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An Introduction to Latent Dirichlet Allocation (LDA) 1 LDA • A generative probabilistic model for collections of discrete data such text corpora. • Is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as finite mixture over an underlying set of topics. • Each topic again is modeled as an infinite mixture of an underlying set of topic probabilities. • It has natural advantages over unigram model and probabilistic LSI model. 2 History (1) – text processing • IR – text to real number vector(Baeza-Yates and Ribeiro-Neto, 1999), tfidf (Salton and McGill, 1983) – Tfidf – shortcoming: (1) Lengthy and (2) Cannot model inter- and intra- document statistical structure • LSI – dimension reduction (Deerwester et al., 1990) – Advantages: achieve significant compression in large collections and capture synonymy and polysemy. • Generative probabilistic model – to study the ability of LSI (Papadimitriou et al., 1998) – Why LSI, we can model the data directly using maximum likelihood or Bayesian methods. 3 History (2) – text processing • Probabilistic LSI – also aspect model. Milestone (Hofmann, 1999). – P(wi|θj), d={w1, …, wN}, and θ={θ1, …, θk}. each word is generated from a single model θj. Document d is considered to be a mixing proportions for these mixture components θ, that is a list of numbers (the mixing proportions for topics). – Disadvantage: no probabilistic model at document level. • The number of parameters grows linearly with the size of corpora. • It is not clear to assign probability to document outside of the collection. (does not make any assumptions about how the mixture weights θ are generated, making it difficult to test the generalizability of the model to new documents. ) 4 Notation • D={d1, …, dM}, d={w1, …, wN}, • and θ={θ1, …, θk}, equivalently D={w1, …, wM}. (Bold variable denotes vector.) • Suppose we have V distinct words in the whole data set. 5 LDA • The basic idea: Documents are represented as random mixtures over latent topics, where each topic is characterized by a distribution over words. • For each document d, we generate as follows: 1. Choose N Poisson( ) 2. Choose θ Dir(α) k topics z 3. For each of the N words wn : β is a k V matrix (a) Choose a topic zn Multinomial(θ) with ij =p(wi | z j ) (b) Choose a word wn from p( wn | zn , β), a multinomial probability conditioned on the topic zn 6 Dirichlet Random Variables θ • A k-dimensional Dirichlet random variables θ can take values in the (k-1)-simplex (a k-vector θ lies in the (k-1)- θi 0, i 1 θi 1 , and thus the probability density k simplex if can be: ( i 1 α i ) k p (θ | α ) θ1 i 1 α θα k 1 k k i 1 (α i ) where α is a k-vector parameter with αi>0, and Γ(x) is a gamma function 7 Multinomial Distribution • Each trial can end in exactly one of k categories • n independent trials • Probability a trial results in category i is pi • Yi is the number of trials resulting in category I • p1+…+pk = 1 • Y1+…+Yk = n 8 Multinomial Distribution 9 Conjugation 10 Entropy 11 Joint Distribution • Given the parameters α and β, the joint distribution of a topic mixture θ, a set of N topics z, and a set of N words w is given by: N p (θ, z , w | α ) p (θ | α ) p ( zn | θ) p( wn | z n , β) n 1 where p(zn|θ) is simply θi for the unique i such that zni=1. Integrating over θ and summing over z we obtain the marginal distribution of a document N p(w | α, β) p(θ | α) p( zn | θ) p( wn | zn , β) dθ n 1 z (a) n 12 Joint Distribution (cont.) • Finally, taking the product of the marginal probabilities of single documents, we can obtain the probability of a corpus: M Nd p( D | α, β) p(θd | α) p( zdn | θd ) p( wdn | zdn , β) dθd n 1 z d 1 nd β Similar model can be referred to hierarchical models (Gelman et al., 1995), or more precisely as conditionally α θ z w independent hierarchical models (Kass N M and Steffey, 1989). 13 Relationships with Other Latent Models N w p (w ) p( wn ) N n 1 M N z w p ( w ) p ( z ) p ( wn | z ) N z n 1 M d z w p(d , wn ) p(d ) p(wn | z) p( z | d ) N z M Problems: (1) p(z|d) only model the documents in the training data set and cannot for the unseen document; and (2) the parameter kV+kM grows linearly in M and thus overfitting. 14 Graphical Interpretation The probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). ( i 1 α i ) k p (θ | α ) θ1 i 1 α θα k 1 k k i 1 (α i ) k θi 0, i 1 θi 1 15 Graphical Interpretation (cont.) • The Dirichlet prior on the topic-word distributions can be interpreted as forces on the topic locations with higher β moving the topic locations away from the corners of the simplex. 