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CS590I: Information Retrieval

- CS-590I
- Information Retrieval
- Retrieval Models: Language models
- Luo Si
- Department of Computer Science
- Purdue University

Retrieval Model: Language Model

- Introduction to language model

- Unigram language model

- Document language model estimation

- Maximum Likelihood estimation
- Maximum a posterior estimation
- Jelinek Mercer Smoothing

- Model-based feedback

Language Models: Motivation

- Vector space model for information retrieval
- Documents and queries are vectors in the term space
- Relevance is measure by the similarity between document vectors and query vector
- Problems for vector space model
- Ad-hoc term weighting schemes
- Ad-hoc similarity measurement

No justification of relationship between relevance and similarity

- We need more principled retrieval models…

Introduction to Language Models:

- Language model can be created for any language sample
- A document
- A collection of documents
- Sentence, paragraph, chapter, query…
- The size of language sample affects the quality of language model
- Long documents have more accurate model
- Short documents have less accurate model
- Model for sentence, paragraph or query may not be reliable

Introduction to Language Models:

- A document language model defines a probability distribution over indexed terms
- E.g., the probability of generating a term
- Sum of the probabilities is 1
- A query can be seen as observed data from unknown models
- Query also defines a language model (more on this later)
- How might the models be used for IR?
- Rank documents by Pr( | )
- Rank documents by language models of and based on kullback-Leibler (KL) divergence between the models (come later)

sport, basketball

Language Model for

Language Model for

Language Model for

sport, basketball, ticket, sport

stock, finance, finance, stock

Language Model for IR: ExampleGenerate retrieval results

Estimate the generation probability of Pr( | )

Estimating language model for each document

basketball, ticket, finance, ticket, sport

Language Models

- Three basic problems for language models
- What type of probabilistic distribution can be used to construct language models?
- How to estimate the parameters of the distribution of the language models?
- How to compute the likelihood of generating queries given the language modes of documents?

Examples:

- Five words in vocabulary (sport, basketball, ticket, finance, stock)
- For a document , its language mode is:
- {Pi(“sport”), Pi(“basketball”), Pi(“ticket”), Pi(“finance”), Pi(“stock”)}

- Language model built by multinomial distribution on single terms (i.e., unigram) in the vocabulary

- Formally:
- The language model is: {Pi(w) for any word w in vocabulary V}

Multinomial Model for

Multinomial Model for

Multinomial Model for

basketball, ticket, finance, ticket, sport

sport, basketball, ticket, sport

stock, finance, finance, stock

Multinomial/Unigram Language ModelsEstimating language model for each document

Maximum Likelihood Estimation (MLE)

- Maximum Likelihood Estimation:
- Find model parameters that make generation likelihood reach maximum:

M*=argmaxMPr(D|M)

There are K words in vocabulary, w1...wK (e.g., 5)

Data: one document with counts tfi(w1), …, tfi(wK), and length | |

Model: multinomial M with parameters {pi(wk)}

Likelihood: Pr( | M)

M*=argmaxMPr( |M)

Maximum Likelihood Estimation (MLE)

Use Lagrange multiplier approach Set partial derivatives to zero Get maximum likelihood estimate

(psp, pb, pt, pf, pst) =

(0.5,0.25,0.25,0,0)

(psp, pb, pt, pf, pst) =

(0.2,0.2,0.4,0.2,0)

(psp, pb, pt, pf, pst) =

(0,0,0,0.5,0.5)

basketball, ticket, finance, ticket, sport

sport, basketball, ticket, sport

stock, finance, finance, stock

Maximum Likelihood Estimation (MLE)Estimating language model for each document

Maximum Likelihood Estimation (MLE)

- Maximum Likelihood Estimation:
- Assign zero probabilities to unseen words in small sample

- A specific example:
- Only two words in vocabulary (w1=sport, w2=business) like (head, tail) for a coin; A document generates sequence of two words or draw a coin for many times

- Only observe two words (flip the coin twice) and MLE estimators are:
- “business sport” Pi(w1)=0.5
- “sport sport” Pi(w1)=1 ?
- “business business” Pi(w1)=0 ?

