topic models

1. Modeling Science
David M. Blei
Department of Computer Science
Princeton University
April 17, 2008
Joint work with John Lafferty (CMU)
D. Blei Modeling Science 1 / 53

2. Modeling Science
Science, August 13, 1886 Science, June 24, 1994
evolution rna disease
water acid disease evolutionary mrna host
milk water blood species site bacteria
food solution cholera organisms splicing diseases
dry experiments bacteria biology rnas new
fed liquid found phylogenetic nuclear bacterial
cows chemical bacillus life sequence resistance
houses action experiments origin introns control
butter copper organisms diversity messenger strains
fat crystals bacilli
groups cleavage infectious
found carbon cases
made alcohol diseases
molecular two malaria
contained made germs animals splice parasites
wells obtained animal two sequences parasite
produced substances koch new polymerase tuberculosis
poisonous nitrogen made living intron health
5
• On-line archives of document collections require better 6
organization. Manual organization is not practical.
• Our goal: To discover the hidden thematic structure with
hierarchical probabilistic models called topic models.
• Use this structure for browsing, search, and similarity.
D. Blei Modeling Science 2 / 53

3. Modeling Science
Science, August 13, 1886 Science, June 24, 1994
evolution rna disease
water acid disease evolutionary mrna host
milk water blood species site bacteria
food solution cholera organisms splicing diseases
dry experiments bacteria biology rnas new
fed liquid found phylogenetic nuclear bacterial
cows chemical bacillus life sequence resistance
houses action experiments origin introns control
butter copper organisms diversity messenger strains
fat crystals bacilli
groups cleavage infectious
found carbon cases
made alcohol diseases
molecular two malaria
contained made germs animals splice parasites
wells obtained animal two sequences parasite
produced substances koch new polymerase tuberculosis
poisonous nitrogen made living intron health
5
• Our data are the pages Science from 1880-2002 (from JSTOR) 6
• No reliable punctuation, meta-data, or references.
• Note: this is just a subset of JSTOR’s archive.
D. Blei Modeling Science 2 / 53

4. Discover topics from a corpus
“Genetics” “Evolution” “Disease” “Computers”
human evolution disease computer
genome evolutionary host models
dna species bacteria information
genetic organisms diseases data
genes life resistance computers
sequence origin bacterial system
gene biology new network
molecular groups strains systems
sequencing phylogenetic control model
map living infectious parallel
information diversity malaria methods
genetics group parasite networks
mapping new parasites software
project two united new
sequences common tuberculosis simulations
D. Blei Modeling Science 3 / 53

5. Model the evolution of topics over time
"Theoretical Physics" "Neuroscience"
FORCE OXYGEN
o o o o LASER o
o o o
o o o o o NERVE o
o o o o o
o o o o
o o o o o o o
o o o
o o o o o o o
RELATIVITY o o o
o
o
o
o o o o
o o o o o o o o
o o
o o o o o
o o o o NEURON
o o o
o o
o o o o o
o
o o o
o o o o o
o o o o
o o o
o o o
o o o o o o o o
o o o o o o
o o
o o o o o o o o o o o o o o
o o o o o o
o o o o o o
o o o o o o o
o o
1880 1900 1920 1940 1960 1980 2000 1880 1900 1920 1940 1960 1980 2000
D. Blei Modeling Science 4 / 53

