\frac{N-n+\frac{1}{2}}{n+\frac{1}{2}}$$

This is approximately log((N - n)/n). Or log(N/n), since n is usually small compared with N. This is called inverse logarithmic weighting, and has been used in IR for many decades, quite independently of the probabilistic theory which underpins it. Weights of this form are in fact the starting point in Xapian when no relevance information is present.

The number n incidentally is often called the frequency of a term. We prefer the phrase term frequency, to better distinguish it from wdf and wqf introduced below.

In extreme cases w(t) can be negative. In Xapian, negative values are disallowed, and simply replaced by a small positive value.

## wdp, wdf, ndl and wqf

Before we see how the weights are used there are a few more ideas to introduce.

As mentioned before, a term t is said to index a document D, or t → D. We have emphasised that D may not be a piece of text in machine-readable form, and that, even when it is, t may not actually occur in the text of D. Nevertheless, it will often be the case that D is made up of a list of words,

D = w1, w2, w3 ... wm

and that many, if not all, of the terms which index D derive from these words (for example, the terms are often lower-cased and stemmed forms of these words).

If a term derives from words w9, w38, w97 and w221 in the indexing process, we can say that the term "occurs" in D at positions 9, 38, 97 and 221, and so for each term a document may have a vector of positional information. These are the within-document positions of t, or the wdp information of t.

The within-document frequency, or wdf, of a term t in D is the number of times it is pulled out of D in the indexing process. Usually this is the size of the wdp vector, but in Xapian it can exceed it, since we can apply extra wdf to some parts of the document text. For example, often this is done for the document title and abstract to attach extra importance to their contents compared to the rest of the document text.

There are various ways in which we might measure the length of a document, but the easiest is to suppose it is made up of m words, w1 to wm, and to define its length as m.

The normalised document length, or ndl, is then m divided by the average length of the documents in the IR system. So the average length document has ndl equal to 1, short documents are less than 1, long documents greater than 1. We have found that very small ndl values create problems, so Xapian actually allows for a non-zero minimum value for the ndl.

In the probabilistic model the query, Q, is itself very much like another document. Frequently indeed Q will be created from a document, either one already in the IR system, or by an indexing process very similar to the one used to add documents into the whole IR system. This corresponds to a user saying "give me other documents like this one". One can therefore attach a similar meaning to within-query position information, within-query frequency, and normalised query length, or wqp, wqf and nql. Xapian does not currently use the concept of wqp.

## Using the weights. The MSet

Now to pull everything together. From the probabilistic term weights we can assign a weight to any document, d, as follows,

$$W(d) = \sum_{t\to d,t \in Q}\:\frac{(k + 1) f_t}{k.L_d + f_t}\:w(t)$$

The sum extends over the terms of Q which index d. ft is the wdf of t in d, Ld is the ndl of d, and k is some suitably chosen constant.

The factor k + 1 is actually redundant, but helps with the interpretation of the equation. In Xapian, this weighting scheme is implemented by the Xapian::TradWeight class and the factor (k + 1) is ignored.

If k is set to zero the factor before w(t) is 1, and the wdfs are ignored. As k tends to infinity, the factor becomes ft/Ld, and the wdfs take on their greatest importance. Intermediate values scale the wdf contribution between these extremes. The best k actually depends on the characteristics of the IR system as a whole, and unfortunately no rule can be given for choosing it. By default, Xapian sets k to 1 which should give reasonable results for most systems. W(d) is merely tweaked a bit by the wdf values, and users observe a simple pattern of retrieval. It is possible to tune k to provide optimal results for a specific system.

Any d in the IR system has a value W(d), but, if no term of the query indexes d, W(d) will be zero. In practice only documents for which W(d) > 0 will be of interest, and these are the documents indexed by at least one term of Q. If we now take these documents and arrange them by decreasing W(d) value, we get a ranked list called the match set, or MSet, of document and weight pairs:

item0: D0, W(D0)

item1: D1, W(D1)

item2: D2, W(D2)

\begin{aligned} \text{. \\ . \\ . } \end{aligned}

itemKDK, W(DK)

where W(Dj) ≥ W(Di) if j > i.

