Machine Learning Summer School, Austin, TX January 08, 2015

Transcription

1 Parametric Department of Information, Risk, and Operations Management Department of Statistics and Data Sciences The University of Texas at Austin Machine Learning Summer School, Austin, TX January 08, / 45

2 for Part of Poisson, gamma, and negative inference for the negative Sparse codes Regression Images for Dictionary counts P N Latent variable X = P K K N modelsφ Θ for discrete data Latent Dirichlet allocation Poisson factor Words Documents Matrix P N X Topics = P K Words Φ Topics Documents K N Θ 2 / 45

3 data is common Nonnegative and discrete: Number of auto insurance claims / highway accidents / crimes Consumer behavior, labor mobility, marketing, voting Photon counting Species sampling Text Infectious diseases, Google Flu Trends Next generation sequencing (statistical genomics) Mixture can be viewed as a count- problem Number of points in a cluster (mixture model, we are a count vector) Number of words assigned to topic k in document j (we are a K J latent count matrix in a topic model/mixed-membership model) 3 / 45

4 data is common Nonnegative and discrete: Number of auto insurance claims / highway accidents / crimes Consumer behavior, labor mobility, marketing, voting Photon counting Species sampling Text Infectious diseases, Google Flu Trends Next generation sequencing (statistical genomics) Mixture can be viewed as a count- problem Number of points in a cluster (mixture model, we are a count vector) Number of words assigned to topic k in document j (we are a K J latent count matrix in a topic model/mixed-membership model) 3 / 45

5 Siméon-Denis Poisson (21 June April 1840) "Life is good for only two things: doing mathematics and teaching it." Poisson 4 / 45

6 Siméon-Denis Poisson (21 June April 1840) "Life is good for only two things: doing mathematics and teaching it." Poisson 4 / 45

7 Poisson x Pois(λ) Probability mass function: P(x λ) = λx e λ, x {0, 1,...} x! The mean and variance is the same: E[x] = Var[x] = λ. Restrictive to model over-dispersed (variance greater than the mean) counts that are commonly observed in practice. A basic building block to construct more flexible count. Overdispersed are commonly observed due to Heterogeneity: difference individuals Contagion: dependence the occurrence of events 5 / 45

8 Mixed Poisson x Pois(λ), λ f Λ (λ) Mixing the Poisson rate parameter with a positive leads to a mixed Poisson. A mixed Poisson is always over-dispersed. Law of total expectation: Law of total variance: E[x] = E[E[x λ]] = E[λ]. Var[x] = Var[E[x λ]] + E[Var[x λ]] = Var[λ] + E[λ]. Thus Var[x] > E[x] unless λ is a constant. The gamma is a popular choice as it is conjugate to the Poisson. 6 / 45

9 Mixing the gamma with the Poisson as ( ) p x Pois(λ), λ Gamma r,, 1 p where p/(1 p) is the gamma scale parameter, leads to the negative x NB(r, p) with probability mass function P(x r, p) = Γ(x + r) x!γ(r) px (1 p) r, x {0, 1,...} 7 / 45

10 Compound Poisson A compound Poisson is the summation of a Poisson random number of i.i.d. random variables. If x = n i=1 y i, where n Pois(λ) and y i are i.i.d. random variable, then x is a compound Poisson random variable. The negative random variable x NB(r, p) can also be generated as a compound Poisson random variable as l x = u i, l Pois[ r ln(1 p)], u i Log(p) i=1 where u Log(p) is the logarithmic with probability mass function P(u p) = 1 p u ln(1 p) u, u {1, 2, }. 8 / 45

11 m NB(r, p) r is the dispersion parameter p is the probability parameter Probability mass function ( ) Γ(r + m) r f M (m r, p) = m!γ(r) pm (1 p) r = ( 1) m p m (1 p) r m It is a gamma-poisson mixture It is a compound Poisson rp Its variance is greater that its mean rp (1 p) 2 1 p Var[m] = E[m] + (E[m])2 r 9 / 45

12 The conjugate prior for the negative probability parameter p is the beta : if m i NB(r, p), p Beta(a 0, b 0 ), then ( ) n (p ) = Beta a 0 + m i, b 0 + nr i=1 The conjugate prior for the negative dispersion parameter r is unknown, but we have a simple data augmentation technique to derive closed-form Gibbs sampling update equations for r. 10 / 45