6. Model connections between topics
neurons
brain stimulus
motor
memory
visual
activated subjects synapses
tyrosine phosphorylation cortical
left ltp
activation
phosphorylation p53 task surface glutamate
kinase cell cycle proteins tip synaptic
activity protein
cyclin binding rna image neurons
regulation domain dna sample materials
computer
domains rna polymerase organic
problem device
receptor cleavage
information
polymer
science amino acids
research
scientists
receptors cdna
site
computers
polymers
funding molecules physicists
support says ligand sequence problems
laser particles
nih research ligands isolated optical physics
program people protein sequence light
apoptosis sequences surface
particle
electrons experiment
genome liquid quantum
wild type dna surfaces stars
mutant enzyme sequencing fluid
mutations enzymes model reaction astronomers
united states mutants
iron
active site
reactions universe
women cells
mutation reduction molecule galaxies
universities
cell molecules
expression magnetic
galaxy
cell lines plants
magnetic field transition state
students bone marrow plant
spin
superconductivity
gene
education genes
superconducting
pressure mantle
arabidopsis
bacteria high pressure crust sun
bacterial pressures upper mantle solar wind
host fossil record core meteorites earth
resistance development birds inner core ratios planets
mice parasite embryos fossils planet
gene dinosaurs
antigen virus drosophila species
disease fossil
t cells hiv genes forest
mutations
antigens aids expression forests
families earthquake co2
immune response infection populations
mutation earthquakes carbon
viruses ecosystems
fault carbon dioxide
ancient images methane
patients genetic found
disease cells population impact
data water
ozone
treatment proteins populations million years ago volcanic atmospheric
drugs differences africa
clinical researchers deposits climate
measurements
variation stratosphere
protein magma ocean
eruption ice concentrations
found volcanism changes
climate change
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7. Outline
1 Introduction
2 Latent Dirichlet allocation
3 Dynamic topic models
4 Correlated topic models
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8. Outline
1 Introduction
2 Latent Dirichlet allocation
3 Dynamic topic models
4 Correlated topic models
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9. Probabilistic modeling
1 Treat data as observations that arise from a generative
probabilistic process that includes hidden variables
• For documents, the hidden variables reflect the thematic
structure of the collection.
2 Infer the hidden structure using posterior inference
• What are the topics that describe this collection?
3 Situate new data into the estimated model.
• How does this query or new document fit into the estimated
topic structure?
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10. Intuition behind LDA
Simple intuition: Documents exhibit multiple topics.
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11. Generative process
• Cast these intuitions into a generative probabilistic process
• Each document is a random mixture of corpus-wide topics
• Each word is drawn from one of those topics
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12. Generative process
• In reality, we only observe the documents
• Our goal is to infer the underlying topic structure
• What are the topics?
• How are the documents divided according to those topics?
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13. Graphical models (Aside)
Y Y
≡
··· Xn
X1 X2 XN N
• Nodes are random variables
• Edges denote possible dependence
• Observed variables are shaded
• Plates denote replicated structure
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14. Graphical models (Aside)
Y Y
≡
··· Xn
X1 X2 XN N
• Structure of the graph defines the pattern of conditional
dependence between the ensemble of random variables
• E.g., this graph corresponds to
N
p(y, x1 , . . . , xN ) = p(y ) p(xn | y)
n=1
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15. Latent Dirichlet allocation
Per-word
Dirichlet
topic assignment
parameter
Per-document Observed Topic
topic proportions word Topics hyperparameter
α θd Zd,n Wd,n βk η
N
D K
Each piece of the structure is a random variable.
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16. Latent Dirichlet allocation
α θd Zd,n Wd,n βk η
N
D K
1 Draw each topic βi ∼ Dir(η), for i ∈ {1, . . . , K }.
2 For each document:
1 Draw topic proportions θd ∼ Dir(α).
2 For each word:
1 Draw Zd,n ∼ Mult(θd ).
2 Draw Wd,n ∼ Mult(βzd,n ).
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17. Latent Dirichlet allocation
α θd Zd,n Wd,n βk η
N
D K
• From a collection of documents, infer
• Per-word topic assignment zd,n
• Per-document topic proportions θd