Maximum Likelihood Estimation (MLE)

- A specific example:

- Only observe two words (flip the coin twice) and MLE estimators are:
- “business sport” Pi(w1)*=0.5
- “sport sport” Pi(w1)*=1 ?
- “business business” Pi(w1)*=0 ?

- Data sparseness problem

Solution to Sparse Data Problems

- Maximum a posterior (MAP) estimation
- Shrinkage
- Bayesian ensemble approach

A specific examples:

- Only two words in vocabulary (sport, business)
- For a document :

Prior Distribution

Maximum A Posterior (MAP) Estimation- Maximum A Posterior Estimation:
- Select a model that maximizes the probability of model given observed data

M*=argmaxMPr(M|D)=argmaxMPr(D|M)Pr(M)

- Pr(M): Prior belief/knowledge
- Use prior Pr(M) to avoid zero probabilities

Maximum A Posterior (MAP) Estimation

- Maximum A Posterior Estimation:
- Introduce prior on the multinomial distribution

- Use prior Pr(M) to avoid zero probabilities, most of coins are more or less unbiased
- Use Dirichlet prior on p(w)

Hyper-parameters

Constant for pK

(x) is gamma function

Maximum A Posterior (MAP) Estimation

- Maximum A Posterior:

M*=argmaxMPr(M|D)=argmaxMPr(D|M)Pr(M)

Pseudo Counts

Maximum A Posterior (MAP) Estimation

- A specific example:

- Only observe two words (flip a coin twice):
- “sport sport” Pi(w1)*=1 ?

P(w1)2 (1-P(w1)2)

times

Maximum A Posterior (MAP) Estimation

- A specific example:

- Only observe two words (flip a coin twice):
- “sport sport” Pi(w1)*=1 ?

MAP EstimationUnigram Language Model

- Maximum A Posterior Estimation:
- Use Dirichlet prior for multinomial distribution
- How to set the parameters for Dirichlet prior

MAP EstimationUnigram Language Model

- Maximum A Posterior Estimation:
- Use Dirichlet prior for multinomial distribution

- There are K terms in the vocabulary:

Hyper-parameters

Constant for pK

MAP EstimationUnigram Language Model

- MAP Estimation for unigram language model:

- Use Lagrange Multiplier; Set derivative to 0

Pseudo counts set by hyper-parameters

MAP EstimationUnigram Language Model

- MAP Estimation for unigram language model:

- Use Lagrange Multiplier; Set derivative to 0

- How to determine the appropriate value for hyper-parameters?

- When nothing observed from a document

- What is most likely pi(wk) without looking at the content of the document?

MAP EstimationUnigram Language Model

- MAP Estimation for unigram language model:

- What is most likely pi(wk) without looking at the content of the document?
- The most likely pi(wk) without looking into the content of the document d is the unigram probability of the collection:
- {p(w1|c), p(w2|c),…, p(wK|c)}

- Without any information, guess the behavior of one member on the behavior of whole population

Constant

MAP EstimationUnigram Language Model

- MAP Estimation for unigram language model:

- Use Lagrange Multiplier; Set derivative to 0

Pseudo counts

Pseudo document length

Maximum A Posterior (MAP) Estimation

- Dirichlet MAP Estimation for unigram language model:

- Step 0: compute the probability on whole collection based collection unigram language model
- Step 1: for each document , compute its smoothed unigram language model (Dirichlet smoothing) as

Maximum A Posterior (MAP) Estimation

- Dirichlet MAP Estimation for unigram language model:

- Step 2: For a given query ={tfq(w1),…, tfq(wk)}

- For each document , compute likelihood
- The larger the likelihood, the more relevant the document is to the query

Dirichlet Smoothing & TF-IDF

- Dirichlet Smoothing:

Dirichlet Smoothing Hyper-Parameter

- Dirichlet Smoothing:

Hyper-parameter

- When is very small, approach MLE estimator

- When is very large, approach probability on whole collection

- How to set appropriate ?

w1

Leave w1 out

Leave wj out

...

wj

...