13 If we assign m customers to tables using a Chinese restaurant process with concentration parameter r, then the random number of occupied tables l follows the Chinese Restaurant Table (CRT) f L (l m, r) = Γ(r) Γ(m + r) s(m, l) r l, l = 0, 1,, m. s(m, l) are unsigned Stirling numbers of the first kind. The joint of the customer count m NB(r, p) and table count is the Poisson-logarithmic bivariate count s(m, l) r l f M,L (m, l r, p) = (1 p) r p m. m! 11 / 45

14 Poisson-logarithmic bivariate count Probability mass function: l s(m, l) r f M,L (m, l; r, p) = (1 p) r p m. m! It is clear that the gamma is a conjugate prior for r to this bivariate count. The joint of the customer count and table count are equivalent: Draw NegBino(r, p) customers Draw Poisson(--r ln (1 -- p)) tables Assign customers to tables using a Chinese restaurant process with concentration parameter r Draw Logarithmic(p) customers on each table 12 / 45

15 inference for the negative count : m i NegBino(r, p), p Beta(a 0, b 0 ), r Gamma(e 0, 1/f 0 ). Gibbs sampling via data augmetantion: (p ) Beta (a 0 + n i=1 m i, b 0 + nr) ; (l i ) = ( ) m i t=1 b t, b t Bernoulli r t+r 1 ; ( (r ) Gamma e 0 + ) n i=1 l 1 i, f 0 n ln(1 p). Expectation-Maximization Variational Bayes 13 / 45

16 inference for the negative count : m i NegBino(r, p), p Beta(a 0, b 0 ), r Gamma(e 0, 1/f 0 ). Gibbs sampling via data augmetantion: (p ) Beta (a 0 + n i=1 m i, b 0 + nr) ; (l i ) = ( ) m i t=1 b t, b t Bernoulli r t+r 1 ; ( (r ) Gamma e 0 + ) n i=1 l 1 i, f 0 n ln(1 p). Expectation-Maximization Variational Bayes 13 / 45

17 Gibbs sampling: E[r] = 1.076, E[p] = Expectation-Maximization: r : 1.025, p : Variational Bayes: E[r] = 0.999, E[p] = For this example, variational Bayes inference correctly identifies the modes but underestimates the posterior variances of model parameters. 14 / 45

18 Gibbs sampling: E[r] = 1.076, E[p] = Expectation-Maximization: r : 1.025, p : Variational Bayes: E[r] = 0.999, E[p] = For this example, variational Bayes inference correctly identifies the modes but underestimates the posterior variances of model parameters. 14 / 45

19 Gibbs sampling: E[r] = 1.076, E[p] = Expectation-Maximization: r : 1.025, p : Variational Bayes: E[r] = 0.999, E[p] = For this example, variational Bayes inference correctly identifies the modes but underestimates the posterior variances of model parameters. 14 / 45

20 gamma chain NegBino-Gamma-Gamma-... Augmentation (CRT, NegBino)-Gamma-Gamma-... Equivalence (Log, Poisson)-Gamma-Gamma-... Marginalization NegBino-Gamma / 45

21 gamma chain NegBino-Gamma-Gamma-... Augmentation (CRT, NegBino)-Gamma-Gamma-... Equivalence (Log, Poisson)-Gamma-Gamma-... Marginalization NegBino-Gamma / 45

22 gamma chain NegBino-Gamma-Gamma-... Augmentation (CRT, NegBino)-Gamma-Gamma-... Equivalence (Log, Poisson)-Gamma-Gamma-... Marginalization NegBino-Gamma / 45

23 gamma chain NegBino-Gamma-Gamma-... Augmentation (CRT, NegBino)-Gamma-Gamma-... Equivalence (Log, Poisson)-Gamma-Gamma-... Marginalization NegBino-Gamma / 45

24 gamma chain NegBino-Gamma-Gamma-... Augmentation (CRT, NegBino)-Gamma-Gamma-... Equivalence (Log, Poisson)-Gamma-Gamma-... Marginalization NegBino-Gamma / 45