• Per-corpus topic distributions βk
• Use posterior expectations to perform the task at hand, e.g.,
information retrieval, document similarity, etc.
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18. Latent Dirichlet allocation
α θd Zd,n Wd,n βk η
N
D K
• Computing the posterior is intractable:
N
p(θ | α) n=1 p(zn | θ )p(wn | zn , β1:K )
N K
θ p(θ | α) n=1 z=1 p(zn | θ )p(wn | zn , β1:K )
• Several approximation techniques have been developed.
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19. Latent Dirichlet allocation
α θd Zd,n Wd,n βk η
N
D K
• Mean field variational methods (Blei et al., 2001, 2003)
• Expectation propagation (Minka and Lafferty, 2002)
• Collapsed Gibbs sampling (Griffiths and Steyvers, 2002)
• Collapsed variational inference (Teh et al., 2006)
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20. Example inference
• Data: The OCR’ed collection of Science from 1990–2000
• 17K documents
• 11M words
• 20K unique terms (stop words and rare words removed)
• Model: 100-topic LDA model using variational inference.
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21. Example inference
0.4
0.3
Probability
0.2
0.1
0.0
1 8 16 26 36 46 56 66 76 86 96
Topics
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22. Example topics
“Genetics” “Evolution” “Disease” “Computers”
human evolution disease computer
genome evolutionary host models
dna species bacteria information
genetic organisms diseases data
genes life resistance computers
sequence origin bacterial system
gene biology new network
molecular groups strains systems
sequencing phylogenetic control model
map living infectious parallel
information diversity malaria methods
genetics group parasite networks
mapping new parasites software
project two united new
sequences common tuberculosis simulations
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23. LDA summary
• LDA is a powerful model for
• Visualizing the hidden thematic structure in large corpora
• Generalizing new data to fit into that structure
• LDA is a mixed membership model (Erosheva, 2004) that builds
on the work of Deerwester et al. (1990) and Hofmann (1999).
• For document collections and other grouped data, this might be
more appropriate than a simple finite mixture
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24. LDA summary
• Modular : It can be embedded in more complicated models.
• E.g., syntax and semantics; authorship; word sense
• General: The data generating distribution can be changed.
• E.g., images; social networks; population genetics data
• Variational inference is fast; lets us to analyze large data sets.
• See Blei et al., 2003 for details and a quantitative comparison.
• Code to play with LDA is freely available on my web-site,
http://www.cs.princeton.edu/∼blei.
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25. LDA summary
• But, LDA makes certain assumptions about the data.
• When are they appropriate?
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26. Outline
1 Introduction
2 Latent Dirichlet allocation
3 Dynamic topic models
4 Correlated topic models
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27. LDA and exchangeability
α θd Zd,n Wd,n βk η
N
D K
• LDA assumes that documents are exchangeable.
• I.e., their joint probability is invariant to permutation.
• This is too restrictive.
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28. Documents are not exchangeable
"Infrared Reflectance in Leaf-Sitting
"Instantaneous Photography" (1890)
Neotropical Frogs" (1977)
• Documents about the same topic are not exchangeable.
• Topics evolve over time.
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29. Dynamic topic model
• Divide corpus into sequential slices (e.g., by year).
• Assume each slice’s documents exchangeable.
• Drawn from an LDA model.
• Allow topic distributions evolve from slice to slice.
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30. Dynamic topic models
α α α
θd θd θd
Zd,n Zd,n Zd,n
Wd,n Wd,n Wd,n
N N N
D D D
...
βk,1 βk,2 βk,T
K
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31. Modeling evolving topics
βk,1 βk,2 βk,T
...
• Use a logistic normal distribution to model evolving topics
(Aitchison, 1980)
• A state-space model on the natural parameter of the topic
multinomial (West and Harrison, 1997)
βt,k | βt−1,k ∼ N (βt−1,k , Iσ 2 )
V −1
p(w | βt,k ) = exp βt,k − log(1 + v =1 exp{βt,k ,v })
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32. Posterior inference
• Our goal is to compute the posterior distribution,
p(β1:T ,1:K , θ1:T ,1:D , z1:T ,1:D | w1:T ,1:D ).
• Exact inference is impossible
• Per-document mixed-membership model
• Non-conjugacy between p(w | βt,k ) and p(βt,k )
• MCMC is not practical for the amount of data.
• Solution: Variational inference
D. Blei Modeling Science 26 / 53

