nips nips2009 nips2009-40 knowledge-graph by maker-knowledge-mining

40 nips-2009-Bayesian Nonparametric Models on Decomposable Graphs

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Author: Francois Caron, Arnaud Doucet

Abstract: Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. These models are attractive because they ensure exchangeability (over samples). We propose here extensions of these models where the dependency between samples is given by a known decomposable graph. These models have appealing properties and can be easily learned using Monte Carlo techniques. 1 Motivation The CRP and IBP have found numerous applications in machine learning over recent years [5, 10]. We consider here the case where the data we are interested in are ‘locally’ dependent; these dependencies being represented by a known graph G where each data point/object is associated to a vertex. These local dependencies can correspond to any conceptual or real (e.g. space, time) metric. For example, in the context of clustering, we might want to propose a prior distribution on partitions enforcing that data which are ‘close’ in the graph are more likely to be in the same cluster. Similarly, in the context of latent feature models, we might be interested in a prior distribution on features enforcing that data which are ‘close’ in the graph are more likely to possess similar features. The ‘standard’ CRP and IBP correspond to the case where the graph G is complete; that is it is fully connected. In this paper, we generalize the CRP and IBP to decomposable graphs. The resulting generalized versions of the CRP and IBP enjoy attractive properties. Each clique of the graph follows marginally a CRP or an IBP process and explicit expressions for the joint prior distribution on the graph is available. It makes it easy to learn those models using straightforward generalizations of Markov chain Monte Carlo (MCMC) or Sequential Monte Carlo (SMC) algorithms proposed to perform inference for the CRP and IBP [5, 10, 14]. The rest of the paper is organized as follows. In Section 2, we review the popular Dirichlet multinomial allocation model and the Dirichlet Process (DP) partition distribution. We propose an extension of these two models to decomposable graphical models. In Section 3 we discuss nonparametric latent feature models, reviewing brieﬂy the construction in [5] and extending it to decomposable graphs. We demonstrate these models in Section 4 on two applications: an alternative to the hierarchical DP model [12] and a time-varying matrix factorization problem. 2 Prior distributions for partitions on decomposable graphs Assume we have n observations. When performing clustering, we associate to each of this observation an allocation variable zi ∈ [K] = {1, . . . , K}. Let Πn be the partition of [n] = {1, . . . , n} deﬁned by the equivalence relation i ↔ j ⇔ zi = zj . The resulting partition Πn = {A1 , . . . , An(Πn ) } 1 is an unordered collection of disjoint non-empty subsets Aj of [n], j = 1, . . . , n(Πn ), where ∪j Aj = [n] and n(Πn ) is the number of subsets for partition Πn . We also denote by Pn be the set of all partitions of [n] and let nj , j = 1, . . . , n(Πn ), be the size of the subset Aj . Each allocation variable zi is associated to a vertex/site of an undirected graph G, which is assumed to be known. In the standard case where the graph G is complete, we ﬁrst review brieﬂy here two popular prior distributions on z1:n , equivalently on Πn . We then extend these models to undirected decomposable graphs; see [2, 8] for an introduction to decomposable graphs. Finally we brieﬂy discuss the directed case. Note that the models proposed here are completely different from the hyper multinomial-Dirichlet in [2] and its recent DP extension [6]. 2.1 Dirichlet multinomial allocation model and DP partition distribution Assume for the time being that K is ﬁnite. When the graph is complete, a popular choice for the allocation variables is to consider a Dirichlet multinomial allocation model [11] θ θ , . . . , ), zi |π ∼ π (1) K K where D is the standard Dirichlet distribution and θ > 0. Integrating out π, we obtain the following Dirichlet multinomial prior distribution π ∼ D( Pr(z1:n ) = K j=1 Γ(θ) Γ(nj + θ K) (2) θ Γ(θ + n)Γ( K )K and then, using the straightforward equality Pr(Πn ) = PK where PK = {Πn ∈ Pn |n(Πn ) ≤ K}, we obtain K! (K−n(Πn ))! Pr(z1:n ) valid for for all Πn ∈ n(Π ) Pr(Πn ) = θ Γ(θ) j=1n Γ(nj + K ) K! . θ (K − n(Πn ))! Γ(θ + n)Γ( K )n(Πn ) (3) DP may be seen as a generalization of the Dirichlet multinomial model when the number of components K → ∞; see for example [10]. In this case the distribution over the partition Πn of [n] is given by [11] n(Π ) θn(Πn ) j=1n Γ(nj ) . (4) Pr(Πn ) = n i=1 (θ + i − 1) Let Π−k = {A1,−k , . . . , An(Π−k ),−k } be the partition induced by removing item k to Πn and nj,−k be the size of cluster j for j = 1, . . . , n(Π−k ). It follows from (4) that an item k is assigned to an existing cluster j, j = 1, . . . , n(Π−k ), with probability proportional to nj,−k / (n − 1 + θ) and forms a new cluster with probability θ/ (n − 1 + θ). This property is the basis of the CRP. We now extend the Dirichlet multinomial allocation and the DP partition distribution models to decomposable graphs. 2.2 Markov combination of Dirichlet multinomial and DP partition distributions Let G be a decomposable undirected graph, C = {C1 , . . . , Cp } a perfect ordering of the cliques and S = {S2 , . . . , Cp } the associated separators. It can be easily checked that if the marginal distribution of zC for each clique C ∈ C is deﬁned by (2) then these distributions are consistent as they yield the same distribution (2) over the separators. Therefore, the unique Markov distribution over G with Dirichlet multinomial distribution over the cliques is deