EM algorithm and its application in probabilistic latent semantic analysis

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EM algorithm and its application in probabilistic latent semantic analysis

  1. 1. EM algorithm and its application in Probabilistic Latent Semantic Analysis (pLSA) Duc-Hieu Tran tdh.net [at] gmail.com Nanyang Technological University July 27, 2010Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 1 / 27
  2. 2. The parameter estimation problem Outline The parameter estimation problem EM algorithm Probabilistic Latent Sematic Analysis ReferenceDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 2 / 27
  3. 3. The parameter estimation problem Introduction Known the prior probabilities P(ωi ), class-conditional densities p(x|ωi ) =⇒ optimal classifier P(ωj |x) ∝ p(x|ωj )p(ωj ) decide ωi if p(ωi |x) > P(ωj |x), ∀j = i In practice, p(x|ωi ) is unknown – just estimated from training samples (e.g., assume p(x|ωi ) ∼ N (µi , Σi )).Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 3 / 27
  4. 4. The parameter estimation problem Frequentist vs. Bayesian schools Frequentist parameters – quantities whose values are fixed but unknown. the best estimate of their values – the one that maximizes the probability of obtaining the observed samples. Bayesian paramters – random variables having some known prior distribution. observation of the samples converts this to a posterior density; revising our opinion about the true values of the parameters.Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 4 / 27
  5. 5. The parameter estimation problem Examples training samples: S = {(x (1) , y (1) ), . . . (x (m) , y (m) )} frequentist: maximum likelihood max p(y (i) |x (i) ; θ) θ i bayesian: P(θ) – prior, e.g., P(θ) ∼ N (0, I) m P(θ|S) ∝ P(y (i) |x (i) , θ) .P(θ) i=1 θMAP = arg max P(θ|S) θDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 5 / 27
  6. 6. EM algorithm Outline The parameter estimation problem EM algorithm Probabilistic Latent Sematic Analysis ReferenceDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 6 / 27
  7. 7. EM algorithm An estimation problem training set of m independent samples: {x (1) , x (2) , . . . , x (m) } goal: fit the paramters of a model p(x, z) to the data the likelihood: m m (i) (θ) = log p(x ; θ) = log p(x (i) , z; θ) i=1 i=1 z explicitly maximize (θ) might be difficult. z - laten random variable if z (i) were observed, then maximum likelihood estimation would be easy. strategy: repeatedly construct a lower-bound on (E-step) and optimize that lower-bound (M-step).Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 7 / 27
  8. 8. EM algorithm EM algorithm (1) digression: Jensen’s inequality. f – convex function; E [f (X )] ≥ f (E [X ]) for each i, Qi – distribution of z: z Qi (z) = 1, Qi (z) ≥ 0 (θ) = log p(x (i) ; θ) i = log p(x (i) , z (i) ; θ) i z (i) p(x (i) , z (i) ; θ) = log Qi (z (i) ) (1) i Qi (z (i) ) z (i) applying Jensen’s inequality, concave function log p(x (i) , z (i) ; θ) ≥ Qi (z (i) )log (2) i Qi (z (i) ) z (i) More detail . . .Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 8 / 27