33. Science data
TECHVIEW: DNA S E Q U E N C I NG
Sequencing the Genome, Fast
James C. Mullikin and Amanda A. McMurray
Genome sequencing projects reveal
the genetic makeup of an organism
by reading off the sequence of the
DNA bases, which encodes all of the infor-
mation necessary for the life of the organ-
ism. The base sequence contains four nu-
cleotides-adenine, thymidine, guanosine,
and cytosine-which are linked together
into long double-helical chains. Over the
last two decades, automated DNA se-
quencers have made the process of obtain-
ing the base-by-base sequence of DNA...
• Analyze JSTOR’s entire collection from Science (1880-2002)
• Restrict to 30K terms that occur more than ten times
• The data are 76M words in 130K documents
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34. Analyzing a document
Original article Topic proportions
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35. Analyzing a document
Original article Most likely words from top topics
sequence devices data
genome device information
genes materials network
sequences current web
human high computer
gene gate language
dna light networks
sequencing silicon time
chromosome material software
regions technology system
analysis electrical words
data fiber algorithm
genomic power number
number based internet
D. Blei Modeling Science 28 / 53

36. Analyzing a topic
1880 1890 1900 1910 1920 1930 1940
electric electric apparatus air apparatus tube air
machine power steam water tube apparatus tube
power company power engineering air glass apparatus
engine steam engine apparatus pressure air glass
steam electrical engineering room water mercury laboratory
two machine water laboratory glass laboratory rubber
machines two construction engineer gas pressure pressure
iron system engineer made made made small
battery motor room gas laboratory gas mercury
wire engine feet tube mercury small gas
1950 1960 1970 1980 1990 2000
tube tube air high materials devices
apparatus system heat power high device
glass temperature power design power materials
air air system heat current current
chamber heat temperature system applications gate
instrument chamber chamber systems technology high
small power high devices devices light
laboratory high flow instruments design silicon
pressure instrument tube control device material
rubber control design large heat technology
D. Blei Modeling Science 29 / 53

37. Visualizing trends within a topic
"Theoretical Physics" "Neuroscience"
FORCE OXYGEN
o o o o LASER o
o o o
o o o o o NERVE o
o o o o o
o o o o
o o o o o o o
o o o
o o o o o o o
RELATIVITY o o o
o
o
o
o o o o
o o o o o o o o
o o
o o o o o
o o o o NEURON
o o o
o o
o o o o o
o
o o o
o o o o o
o o o o
o o o
o o o
o o o o o o o o
o o o o o o
o o
o o o o o o o o o o o o o o
o o o o o o
o o o o o o
o o o o o o o
o o
1880 1900 1920 1940 1960 1980 2000 1880 1900 1920 1940 1960 1980 2000
D. Blei Modeling Science 30 / 53

38. Time-corrected document similarity
• Consider the expected Hellinger distance between the topic
proportions of two documents,
K
dij = E ( θi,k − θj,k )2 | wi , wj
k =1
• Uses the latent structure to define similarity
• Time has been factored out because the topics associated to the
components are different from year to year.
• Similarity based only on topic proportions
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39. Time-corrected document similarity
The Brain of the Orang (1880)
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40. Time-corrected document similarity
Representation of the Visual Field on the Medial Wall of
Occipital-Parietal Cortex in the Owl Monkey (1976)
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41. Browser of Science
D. Blei Modeling Science 34 / 53

42. Quantitative comparison
• Compute the probability of each year’s documents conditional on
all the previous year’s documents,
p(wt | w1 , . . . , wt−1 )
• Compare exchangeable and dynamic topic models
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43. Quantitative comparison
q
LDA
25
DTM
Per−word negative log likelihood
20
q
15
q
q
q
q
10
q q q
q q
q q
q q q
q q q
q q q q q
q q q
q q q
q q q q
q q q q q
q
1920 1940 1960 1980 2000
Year
D. Blei Modeling Science 36 / 53

44. Outline
1 Introduction
2 Latent Dirichlet allocation
3 Dynamic topic models
4 Correlated topic models
D. Blei Modeling Science 37 / 53

45. The hidden assumptions of the Dirichlet distribution
• The Dirichlet is an exponential family distribution on the simplex,
positive vectors that sum to one.
• However, the near independence of components makes it a poor
choice for modeling topic proportions.
• An article about fossil fuels is more likely to also be about
geology than about genetics.
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46. The logistic normal distribution
• The logistic normal is a distribution on the simplex that can
model dependence between components.
• The natural parameters of the multinomial are drawn from a
multivariate Gaussian distribution.
X ∼ NK −1 (µ, )
K −1
θi = exp{xi − log(1 + j=1 exp{xj })}
D. Blei Modeling Science 39 / 53