Dirichlet Smoothing Hyper-Parameter

- Leave One out Validation:

...

...

w1

...

wj

...

Dirichlet Smoothing Hyper-Parameter

- Leave One out Validation:

Leave all words out one by one for a document

Do the procedure for all documents in a collection

Find appropriate

Dirichlet Smoothing Hyper-Parameter

- What type of document/collection would get large ?
- Most documents use similar vocabulary and wording pattern as the whole collection
- What type of document/collection would get small ?
- Most documents use different vocabulary and wording pattern than the whole collection

U.S.

Indiana

West Lafayette

Shrinkage

- Maximum Likelihood (MLE) builds model purely on document data and generates query word
- Model may not be accurate when document is short (many unseen words)
- Shrinkage estimator builds more reliable model by consulting more general models (e.g., collection language model)

Example: Estimate P(Lung_Cancer|Smoke)

Jelinek Mercer Smoothing

- Assume for each word, with probability , it is generated from document language model (MLE), with probability 1- , it is generated from collection language model (MLE)
- Linear interpolation between document language model and collection language model

Shrinkage

JM Smoothing:

Model Based Feedback

- Equivalence of retrieval based on query generation likelihood and Kullback-Leibler (KL) Divergence between query and document language models

- Kullback-Leibler (KL) Divergence between two probabilistic distributions

- It is the distance between two probabilistic distributions
- It is always larger than zero

How to prove it ?

Model Based Feedback

- Equivalence of retrieval based on query generation likelihood and Kullback-Leibler (KL) Divergence between query and document language models

Loglikelihood of query generation probability

Document independent constant

- Generalize query representation to be a distribution (fractional term weighting)

Retrieval results

Retrieval results

Calculate KL Divergence

Estimate the generation probability of Pr( | )

Language Model for

Language Model for

Language Model for

Estimating query language model

Estimating language model

Estimating document language model

Model Based Feedback

Language Model for

Language Model for

Language Model for

No feedback

Full feedback

Estimating document language model

Model Based FeedbackFeedback Documents from initial results

Retrieval results

Calculate KL Divergence

New Query Model

Estimating query language model

Model Based Feedback: Estimate

- Assume there is a generative model to produce each word within feedback document(s)

- For each word in feedback document(s), given

Background model

Feedback Documents

1-

PC(w)

w

Flip a coin

Topic words

w

qF(w)

Model Based Feedback: Estimate

MLE

Estimator

- For each word, there is a hidden variable telling which language model it comes from

the 0.12

to 0.05

it 0.04

a 0.02

…

sport 0.0001

basketball 0.00005

…

Feedback

Documents

Background

Model

pC(w|C)

1-=0.8

Unknown

query topic

p(w|F)=?

“Basketball”

…

sport =?

basketball =?

game =?

player =?

…

=0.2

If we know the value of hidden variable of each word ...

Model Based Feedback: Estimate

E-step

- For each word, the hidden variable Zi = {1 (feedback), 0 (background}

- Step1: estimate hidden variable based current on model parameter (Expectation)

the (0.1) basketball (0.7) game (0.6) is (0.2) ….

- Step2: Update model parameters based on the guess in step1 (Maximization)

M-Step

Model Based Feedback: Estimate

- Expectation-Maximization (EM) algorithm

- Step 0: Initialize values of

- Step1: (Expectation)

- Step2: (Maximization)

Give =0.5

Model Based Feedback: Estimate

- Properties of parameter

- If is close to 0, most common words can be generated from collection language model, so more topic words in query language mode

- If is close to 1, query language model has to generate most common words, so fewer topic words in query language mode

Retrieval Model: Language Model

- Introduction to language model

- Unigram language model

- Document language model estimation

- Maximum Likelihood estimation
- Maximum a posterior estimation
- Jelinek Mercer Smoothing

- Model-based feedback

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