  9. 9. EM algorithm EM algorithm (2) for any set of distribution Qi , formula (2) gives a lower-bound on (θ) how to choose Qi ? strategy: make the inequality hold with equality at our particular value of θ. require: p(x (i) , z (i) ; θ) =c Qi (z (i) ) c – constant not depend on z (i) choose: Qi (z (i) ) ∝ p(x (i) , z (i) ; θ) we know z Qi (z (i) ) = 1, so p(x (i) , z (i) ; θ) p(x (i) , z (i) ; θ) Qi (z (i) ) = = = p(z (i) |x (i) ; θ) z p(x (i) , z; θ) p(x (i) ; θ)Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 9 / 27
  10. 10. EM algorithm EM algorithm (3) Qi – posterior distribution of z (i) given x (i) and the parameter θ EM algorithm: repeat until convergence E-step: for each i Qi (z (i) ) := p(z (i) |x (i) ; θ) M-step: p(x (i) , z (i) ; θ) θ := arg max Qi (z (i) ) log θ i Qi (z (i) ) z (i) The algorithm will converge, since (θ(t) ) ≤ (θ(t+1) )Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 10 / 27
  11. 11. EM algorithm EM algorithm (4) Digression: coordinate ascent algorithm. maxW (α1 , . . . αm ) α loop until converge: for i ∈ 1, . . . , m: αi = arg max W (α1 , . . . , αi , . . . , αm ) ˆ αi ˆ EM-algorithm as coordinate ascent algorithm p(x (i) , z (i) ; θ) J(Q, θ) = Qi (z (i) ) log i Qi (z (i) ) z (i) (θ) ≥ J(Q, θ) EM algorithm can be viewed as coordinate ascent on J E-step: maximize w.r.t Q M-step: maximize w.r.t θDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 11 / 27
  12. 12. Probabilistic Latent Sematic Analysis Outline The parameter estimation problem EM algorithm Probabilistic Latent Sematic Analysis ReferenceDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 12 / 27
  13. 13. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (1) set of documents D = {d1 , . . . , dN } set of words W = {w1 , . . . , wM } set of unobserved classes Z = {z1 , . . . , zK } conditional independence assumption: P(di , wj |zk ) = P(di |zk )P(wj |zk ) (3) so, K P(wj |di ) = P(zk |di )P(wj |zk ) (4) k=1 K P(di , wj ) = P(di ) P(wj |zk )P(zk |di ) k=1 More detail . . .Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 13 / 27
  14. 14. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (2) n(di , wj ) – # word wj in doc. di Likelihood N N M L= P(di ) = [P(di , wj )]n(di ,wj ) i=1 i=1 j=1 N M K n(di ,wj ) = P(di ) P(wj |zk )P(zk |di ) i=1 j=1 k=1 log-likelihood = log(L) N M K = n(di , wj ) log P(di ) + n(di , wj ) log P(wj |zk )P(zk |di ) i=1 j=1 k=1Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 14 / 27
  15. 15. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (3) maximize w.r.t P(wj |zk ), P(zk |di ) ≈ maximize N M K n(di , wj ) log P(wj |zk )P(zk |di ) i=1 j=1 k=1 N M K P(wj |zk )P(zk |di ) = n(di , wj ) log Qk (zk ) Qk (zk ) i=1 j=1 k=1 N M K P(wj |zk )P(zk |di ) ≥ n(di , wj ) Qk (zk ) log Qk (zk ) i=1 j=1 k=1 choose P(wj |zk )P(zk |di ) Qk (zk ) = K = P(zk |di , wj ) l=1 P(wj |zl )P(zl |di ) More detail . . .Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 15 / 27
  16. 16. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (4) ≈ maximize (w.r.t P(wj |zk ), P(zk |di )) N M K P(wj |zk )P(zk |di ) n(di , wj ) P(zk |di , wj ) log P(zk |di , wj ) i=1 j=1 k=1 ≈ maximize N M K n(di , wj ) P(zk |di , wj ) log[P(wj |zk )P(zk |di )] i=1 j=1 k=1Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 16 / 27
  17. 17. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (5) EM-algorithm E-step: update P(wj |zk )P(zk |di ) P(zk |di , wj ) = K l=1 P(wj |zl )P(zl |di ) M-step: maximize w.r.t P(wj |zk ), P(zk |di ) N M K n(di , wj ) P(zk |di , wj ) log[P(wj |zk )P(zk |di )] i=1 j=1 k=1 subject to M P(wj |zk ) = 1, k ∈ {1 . . . K } j=1 K P(zk |di ) = 1, i ∈ {1 . . . N} k=1Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 17 / 27