47. Correlated topic model (CTM)
Σ βk
ηd Zd,n Wd,n K
N
µ D
• Draw topic proportions from a logistic normal, where topic
occurrences can exhibit correlation.
• Use for:
• Providing a “map” of topics and how they are related
• Better prediction via correlated topics
D. Blei Modeling Science 40 / 53

48. neurons
brain stimulus
motor
memory
visual
activated subjects synapses
tyrosine phosphorylation cortical
left ltp
activation
phosphorylation p53 task surface glutamate
kinase cell cycle proteins tip synaptic
activity protein
cyclin binding rna image neurons
regulation domain dna sample materials
computer
domains rna polymerase organic
problem device
receptor cleavage
information
polymer
science amino acids
research
scientists
receptors cdna
site
computers
polymers
funding molecules physicists
support says ligand sequence problems
laser particles
nih research ligands isolated optical physics
program people protein sequence light
apoptosis sequences surface
particle
electrons experiment
genome liquid quantum
wild type dna surfaces stars
mutant enzyme sequencing fluid
mutations enzymes model reaction astronomers
united states mutants
iron
active site
reactions universe
women cells
mutation reduction molecule galaxies
universities
cell molecules
expression magnetic
galaxy
cell lines plants
magnetic field transition state
students bone marrow plant
spin
superconductivity
gene
education genes
superconducting
pressure mantle
arabidopsis
bacteria high pressure crust sun
bacterial pressures upper mantle solar wind
host fossil record core meteorites earth
resistance development birds inner core ratios planets
mice parasite embryos fossils planet
gene dinosaurs
antigen virus drosophila species
disease fossil
t cells hiv genes forest
mutations
antigens aids expression forests
families earthquake co2
immune response infection populations
mutation earthquakes carbon
viruses ecosystems
fault carbon dioxide
ancient images methane
patients genetic found
disease cells population impact
data water
ozone
treatment proteins populations million years ago volcanic atmospheric
drugs differences africa
clinical researchers deposits climate
measurements
variation stratosphere
protein magma ocean
eruption ice concentrations
found volcanism changes
climate change
D. Blei Modeling Science 41 / 53

49. Summary
• Topic models provide useful descriptive statistics for analyzing
and understanding the latent structure of large text collections.
• Probabilistic graphical models are a useful way to express
assumptions about the hidden structure of complicated data.
• Variational methods allow us to perform posterior inference to
automatically infer that structure from large data sets.
• Current research
• Choosing the number of topics
• Continuous time dynamic topic models
• Topic models for prediction
• Inferring the impact of a document
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50. “We should seek out unfamiliar summaries of observational material,
and establish their useful properties... And still more novelty can
come from finding, and evading, still deeper lying constraints.”
(John Tukey, The Future of Data Analysis, 1962)
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51. Supervised topic models (with Jon McAuliffe)
• Most topic models are unsupervised. They are fit by maximizing
the likelihood of a collection of documents.
• Consider documents paired with response variables.
For example:
• Movie reviews paired with a number of stars
• Web pages paired with a number of “diggs”
• We develop supervised topic models, models of documents and
responses that are fit to find topics predictive of the response.
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52. Supervised LDA
α θd Zd,n Wd,n βk K
N
Yd D η, σ 2
1 Draw topic proportions θ | α ∼ Dir(α).
2 For each word
1 Draw topic assignment zn | θ ∼ Mult(θ ).
2 Draw word wn | zn , β1:K ∼ Mult(βzn ).
3 Draw response variable y | z1:N , η, σ 2 ∼ N η z, σ 2 , where
¯
N
z = (1/N)
¯ n=1 zn .
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53. Comments
• SLDA is used as follows.
• Fit coefficients and topics from a collection of
document-response pairs.
• Use the fitted model to predict the responses of previously
unseen documents,
E[Y | w1:N , α, β1:K , η, σ 2 ] = η E[Z | w1:N , α, β1:K ].
¯
• The process enforces that the document is generated first,
followed by the response. The response is generated from the
particular topics that were realized in generating the document.
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54. Example: Movie reviews
least bad more awful his both
problem guys has featuring their motion
unfortunately watchable than routine character simple
supposed its films dry many perfect
worse not director offered while fascinating
flat one will charlie performance power
dull movie characters paris between complex
● ● ● ●● ● ● ● ●
−30 −20 −10 0 10 20
have not one however
like about from cinematography
you movie there screenplay
was all which performances
just would who pictures
some they much effective
out its what picture
• We fit a 10-topic sLDA model to movie review data (Pang and
Lee, 2005).
• The documents are the words of the reviews.
• The responses are the number of stars associated with
each review (modeled as continuous).
• Each component of coefficient vector η is associated with a topic.
D. Blei Modeling Science 47 / 53