25 Poisson and multinomial Suppose that x 1,..., x K are independent Poisson random variables with x k Pois(λ k ), x = K k=1 x k. Set λ = K k=1 λ k; let (y, y 1,..., y K ) be random variables such that ( ) y Pois(λ), (y 1,..., y k ) y Mult y; λ 1 λ,..., λ K λ. Then the of x = (x, x 1,..., x K ) is the same as the of y = (y, y 1,..., y K ). 16 / 45

26 Model: Multinomial and Dirichlet (x i1,..., x ik ) Multinomial(n i, p 1,..., p k ), (p 1,..., p k ) Dirichlet(α 1,..., α k ) = Γ( k j=1 α j) k j=1 Γ(α j) The conditional posterior of (p 1,..., p k ) is Dirichlet distributed as ( (p 1,..., p k ) Dirichlet α 1 + i k j=1 x i1,..., α k + i p α j 1 j x ik ) 17 / 45

27 Gamma and Dirichlet Suppose that random variables y and (y 1,..., y K ) are independent with y Gamma(γ, 1/c), (y 1,..., y K ) Dir(γp 1,, γp K ) where K k=1 p k = 1; Let x k = yy k then {x k } 1,K are independent gamma random variables with x k Gamma(γp k, 1/c). The proof can be found in arxiv: v1 18 / 45

28 various Modeling Mixture Modeling Gaussian Logit Logarithmic Polya-Gamma Binomial Chinese Restaurant Beta Bernoulli Latent Gaussian Poisson Multinomial Gamma Dirichlet 19 / 45

29 Poisson regression Model: y i Pois(λ i ), λ i = exp(x T i β) Model assumption: Var[y i x i ] = E[y i x i ] = exp(x T i β). Poisson regression does not model over-dispersion. 20 / 45

30 Model: Model property: Poisson regression with multiplicative random effects y i Pois(λ i ), λ i exp(x T i β)ɛ i Var[y i x i ] = E[y i x i ] + Var[ɛ i] E 2 [ɛ i ] E2 [y i x i ]. regression (gamma random effect): ɛ i Gamma(r, 1/r) = r r Γ(r) ɛ i r 1 e rɛ i. Lognormal-Poisson regression (lognormal random effect): ɛ i ln N (0, σ 2 ). 21 / 45

31 Model: Model property: Poisson regression with multiplicative random effects y i Pois(λ i ), λ i exp(x T i β)ɛ i Var[y i x i ] = E[y i x i ] + Var[ɛ i] E 2 [ɛ i ] E2 [y i x i ]. regression (gamma random effect): ɛ i Gamma(r, 1/r) = r r Γ(r) ɛ i r 1 e rɛ i. Lognormal-Poisson regression (lognormal random effect): ɛ i ln N (0, σ 2 ). 21 / 45

32 Model: Model property: Poisson regression with multiplicative random effects y i Pois(λ i ), λ i exp(x T i β)ɛ i Var[y i x i ] = E[y i x i ] + Var[ɛ i] E 2 [ɛ i ] E2 [y i x i ]. regression (gamma random effect): ɛ i Gamma(r, 1/r) = r r Γ(r) ɛ i r 1 e rɛ i. Lognormal-Poisson regression (lognormal random effect): ɛ i ln N (0, σ 2 ). 21 / 45

33 Lognormal and gamma mixed negative regression Model (Zhou et al., ICML2012): y i NegBino (r, p i ), r Gamma(a 0, 1/h) inference with the Polya-Gamma. Model properties: ( ) Var[y i x i ] = E[y i x i ] + e σ2 (1 + r 1 ) 1 E 2 [y i x i ]. Special cases: regression: σ 2 = 0; Lognormal-Poisson regression: r ; Poisson regression: σ 2 = 0 and r. 22 / 45

34 Lognormal and gamma mixed negative regression Model (Zhou et al., ICML2012): y i NegBino (r, p i ), r Gamma(a 0, 1/h) ( ) pi ψ i = log = x T i β + ln ɛ i, ɛ i ln N (0, σ 2 ) 1 p i inference with the Polya-Gamma. Model properties: ( ) Var[y i x i ] = E[y i x i ] + e σ2 (1 + r 1 ) 1 E 2 [y i x i ]. Special cases: regression: σ 2 = 0; Lognormal-Poisson regression: r ; Poisson regression: σ 2 = 0 and r. 22 / 45