16 Matrix Interpretation • In the topic model, the word-document co-occurrence matrix is split into two parts: a topic matrix Φ and a document matrix Θ. Note that the diagonal matrix D in LSA can be absorbed in the matrix U or V, making the similarity between the two representations even clearer. 17 Inference and Parameter Estimation • The key inferential problem is that of computing the posterior distribution of the hidden variables given a document: p (θ, z , w | α, β) p (θ, z | w , α, β) p ( w | α, β) • However, such distribution is intractable to compute in general. For normalization in the above distribution, we have to marginalize over the hidden variables and write the Equation (a) in terms of the model parameters: ( i αi ) k αi 1 N k V p ( w | α, β) θi (θiβij ) wnj dθ i (αi ) i 1 n 1 i 1 j 1 18 Inference and Parameter Estimation • The key inferential problem is that of computing the posterior distribution of the hidden variables given a document: p (θ, z , w | α, β) p (θ, z | w , α, β) p ( w | α, β) • However, such distribution is intractable to compute in general. For normalization in the above distribution, we N have to marginalize over the hidden variables and write the p(w | α, β) p(θ | α) p( zn | θ) p( wn | zn , β) dθ Equation (a) in terms of the model parameters: n 1 z n ( i αi ) k αi 1 N k V p ( w | α, β) θi (θiβij ) wnj dθ i (αi ) i 1 n 1 i 1 j 1 19 Inference and Parameter Estimation (cont.) ( i αi ) k αi 1 N k V p ( w | α, β) θi (θiβij ) wnj dθ i (αi ) i 1 n 1 i 1 j 1 • This function is intractable due to the coupling between θ and β in the summation over latent topics (Dickey, 1983). • Rather than the intractable exact inference, we can use some other approximate inference algorithms, e.g., Laplace approximation, variational approximation, and Markov chain Monte Carlo (Jordan, 1999). 20 Variational Inference • Here, we introduce a simple convexity-based variational algorithm for inference in LDA. • The basic idea here is to make use of Jensen’s inequality to obtain an adjustable lower bound on the log likelihood (Jordan, 1999). p ( , z , w | , )q( , z ) log p(w | α, β) log d z q ( , z ) q ( , z )log p ( , z , w | , )d q ( , z )log q ( , z )d z z • A simple way to obtain a tractable family of lower bounds is to consider simple modifications of the original graphical model in which some of the edges and nodes are removed. 21 Variational Inference (cont.) • Hence, by dropping edges between θ, z, and w, and w nodes, and also endow the resulting simplified graphical model with free variational parameters, we obtain a family of distributions on the latent variables: N q (θ, z | γ , ) q (θ | γ ) q ( zn | n ) n 1 • where the Dirichlet parameter γ and the multinomial parameters (Φ1, …, ΦN) are the free variational parameters. β γ Φ α θ z w θ z N N M M 22 How to determine the vatiational parameters • We can set up an optimization problem to determine the values of the variational parameters γ and Φ. • We can define the optimization function as minimizing the Kullback-Leibler (KL) divergence between the variational distribution and the true posterior p(θ,z|w,α,β): (γ* , * ) arg min D(q(θ, z | γ, ) || p(θ, z | w, α, β) (5) ( γ , ) • This minimization can be achieved by an iterative fixed- point method. 23 Variational Inference • We now discuss how to set the parameter γ and Φ via an optimization procedure. • Following Jordan et al. (1999), we have a lower bound of the log likelihood of a document using Jensen’s inequality: q(w | α, β) log p(θ, z, w | α, β)d z p(θ, z, w | α, β)q(θ, z ) log d z q(θ, z ) q (θ, z )log p (θ, z , w | α, β)d q (θ, z )log q(θ, z )d z z E q [log p (θ, z , w | α, β)] E q [log q (θ, z )] 24 Variational Inference (cont.) • Then from the above formula, we see that the Jensen’s inequality provides us with a lower bound on the log likelihood for an arbitrary variational distribution q(θ,z|γ,Φ). • It can be easily verified that the difference between the left-hand side and the right-hand side of the above equation is the KL divergence between the variational posterior probability and the true posterior probability. 25 Variational Inference (cont.) • That is, letting L(, γ; α, β) denote the right-hand side of the above equation we have: log p(w | α, β) L(γ, ; α, β) D(q(θ, z | γ, ) || p(θ, z | w, α, β)) • This means that maximizing the lower bound L(, γ; α, β) w.r.t. γ and Φ is equivalent to minimizing the KL divergence between the variational posterior probability and the true posterior probability, the optimization problem in equation (5). 