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Latent Dirichlet Allocation
Presenter: Hsuan-Sheng Chiu
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Reference
• D. M. Blei, A. Y. Ng and M. I. Jordan, “Latent Dirichlet allocation”, Journal of Machine Learning Research, vol. 3, no. 5, pp. 993-1022, 2003.
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Outline
• Introduction
• Notation and terminology
• Latent Dirichlet allocation
• Relationship with other latent variable models
• Inference and parameter estimation
• Discussion
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Introduction
• We consider with the problem of modeling text corpora and other collections of discrete data– To find short description of the members a collection
• Significant process in IR– tf-idf scheme (Salton and McGill, 1983)– Latent Semantic Indexing (LSI, Deerwester et al., 1990) – Probabilistic LSI (pLSI, aspect model, Hofmann, 1999)
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Introduction (cont.)
• Problem of pLSI: – Incomplete: Provide no probabilistic model at the level of docum
ents– The number of parameters in the model grows linear with the siz
e of the corpus– It is not clear how to assign probability to a document outside of t
he training data
• Exchangeability: bag of words
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Notation and terminology
• A word is the basic unit of discrete data ,from vocabulary indexed by {1,…,V}. The vth word is represented by a V-vector w such that wv = 1 and wu = 0 for u≠v
• A document is a sequence of N words denote by w = (w1,w2,…,wN)
• A corpus is a collection of M documents denoted by D = {w1,w2,…,wM}
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Latent Dirichlet allocation
• Latent Dirichlet allocation (LDA) is a generative probabilistic model of a corpus.
• Generative process for each document w in a corpus D:– 1. Choose N ~ Poisson(ξ)– 2. Choose θ ~ Dir(α)– 3. For each of the N words wn
(a) Choose a topic zn ~ Multinomial(θ)
(b) Choose a word wn from p(wn|zn, β), a multinomial probability conditioned on the topic zn
βij is a a element of k×V matrix = p(wj = 1| zi = 1)
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Latent Dirichlet allocation (cont.)
• Representation of a document generation:
z1 z2 … … zN
w1 w2 … … wNw
N ~ Poisson
θ~ Dir(α) → {z1,z2,…,zk}
β(z) →{w1,w2,…,wn}
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Latent Dirichlet allocation (cont.)
• Several simplifying assumptions:– 1. The dimensionality k of Dirichlet distribution is known and fixe
d– 2. The word probabilities β is fixed quantity that is to be estimate
d– 3. Document length N is independent of all the other data genera
ting variable θ and z
• A k-dimensional Dirichlet random variable θ can take values in the (k-1)-simplex
111
1
1 ...| 1
k
kk
i i
k
i ip
http://www.answers.com/topic/dirichlet-distribution
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Latent Dirichlet allocation (cont.)
• Simplex:
The above figures show the graphs for the n-simplexes with n =2 to 7.(from mathworld, http://mathworld.wolfram.com/Simplex.html)
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Latent Dirichlet allocation (cont.)
• The joint distribution of a topic θ, and a set of N topic z, and a set of N words w:
• Marginal distribution of a document:
• Probability of a corpus:
dzwpzpppN
n znnn
n
w
1
,|||,|
N
nnnn zwpzppp
1
,|||,| wz,,
M
dd
N
n zdndndnd dzwpzppDp
d
dn1 1
,|||,|
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Latent Dirichlet allocation (cont.)
• There are three levels to LDA representation– αβ are corpus-level parameters– θd are document-level variables
– zdn, wdn are word-level variables
corpus document
Refer to as hierarchical models, conditionally independent hierarchical models and parametric empirical Bayes models
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Latent Dirichlet allocation (cont.)
• LDA and exchangeability– A finite set of random variables {z1,…,zN} is said exchangeable if the joint
distribution is invariant to permutation (πis a permutation)
– A infinite sequence of random variables is infinitely exchangeable if every finite subsequence is exchangeable
– De Finetti’s representation theorem states that the joint distribution of an infinitely exchangeable sequence of random variables is as if a random parameter were drawn from some distribution and then the random variables in question were independent and identically distributed, conditioned on that parameter
– http://en.wikipedia.org/wiki/De_Finetti's_theorem
NN zzpzzp ,...,,..., 11
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Latent Dirichlet allocation (cont.)
• In LDA, we assume that words are generated by topics (by fixed conditional distributions) and that those topics are infinitely exchangeable within a document
dzwpzpppN
nnnn zw,
1
||
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Latent Dirichlet allocation (cont.)
• A continuous mixture of unigrams– By marginalizing over the hidden topic variable z, we can under
stand LDA as a two-level model
• Generative process for a document w– 1. choose θ~ Dir(α)– 2. For each of the N word wn
(a) Choose a word wn from p(wn|θ, β)– Marginal distribution od a document
z
zpzwpwp |,|,|
dwppwpN
nn
1
,||,|
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Latent Dirichlet allocation (cont.)
• The distribution on the (V-1)-simplex is attained with only k+kV parameters.