35 Lognormal and gamma mixed negative regression Model (Zhou et al., ICML2012): y i NegBino (r, p i ), r Gamma(a 0, 1/h) ( ) pi ψ i = log = x T i β + ln ɛ i, ɛ i ln N (0, σ 2 ) 1 p i inference with the Polya-Gamma. Model properties: ( ) Var[y i x i ] = E[y i x i ] + e σ2 (1 + r 1 ) 1 E 2 [y i x i ]. Special cases: regression: σ 2 = 0; Lognormal-Poisson regression: r ; Poisson regression: σ 2 = 0 and r. 22 / 45

36 Lognormal and gamma mixed negative regression Model (Zhou et al., ICML2012): y i NegBino (r, p i ), r Gamma(a 0, 1/h) ( ) pi ψ i = log = x T i β + ln ɛ i, ɛ i ln N (0, σ 2 ) 1 p i inference with the Polya-Gamma. Model properties: ( ) Var[y i x i ] = E[y i x i ] + e σ2 (1 + r 1 ) 1 E 2 [y i x i ]. Special cases: regression: σ 2 = 0; Lognormal-Poisson regression: r ; Poisson regression: σ 2 = 0 and r. 22 / 45

37 Lognormal and gamma mixed negative regression Model (Zhou et al., ICML2012): y i NegBino (r, p i ), r Gamma(a 0, 1/h) ( ) pi ψ i = log = x T i β + ln ɛ i, ɛ i ln N (0, σ 2 ) 1 p i inference with the Polya-Gamma. Model properties: ( ) Var[y i x i ] = E[y i x i ] + e σ2 (1 + r 1 ) 1 E 2 [y i x i ]. Special cases: regression: σ 2 = 0; Lognormal-Poisson regression: r ; Poisson regression: σ 2 = 0 and r. 22 / 45

38 example on the NASCAR dataset: Model Poisson NB LGNB LGNB Parameters (MLE) (MLE) (VB) (Gibbs) σ 2 N/A N/A r N/A β β 1 (Laps) β 2 (Drivers) β 3 (TrkLen) Using Variational Bayes inference, we can calculate the correlation matrix for (β 1, β 2, β 3 ) T as / 45

39 Latent Dirichlet allocation Poisson factor Latent Dirichlet allocation (Blei et al., 2003) Hierarchical model: x ji Mult(φ zji ) z ji Mult(θ j ) φ k Dir(η,..., η) ( α θ j Dir K,..., α ) K There are K topics {φ k } 1,K, each of which is a over the V words in the vocabulary. There are N documents in the corpus and θ j represents the proportion of the K topics in the jth document. x ji is the ith word in the jth document. z ji is the index of the topic selected by x ji. 24 / 45

40 Latent Dirichlet allocation Poisson factor Denote n vjk = i δ(x ji = v)δ(z ji = k), n v k = j n vjk, n jk = v n vjk, and n k = j n jk. Blocked Gibbs sampling: P(z ji = k ) φ xji kθ jk, k {1,..., K} (φ k ) Dir(η + n 1 k,..., η + n V k ) ( α (θ j ) Dir K + n j1,..., α ) K + n jk Variational Bayes inference (Blei et al., 2003). 25 / 45

41 Latent Dirichlet allocation Poisson factor Collapsed Gibbs sampling (Griffiths and Steyvers, 2004): Marginalizing out both the topics {φ k } 1,K and the topic proportions {θ j } 1,N. Sample z ji conditioning on all the other topic assignment indices z ji : P(z ji = k z ji ) η + n ji ( x ji k V η + n ji n ji jk + α ), k {1,..., K} K k This is easy to understand as P(z ji = k φ k, θ j ) φ xji kθ jk P(z ji = k z ji ) = P(z ji = k φ k, θ j )P(φ k, θ j z ji )dφ k dθ j P(φ k z ji ) = Dir(η + n ji 1 k,..., η + n ji V k ) ( α P(θ j z ji ) = Dir K + n ji j1,..., α ) K + n ji jk P(φ k, θ j z ji ) = P(φ k z ji )P(θ j z ji ) 26 / 45