55. Simulations
Movie corpus
0.5
● ● ● ● ●
● ● ●
−6.37
●
● ●
● ●
Per−word held out log likelihood
0.4
−6.38
● ● ●
●
●
●
Predictive R2
●
0.3
−6.39
●
●
●
● ●
● ● ● ● ● ●
● ●
●
−6.40
0.2
●
●
● ●
−6.41
0.1
●
sLDA
LDA
−6.42
●
0.0
5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50
Number of topics Number of topics
Digg corpus
−8.0
0.12
−8.1
0.10
Per−word held out log likelihood
●
−8.2
●
0.08
● ●
● ● ●
●
Predictive R2
−8.3
●
●
●
0.06
●
●
−8.4
●
●
0.04
●
●
−8.5
0.02
●
● ●
−8.6
●
●
●
0.00
● ●
●
●
2 4 10 20 30 2 4 10 20 30
Number of topics Number of topics
D. Blei Modeling Science 48 / 53

56. Diversion: Variational inference
• Let x1:N be observations and z1:M be latent variables
• Our goal is to compute the posterior distribution
p(z1:M , x1:N )
p(z1:M | x1:N ) =
p(z1:M , x1:N )dz1:M
• For many interesting distributions, the marginal likelihood of the
observations is difficult to efficiently compute
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57. Variational inference
• Use Jensen’s inequality to bound the log prob of the
observations:
log p(x1:N ) ≥ Eqν [log p(z1:M , x1:N )] − Eqν [log qν (z1:M )].
• We have introduced a distribution of the latent variables with free
variational parameters ν.
• We optimize those parameters to tighten this bound.
• This is the same as finding the member of the family qν that is
closest in KL divergence to p(z1:M | x1:N ).
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58. Mean-field variational inference
• Complexity of optimization is determined by factorization of qν
• In mean field variational inference qν is fully factored
M
qν (z1:M ) = qνm (zm ).
m=1
• The latent variables are independent.
• Each is governed by its own variational parameter νm .
• In the true posterior they can exhibit dependence
(often, this is what makes exact inference difficult).
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59. MFVI and conditional exponential families
• Suppose the distribution of each latent variable conditional on
the observations and other latent variables is in the exponential
family:
p(zm | z−m , x) = hm (zm ) exp{gm (z−m , x)T zm − am (gi (z−m , x))}
• Assume qν is fully factorized and each factor is in the same
exponential family:
qνm (zm ) = hm (zm ) exp{νm zm − am (νm )}
T
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60. MFVI and conditional exponential families
• Variational inference is the following coordinate ascent algorithm
νm = Eqν [gm (Z−m , x)]
• Notice the relationship to Gibbs sampling
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61. Variational family for the DTM
βk,1 βk,2 βk,T
...
ˆ
βk,1 ˆ
βk,2 ˆ
βk,T
• Distribution of θ and z is fully-factorized (Blei et al., 2003)
• Distribution of {β1,k , . . . , βT ,k } is a variational Kalman filter
• Gaussian state-space model with free observations βk ,t .
ˆ
• Fit observations such that the corresponding posterior over the
chain is close to the true posterior.
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62. Variational family for the DTM
βk,1 βk,2 βk,T
...
ˆ
βk,1 ˆ
βk,2 ˆ
βk,T
• Given a document collection, use coordinate ascent on all the
variational parameters until the KL converges.
• Yields a distribution close to the true posterior of interest
• Take expectations w/r/t the simpler variational distribution
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