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Relationship with other latent variable models
• Unigram model
• Mixture of unigrams– Each document is generated by first choosing a topic z and then
generating N words independently form conditional multinomial– k-1 parameters
N
nnwpwp
1
z
N
nn zwpzpwp
1
|
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Relationship with other latent variable models (cont.)
• Probabilistic latent semantic indexing– Attempt to relax the simplifying assumption made in the mixture
of unigrams models– In a sense, it does capture the possibility that a document may c
ontain multiple topics– kv+kM parameters and linear growth in M
z
nn dzpzwpdpwdp ||,
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Relationship with other latent variable models (cont.)
• Problem of PLSI– There is no natural way to use it to assign probability to a previou
sly unseen document– The linear growth in parameters suggests that the model is pron
e to overfitting and empirically , overfitting is indeed a serious problem
• LDA overcomes both of there problems by treating the topic mixture weights as a k-parameter hidden random variable
• The k+kV parameters in a k-topic LDA model do not grow with the size of the training corpus.
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Relationship with other latent variable models (cont.)
• A geometric interpretation: three topics and three words
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Relationship with other latent variable models (cont.)
• The unigram model find a single point on the word simplex and posits that all word in the corpus come from the corresponding distribution.
• The mixture of unigram models posits that for each documents, one of the k points on the word simplex is chosen randomly and all the words of the document are drawn from the distribution
• The pLSI model posits that each word of a training documents comes from a randomly chosen topic. The topics are themselves drawn from a document-specific distribution over topics.
• LDA posits that each word of both the observed and unseen documents is generated by a randomly chosen topic which is drawn from a distribution with a randomly chosen parameter
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Inference and parameter estimation
• The key inferential problem is that of computing the posteriori distribution of the hidden variable given a document
,|
,|,,,,|,w
wzwzp
pp
dp
N
n
k
i
V
j
wiji
k
iik
i i
k
i i jni w
1 1 11
1
1
1,|
Unfortunately, this distribution is intractable to compute in general.A function which is intractable due to the coupling between θ and β in the summation over latent topics
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Inference and parameter estimation (cont.)
• The basic idea of convexity-based variational inference is to make use of Jensen’s inequality to obtain an adjustable lower bound on the log likelihood.
• Essentially, one considers a family of lower bounds, indexed by a set of variational parameters.
• A simple way to obtain a tractable family of lower bound is to consider simple modifications of the original graph model in which some of the edges and nodes are removed.
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Inference and parameter estimation (cont.)
• Drop some edges and the w nodes
N
nnnzqqq
1
||,|, z
,|
,|,,,,|,w
wzwzp
pp
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Inference and parameter estimation (cont.)
• Variational distribution:– Lower bound on Log-likelihood
– KL between variational posteriori and true posteriori
,|,,|,,log,|,
,|,,log,|,
,|,,|,,,|,log,|,,log,|log
zwzzwzz
zwzz wzw
z
z
qEpEdqpq
dqpqdpp
z
,log,,,,|,,
,,,log,|,,|,log,|,
,,log,|,,|,log,|,,,||,|,
,pE,pEqE
d,p,pqdqq
d,|pqdqq,|pqD
qqq wwzzw
wzzzz
wzzzzwzz
zz
zz
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Inference and parameter estimation (cont.)
• Finding a tight lower bound on the log likelihood
• Maximizing the lower bound with respect to γand φ is equivalent to minimizing the KL divergence between the variational posterior probability and the true posterior probability
,,||,|,
,|,log,|,,log,|log
,|pqD
qEpEp qq
wzz
zwzw
,,||,|,minarg,,
** ,|pqD wzz
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Inference and parameter estimation (cont.)
• Expand the lower bound:
|log
|log
,|log
|log
|log
,|,log,|,,log,;,
z
zw
z
zwz
pE
pE
pE
pE
pE
qEpEL
q
q
q
q
q
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Inference and parameter estimation (cont.)
• Then
N
n
k
inini
k
i
k
j jiii
k
i
k
j j
N
n
k
iij
jnni
N
n
k
i
k
j jini
k
i
k
j jiii
k
i
k
j j
w
L
1 1
11
11
1 1
1 11
11
11
log
1loglog
log
1loglog
,;,
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Inference and parameter estimation (cont.)
• We can get variational parameters by adding Lagrange multipliers and setting this derivative to zero:
N
n niii
k
j jiivni
1
1exp
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Inference and parameter estimation (cont.)
• Parameter estimation– Maximize log likelihood of the data:
– Variational inference provide us with a tractable lower bound on the log likelihood, a bound which we can maximize with respect α and β
• Variational EM procedure– 1. (E-step) For each document, find the optimizing values of the
variational parameters {γ, φ}– 2. (M-step) Maximize the result lower bound on the log likelihood
with respect to the model parameters α and β
M
ddp
1
,|log, w
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Inference and parameter estimation (cont.)
• Smoothed LDA model:
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Discussion
• LDA is a flexible generative probabilistic model for collection of discrete data.
• Exact inference is intractable for LDA, but any or a large suite of approximate inference algorithms for inference and parameter estimation can be used with the LDA framework.
• LDA is a simple model and is readily extended to continuous data or other non-multinomial data.