42 Latent Dirichlet allocation Poisson factor In latent Dirichlet allocation, the words in a document are assumed to be exchangeable (bag-of-words assumption). Sparse codes Below we will Images relate latent Dirichlet Dictionary allocation to Poisson P N factor X and show = P K K N it essentially Φ tries Θ to factorize the term-document word count matrix under the Poisson likelihood: Words Documents Matrix P N X Topics = P K Words Φ Topics Documents K N Θ 27 / 45

43 Latent Dirichlet allocation Poisson factor Poisson factor alaysis Factorize the term-document word count matrix M Z V N + under the Poisson likelihood as M Pois(ΦΘ) where Z + = {0, 1,...} and R + = {x : x > 0}. m vj is the number of times that term v appears in document j. Factor loading matrix: Φ = (φ 1,..., φ K ) R V K +. Factor score matrix: Θ = (θ 1,..., θ N ) R K N +. A large number of discrete latent variable models can be united under the Poisson factor framework, with the main differences on how the priors for φ k and θ j are constructed. 28 / 45

44 Latent Dirichlet allocation Poisson factor Two equivalent augmentations Poisson factor ( K ) m vj Pois φ vk θ jk Augmentation 1: m vj = k=1 K n vjk, n vjk Pois(φ vk θ jk ) k=1 Augmentation 2: ( K ) φ vk θ jk m vj Pois φ vk θ jk, ζ vjk = K k=1 φ vkθ jk k=1 [n vj1,, n vjk ] Mult (m vj ; ζ vj1,, ζ vjk ) 29 / 45

45 Latent Dirichlet allocation Poisson factor Nonnegative matrix and gamma-poisson factor Gamma priors on Φ and( Θ: K ) m vj = Pois k=1 φ vkθ jk φ vk Gamma(a φ, 1/b φ ), θ jk Gamma(a θ, g k /a θ ). Expectation-Maximization (EM) algorithm: φ vk = φ vk θ jk = θ jk a φ 1 φ vk a θ 1 θ jk + N i=1 b φ + θ k m vj θ jk K k=1 φ vkθ jk m vj φ vk K k=1 φ vkθ jk + P p=1. a θ /g k + φ k If we set b φ = 0, a φ = a θ = 1 and g k =, then the EM algorithm is the same as those of non-negative matrix (Lee and Seung, 2000) with an objective function of minimizing the KL divergence D KL (M ΦΘ). 30 / 45

46 Latent Dirichlet allocation and Dirichlet-Poisson factor Latent Dirichlet allocation Poisson factor Dirichlet priors on Φ and( Θ: K ) m vj = Pois k=1 φ vkθ jk φ k Dir(η,..., η), θ j Dir(α/K,..., α/k). One may show that both the block Gibbs sampling inference and variational Bayes inference of the Dirichlet-Poisson factor model are the same as that of the Latent Dirichlet allocation. 31 / 45

47 Latent Dirichlet allocation Poisson factor Beta-gamma-Poisson factor Hierachical model (Zhou et al., 2012, Carin, 2014): m vj = K n vjk, n vjk Pois(φ vk θ jk ) k=1 φ k Dir (η,, η), θ jk Gamma [r j, p k /(1 p k )], r j Gamma(e 0, 1/f 0 ), p k Beta[c/K, c(1 1/K)]. n jk = V v=1 n vjk NB(r j, p k ) This parametric model becomes a nonparametric model governed by the beta-negative process as K. 32 / 45

48 Latent Dirichlet allocation Poisson factor Gamma-gamma-Poisson factor Hierachical model ( Carin, 2014): m vj = n jk NB(r k, p j ) K n vjk, n vjk Pois(φ vk θ jk ) k=1 φ k Dir (η,, η), θ jk Gamma [r k, p j /(1 p j )], p j Beta(a 0, b 0 ), r k Gamma(γ 0 /K, 1/c). This parametric model becomes a nonparametric model governed by the gamma-negative process as K. 33 / 45