26 Variational Inference (cont.) • We can expand the above equation L(, γ; α, β) E q [log p(θ | α)] E q [log p(z | θ)] E q [log p( w | z, β)] Eq [log q(θ)] Eq [log q(z )] • By extending it again, we can have k k L(, γ; α, β) log ( j 1 α j ) log (α i ) (α i 1)( ( γ i ) ( j 1 γ j )) k k i 1 i 1 N k ni ( ( γ i ) ( j 1 γ j )) k n 1 i 1 N k V ni wnj log βij n 1 i 1 j 1 k k log ( j 1 γ j ) log ( γ i ) ( γ i 1)( ( γ i ) ( j 1 γ j )) k k i 1 i 1 N k (15) ni log ni 27 n 1 i 1 Entropy 28 Variaitonal Multinomial • We first maximize Eq. (15) w.r.t. Φni, the probability that the n-th word is generated by latent topic i. • We form the Lagrangian by isolating the terms which contain Φni and adding the appropriate Lagrange multipliers. Let βiv be p(wvn=1|zi=1) for the appropriate v. (recall that each wn is a vector of size V with exactly one component equal to one; we can select the unique v such that wvn=1): L[ ni ] ni ( ( γ i ) ( j 1 γ j )) ni log βiv ni log ni n ( j 1 ni 1) k k 29 Variaitonal Multinomial (cont.) • Taking derivatives w.r.t. Φni, we obtain: L ( γ i ) ( j 1 γ j ) log βiv log ni 1 k ni • Setting this to zero yields the maximizing value of the variational parameter Φni : ni βiv exp( ( γ i ) ( j 1 γ j )) k 30 Variational Dirichlet • Next we maximize equation (15) w.r.t. γi, the i-th component of the posterior Dirichlet parameters, the terms containing γi are: k N L[ γ ] (α i 1)( ( γ i ) ( j 1 γ j )) ni ( ( γ i ) ( j 1 γ j )) k k i 1 n 1 k log ( j 1 γ j ) log ( γ i ) ( γ i 1)( ( γ i ) ( j 1 γ j )) k k i 1 • By simplifying k L[ γ ] (α i n 1 ni γ i )( ( γ i ) ( j 1 γ j )) log ( j 1 γ j ) log ( γ i ) N k k i 1 31 Variational Dirichlet (cont.) • Taking the derivative w.r.t. γi: L k (γ i )(αi n 1 ni γ i ) ( j 1 γ j ) (αi n 1 ni γi ) ' N ' k N γ i i 1 • Setting this equation to zero yields a maximum at: γ i α i n1 ni N 32 Solve the Optimization Problem • Derivate the KL divergence and setting them equal to zero, we obtain the following update equations: ni βiwn exp{Eq [log(θi ) | γ]} (6) γ i α i n1 ni N (7) where the expectation in the multinomial update can be computed as follows: E q [log(θi ) | γ ] ( γ i ) ( j 1 γ j ) k (8) where ψ is the first derivative of the logΓ function which is computable via Taylor approximations 33 (Abramowitz and Stegun, 1970). Computing E[log(θi|α)] • Recall that a distribution is in the exponential family if it can be written in the form: p( x | ) h( x)exp{TT ( x) A()} where η is the natural parameter, T(x) is the sufficient statistic, and A(η) is the log of the normalization factor. • As we can write the Dirichlet in this form by exponentiating the log of Eq.: p(θ|α) p(θ | α) exp{( i 1 (α i 1)log θi ) log ( i 1 α i ) i 1 log (α i )} k k k 34 Computing E[log(θi|α)] (cont.) • From this form we see that the natural parameter of the Dirichlet is ηi=αi-1 and the sufficient statistic is T(θi)=logθi. Moreover, based on the general fact that the derivative of the log normalization factor w.r.t. the natural parameter is equal to the expectation of the sufficient statistic, we obtain: E[log(θi ) | α ] (α i ) ( j 1 α j ) k where ψ is the digamma function, the first derivative of the log Gamma function. 35 Variational Inference Algorithm 1 (1) initialize for all i and n 0 ni Each iteration requires k N O((N+1)k) operations (2) initialize γ i α i for all i k For a single document (3) repeat the iteration number (4) for n =1 to N is on the order of the (5) for i =1 to k number of words in it (6) ni1 βiwn exp( (γ ti )) t Thus, the total number (7) normalize ni1 to sum to 1 t of operations roughly γ ti α n 1 ni1 N t (8) on the order of N2k (9) until convergence 36 Parameter Estimation • We can use a empirical Bayes method for parameter estimation. In particular, we wish to find parameters α and β that maximize the marginal log likelihood: M log(α, β) log p ( w d | α, β) d 1 • The quantity p(w|α, β) can be computed by the variational inference as described above. Thus, we can find approximate empirical Bayes estimates for the LDA model via an alternating variational EM procedure that maximizes a lower bound w.r.t. the variational parameters γ and Φ, and then fixed values of the variational parameters, maximizes the lower bound w.r.t. the model parameter α and β. 37 Variational EM 1. (E-step) For each document, find the optimizing values of the variational parameters * * . This is done as described in {γ d , d : d D} the previous section. 