And according to the probabilistic model, the documents D0, D1, D2... are ranked by decreasing order of probability of relevance. So D0 has highest probability of being relevant, then D1 and so on.

Xapian creates the MSet from the posting lists of the terms of the query. This is the central operation of any IR system, and will be familiar to anyone who has used one of the Internet's major search engines, where the query is what you type in the query box, and the resulting hit list corresponds to the top few items of the MSet.

The cutoff point, K, is chosen when the MSet is created. The candidates for inclusion in the MSet are all documents indexed by at least one term of Q, and their number will usually exceed the choice of K (K is typically set to be 1000 or less). So the MSet is actually the best K documents found in the match process.

A modification of this weighting scheme can be employed that takes into account the query itself:

$$W(d) = \sum_{t \to d,\:t \in Q}\frac{(k_3+1)q_t}{(k_3L'+q_t)}\:\:\frac{(k+1)f_t}{(kL_d+f_t)}\:w(t)$$

where qt is the wqf of t in Q, L is the nql, or normalised query length, and k3 is a further constant. In computing W(d) across the document space, this extra factor may be viewed as just a modification to the basic term weights, w(t). Like k and k3, we will need to make an inspired guess for L. In fact the choices for k3 and L will depend on the broader context of the use of this formula, and more advice will be given as occasion arises.

Xapian's default weighting scheme is a generalised form of this weighting scheme modification, known as BM25. In BM25, L is always set to 1.

## Using the weights: the ESet

But as well as ranking documents, Xapian can rank terms, and this is most important. The higher up the ranking the term is, the more likely it is to act as a good differentiator between relevant and non-relevant documents. It is therefore a candidate for adding back into the query. Terms from this list can therefore be used to expand the size of the query, after which the query can be re-run to get a better MSet. Because this list of terms is mainly used for query expansion, it is called the expand set or ESet.

The term expansion weighting formula is as follows,

W(t) = r w(t)

in other words we multiply the term weight by the number of relevant documents that have been indexed by the term.

The ESet then has this form,

item0: t0, W(t0)

item1: t1, W(t1)

item2: t2, W(t2)

\begin{aligned} \text{. \\ . \\ . } \end{aligned}

itemKtK, W(tK)

where W(tj) ≥ W(ti) if j > i.

Since the main function of the ESet is to find new terms to be added to Q, we usually omit from it terms already in Q.

The W(t) weight is applicable to any term in the IR system, but has a value zero when t does not index a relevant document. The ESet is therefore confined to be a ranking of the best K terms which index relevant documents.

This simple form of W(t) is traditional in the probabilistic model, but seems less than optimal because it does not take into account wdf information. One can if fact try to generalise it to:

$$W(t) = \sum_{t \to d,d \in R}\:\frac{(k+1)f_t}{kL+f_t}w(t)$$

k is again a constant, but it does not need to have the same value as the k used in the probabilistic term weights above. In Xapian, k defaults to 1.0 for ESet generation.

This reduces to W(t) = r w(t) when k = 0. Certainly this form can be recommended in the very common case where r = 1, that is, we have a single document marked relevant.

## The progress of a query

Below we describe the general case of the IR model supported, including use of a relevance set (RSet), query expansion, improved term weights and reranking. You don't have to use any of these for Xapian to be useful, but they are available should you need them.

The user enters a query. This is parsed into a form the IR system understands, and run by the IR system, which returns two lists, a list of captions, derived from the MSet, and a list of terms, from the ESet. If the RSet is empty, the first few documents of the MSet can be used as a stand-in - after all, they have a good chance of being relevant! You can read a document by clicking on the caption. (We assume the usual screen/mouse environment.) But you can also mark a document as relevant (change R) or cause a term to be added from the ESet to the query (change Q). As soon as any change is made to the query environment the query can be rerun, although you might have a front-end where nothing happens until you click on some "Run Query" button.

In any case rerunning the query leads to a new MSet and ESet, and so to a new display. The IR process is then an iterative one. You can delete terms from the query or add them in; mark or unmark documents as being relevant. Eventually you converge on the answer to the query, or at least, the best answer the IR system can give you.