Topic Modelling and Latent Dirichlet Allocation
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1 Topic Modelling and Latent Dirichlet Allocation Stephen Clark (with thanks to Mark Gales for some of the slides) Lent 2013 Machine Learning for Language Processing: Lecture 7 MPhil in Advanced Computer Science MPhil in Advanced Computer Science
2 Introduction to Probabilistic Topic Models We want to find themes (or topics) in documents useful for e.g. search or browsing We don t want to do supervised topic classification rather not fix topics in advance nor do manual annotation Need an approach which automatically teases out the topics This is essentially a clustering problem- can think of both words and documents as being clustered MPhil in Advanced Computer Science 1
3 Key Assumptions behind the LDA Topic Model Documents exhibit multiple topics (but typically not many) LDA is a probabilistic model with a corresponding generative process each document is assumed to be generated by this (simple) process A topic is a distribution over a fixed vocabulary these topics are assumed to be generated first, before the documents Only the number of topics is specified in advance MPhil in Advanced Computer Science 2
4 The Generative Process To generate a document: 1. Randomly choose a distribution over topics 2. For each word in the document a. randomly choose a topic from the distribution over topics b. randomly choose a word from the corresponding topic (distribution over the vocabulary) Note that we need a distribution over a distribution (for step 1) Note that words are generated independently of other words (unigram bag-ofwords model) MPhil in Advanced Computer Science 3
5 The Generative Process more Formally Some notation: β 1:K are the topics where each β k is a distribution over the vocabulary θ d are the topic proportions for document d θ d,k is the topic proportion for topic k in document d z d are the topic assignments for document d z d,n is the topic assignment for word n in document d w d are the observed words for document d The joint distribution (of the hidden and observed variables): p(β 1:K,θ 1:D,z 1:D,w 1:D ) = K D N p(β i ) p(θ d ) p(z d,n θ d )p(w d,n β 1:K,z d,n ) i=1 d=1 n=1 MPhil in Advanced Computer Science 4
6 Module L101: Machine Learning for Language Processing Plate Diagram of the Graphical Model Note that only the words are observed (shaded) α and η are the parameters of the respective dirichlet distributions (more later) Note that the topics are generated (not shown in earlier pseudo code) Plates indicate repetition Picture from Blei 2012 MPhil in Advanced Computer Science 5
7 Multinomial Distribution Multinomial distribution: x i {0,...,n} P(x θ) = n! d i=1 x i! d i=1 θ x i i, d n = x i, i=1 d θ i = 1, θ i 0 i=1 When n = 1 the multinomial distribution simplifies to P(x θ) = d i=1 θ x i i, d θ i = 1, θ i 0 i=1 a unigram language model with 1-of-V coding (d = { V the vocabulary size) 1, word i observed x i indicateswordiofthevocabularyobserved,x i = 0, otherwise θ i = P(w i ) the probability that word i is seen MPhil in Advanced Computer Science 6
8 The Dirichlet Distribution Dirichlet (continuous) distribution with parameters α p(x α) = Γ( d i=1 α i) d i=1 Γ(α i) d i=1 x α i 1 i ; for observations : d x i = 1, x i 0 i=1 Γ() is the Gamma distribution Conjugate prior to the multinomial distribution (form of posterior p(θ D,M) is the same as the prior p(θ M)) MPhil in Advanced Computer Science 7
9 Dirichlet Distribution Example (6,2,2) (3,7,5) (2,3,4) Parameters: (α 1,α 2,α 3 ) (6,2,6) MPhil in Advanced Computer Science 8
10 Parameter Estimation Main variables of interest: β k : distribution over vocabulary for topic k θ d,k : topic proportion for topic k in document d Could try and get these directly, eg using EM (Hoffmann, 1999), but this approach not very successful One common technique is to estimate the posterior of the word-topic assignments, given the observed words, directly (whilst marginalizing out β and θ) MPhil in Advanced Computer Science 9
11 Gibbs Sampling Gibbs sampling is an example of a Markov Chain Monte Carlo (MCMC) technique Markov chain in this instance means that we sample from each variable one at a time, keeping the current values of the other variables fixed MPhil in Advanced Computer Science 10
12 Posterior Estimate The Gibbs sampler produces the following estimate, where, following Steyvers and Griffiths: z i is the topic assigned to the ith token in the whole collection; d i is the document containing the ith token; w i is the word type of the ith token; z i is the set of topic assignments of all other tokens; is any remaining information such as the α and η hyperparameters: P(z i = j z i,w i,d i, ) C WT w i j +η W w=1 CWT wj +Wη C DT d i j +α T t=1 CDT d i t +Tα where C WT and C DT are matrices of counts (word-topic and document-topic) MPhil in Advanced Computer Science 11
13 Posterior Estimates of β and θ β ij = Cij WT +η W k=1 CWT kj +Wη θ dj = C DT dj +α T k=1 CDT dk +Tα Using the count matrices as before, where β ij is the probability of word type i for topic j, and θ dj is the proportion of topic j in document d MPhil in Advanced Computer Science 12
14 References David Blei s webpage is a good place to start A good introductory paper: D. Blei. Probabilistic topic models. Communications of the ACM, 55(4):7784, Introduction to Gibbs sampling for LDA: Steyvers, M., Griffiths, T. Probabilistic topic models. Latent Semantic Analysis: A Road to Meaning. T. Landauer, D. McNamara, S. Dennis, and W. Kintsch, eds. Lawrence Erlbaum, MPhil in Advanced Computer Science 13
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