  18. 18. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (6) Solution of maximization problem in M-step: N i=1 n(di , wj )P(zk |di , wj ) P(wj |zk ) = M N m=1 n=1 n(dn , wm )P(zk |dn , wm ) M j=1 n(di , wj )P(zk |di , wj ) P(zk |di ) = n(di ) M where, n(di ) = j=1 n(di , wj ) More detail . . .Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 18 / 27
  19. 19. Probabilistic Latent Sematic Analysis Probabilistic Latent Semantic Analysis (7) All together E-step: P(wj |zk )P(zk |di ) P(zk |di , wj ) = K l=1 P(wj |zl )P(zl |di ) M-step: N i=1 n(di , wj )P(zk |di , wj ) P(wj |zk ) = M N m=1 n=1 n(dn , wm )P(zk |dn , wm ) M j=1 n(di , wj )P(zk |di , wj ) P(zk |di ) = n(di )Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 19 / 27
  20. 20. Reference Outline The parameter estimation problem EM algorithm Probabilistic Latent Sematic Analysis ReferenceDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 20 / 27
  21. 21. Reference R.O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, Wiley-Interscience, 2001. T. Hofmann, ”Unsupervised learning by probabilistic latent semantic analysis,” Machine Learning, vol. 42, 2001, p. 177–196. Course: ”Machine Learning CS229”, Andrew Ng, Stanford UniversityDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 21 / 27
  22. 22. Appendix Generative model for word/document co-occurence select a document di with probability (w.p) P(di ) pick a latent class zk w.p P(zk |di ) generate a word wj w.p P(wj |zk ) K K P(di , wj ) = P(di , wj |zk )P(zk ) = P(wj |zk )P(di |zk )P(zk ) k=1 k=1 K = P(wj |zk )P(zk |di )P(di ) k=1 K = P(di ) P(wj |zk )P(zk |di ) k=1 P(di , wj ) = P(wj |di )P(di ) K =⇒ P(wj |di ) = P(zk |di )P(wj |zk ) k=1Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 22 / 27
  23. 23. Appendix K P(wj |di ) = P(zk |di )P(wj |zk ) k=1 K since k=1 P(zk |di ) = 1, P(wj , di ) is convex combination of P(wj |zk ) ≈ each document is modelled as a mixture of topics ReturnDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 23 / 27
  24. 24. Appendix P(di , wj |zk )P(zk ) P(zk |di , wj ) = (5) P(di , wj ) P(wj |zk )P(di |zk )P(zk ) = (6) P(di , wj ) P(wj |zk )P(zk |di ) = (7) P(wj |di ) P(wj |zk )P(zk |di ) = K (8) l=1 P(wj |zl )P(zl |di ) From (5) to (6) by conditional independence assumption (3). From (7) to (8) by (4). ReturnDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 24 / 27
  25. 25. Appendix Lagrange multipliers τk , ρi N M K H= n(di , wj ) P(zk |di , wj ) log[P(wj |zk )P(zk |di )] i=1 j=1 k=1   K M N K + τk 1 − P(wj |di ) + ρi 1 − P(zk |di ) k=1 j=1 i=1 k=1 N ∂H i=1 P(zk |di , wj )n(di , wj ) = − τk = 0 ∂P(wj |zk ) P(wj |zk ) M ∂H j=1 n(di , wj )P(zk |di , wj ) = − ρi = 0 ∂P(zk |di ) P(zk |di )Duc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 25 / 27
  26. 26. Appendix M from j=1 P(wj |zk ) =1 M N τk = P(zk |di , wj )n(di , wj ) j=1 i=1 K from k=1 P(zk |di , wj ) =1 ρi = n(di ) =⇒ P(wj |zk ), P(zk |di ) ReturnDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 26 / 27
  27. 27. Appendix Applying the Jensen’s inequality f (x) = log (x), concave function p(x (i) , z (i) ; θ) p(x (i) , z (i) ; θ) f Ez (i) ∼Qi ≥ Ez (i) ∼Qi f Qi (z (i) ) Qi (z (i) ) ReturnDuc-Hieu Trantdh.net [at] gmail.com (NTU) EM in pLSA July 27, 2010 27 / 27

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