2. (M-step) Maximize the resulting lower bound on the log likelihood w.r.t. the model parameters α and β. This corresponds to finding maximum likelihood estimates with expected sufficient statistics for each document under the approximate posterior which is computed in the E-step. Actually, the update for the conditional multinomial parameter β can be written out analytically: M Nd β ij dni wdn * j (9) d 1 n 1 The update for α can be implemented using an efficient Newton- Raphson method. These two steps are repeated until converges. 38 Parameter Estimation • We now consider how to obtain empirical Bayes estimates of the model parameters α and β. • We solve this problem by using the variational lower bound as a surrogate for the marginal log likelihood, with the variational parameters Φ and γ fixed to the values found by variational inference. • We then obtain empirical Bayes estimates by maximizing this lower bound w.r.t. the model parameters. 39 Parameter Estimation (cont.) • Recall our approach for finding the empirical Bayes estimates is based on a variational EM procedure. • In the variational E-step, we maximize the bound L(, γ; α, β) w.r.t. the variational parameter γ and Φ. In the M-step, which we describe in this section, we maximize the bound w.r.t. the model parameters α and β. The overall procedure can thus be viewed as coordinate ascent in L. 40 Conditional Multinomials • To maximize w.r.t. β, we isolate terms and add Lagrange multipliers: M Nd k V k L[β] dni w log βij i ( j 1 βij 1) j V dn d 1 n 1 i 1 j 1 i 1 • Taking the derivative w.r.t. βij and set it to zero, we have M Nd βij dni wdn j d 1 n 1 41 Dirichlet • First, we have M k k L[α ] log ( j 1α j ) log (αi ) (αi 1)((γ di ) ( j 1 γ dj )) k k d 1 i 1 i 1 • Taking derivative w.r.t. αi, we obtain: L M M (( j 1 α j ) (αi )) ( γ di ) ( j 1 γ dj ) k k αi d 1 • This derivative depends on αi, where j<>i, and we therefore must use an iterative method to find the maximal α. In particular, the Hessian is in the form found in equation (10): L (i, j )M (αi ) ( j 1α j ) ' ' k αiα j 42 Dirichlet (cont.) • Finally, we can use the same algorithm to find an empirical Bayes point estimate of η, the scalar parameter for the exchangeable Dirichlet in the smoothed LDA model as mentioned above. 43 Smoothing • Simple Laplace smoothing is no longer justified as a maximum a posteriori method in LDA setting. • We can then assume that each row in βkxV is independently drawn from an exchangeable Dirichlet distribution. That is to treat βi as random variables that are endowed with a posterior distribution, conditioned on the data. η β k α θ z w N M 44 Smoothing Model • Thus we obtain a variational approach to Bayesian inference: k N q (β1:k , θ1:M , z1:M | η, γ, ) Dir (βi | ηi ) qd (θd , z n | n , γ d ) i 1 n 1 where qd (θd , zn | n , γd ) is the variational distribution defined for LDA as above and the update for the new variational parameter η is as follow: M Nd ηij η dni wdn * j d 1 n 1 45 Applications 46 Document Modeling • Perplexity is used to indicate the generalization performance of a method. • Specifically, we estimate a document modeling and use this model to describe the new data set. M log p( wd ) perplexity ( Dtest ) exp{ d 1 } M d 1 Nd • LDA outperforms the other models including pLSI, Smoothed Unigram, and Smoothed Mixt. Unigrams. 47 Document Classification • We can use the LDA model results as the features for classifiers. In this way, say 50 topics, we can reduce the feature space by 99.6%. • The experimental results show that such feature reduction may decrease the accuracy only a little. 48 Collaborative Filtering • We can learn a model on a fully observed set of users. Then for each unobserved user, we are shown all but one of the movies preferred by that user and are asked to predict what the held-out movie is. • Precisely, define the predictive perplexity on M test uses as: M log p( wd , Nd | w d ,1:Nd 1 ) predictive perplexity( Dtest ) exp{ d 1 } M 49 Other Applications 50 The Naïve Bayes model c w N N c arg max p(c | w) p(c) p( w | c) p (c) p ( wn | c) c n 1 Object class Prior prob. of Image likelihood decision the object classes given the class 51 Csurka et al. 2004 52 Csurka et al. 2004 Hierarchical Bayesian text models Probabilistic Latent Semantic Analysis (pLSA) d z w N D “face” 53 Sivic et al. ICCV 2005 Hierarchical Bayesian text models “beach” Latent Dirichlet Allocation (LDA) c z w N D 54 Fei-Fei et al. ICCV 2005 Summary • Based on the exchangeability assumption • Can be viewed as a dimensionality reduction technique • Exact inference is intractable, we can approximate instead • Applications in other collection – images and caption for example. 55 End of The Talk ! 56