49 Latent Dirichlet allocation Poisson factor Poisson factor and mixed-membership We may represent the Poisson factor m vj = K n vjk, n vjk Pois(φ vk θ jk ) k=1 in terms of a mixed-membership model, whose group sizes are randomized, as ( ) K θ jk x ji Mult(φ zji ), z ji k θ δ k, m j Pois θ jk, jk k k=1 where i = 1,..., m j in the jth document, and n vjk = m j i=1 δ(x ji = v)δ(z ji = k). The likelihoods of the two representations are different update to a multinomial coefficient (Zhou, 2014). 34 / 45

50 Latent Dirichlet allocation Poisson factor Connections to previous approaches Nonnegative matrix (K-L divergence) (NMF) Latent Dirichlet allocation (LDA) GaP: gamma-poisson factor model (GaP) (Canny, 2004) Hierarchical Dirichlet process LDA (HDP-LDA) (Teh et al., 2006) Poisson factor Infer Infer Support Related priors on θ jk (p k, r j ) (p j, r k ) K algorithms gamma NMF Dirichlet LDA beta-gamma GaP gamma-gamma HDP-LDA 35 / 45

51 Latent Dirichlet allocation Poisson factor Blocked Gibbs sampling Sample z ji from multinomial; n vjk = m j i=1 δ(x ji = v)δ(z ji = k). Sample φ k from Dirichlet For the beta-negative model (beta-gamma-poisson factor ) Sample l jk from CRT(n jk, r j ) Sample r j from gamma Sample p k from beta Sample θ jk from Gamma(r j + n jk, p k ) For the gamma-negative model (gamma-gamma-poisson factor ) Sample l jk from CRT(n jk, r k ) Sample r k from gamma Sample p j from beta Sample θ jk from Gamma(r k + n jk, p j ) Collapsed Gibbs sampling for the beta-negative model can be found in (Zhou, 2014). 36 / 45

52 Latent Dirichlet allocation Poisson factor Example application Example Topics of United Nation General Assembly Resolutions inferred by the gamma-gamma-poisson factor : Topic 1 trade world conference organization negotiations Topic 2 rights human united nations commission Topic 3 environment management protection affairs appropriate Topic 4 women gender equality including system Topic 5 economic summits outcomes conferences major The gamma-negative and beta-negative models have distinct mechanisms on controlling the number of inferred factors. They produce state-of-the-art perplexity results when used for topic of a document corpus (Zhou et al, 2012, Zhou and Carin 2014, Zhou 2014). 37 / 45

53 Stochastic blockmodel A relational (graph) is commonly used to describe the relationship nodes, where a node could represent a person, a movie, a protein, etc. Two nodes are connected if there is an edge (link) them. An undirected unweighted relational with N nodes can be equivalently represented with a sysmetric binary affinity matrix B {0, 1} N N, where b ij = b ji = 1 if an edge exists nodes i and j and b ij = b ji = 0 otherwise. 38 / 45

54 Stochastic blockmodel Each node is assigned to a cluster. Stochastic blockmodel The probability for an edge to exist two nodes is solely decided by the clusters that they are assigned to. Hierachical model: b ij Bernoulli(p zi z j ), p k1 k 2 Beta(a 0, b 0 ), z i Mult(π 1,..., π K ), (π 1,..., π K ) Dir(α/K,..., α/k) Blocked Gibbs sampling: P(z i = k ) = π k j i p b ij for j > i kz j (1 p kzj ) 1 b ij 39 / 45

55 Infinite relational model (Kemp et al., 2006) Stochastic blockmodel As K, the stochastic block model becomes a nonparametric model governed by the Chinese restaurant process (CRP) with concentration parameter α: b ij Bernoulli(p zi z j ), p k1 k 2 Beta(a 0, b 0 ), (z 1,..., z N ) CRP(α) for i > j Collapsed Gibbs sampling can be derived by marginalizing out p k1 k 2 and using the prediction rule of the Chinese restaurant process. 40 / 45

56 Stochastic blockmodel The coauthor of the top 234 NIPS authors / 45

57 Stochastic blockmodel The reordered using the stochastic blockmodel / 45

58 The estimated link probabilities within and blocks Stochastic blockmodel / 45

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60 M. L. Carin. Augment-and-conquer negative processes. In NIPS, M. L. Carin. process count and mixture. IEEE TPAMI, M. Zhou. Beta-negative process and exchangeable random partitions for mixed-membership. In NIPS, / 45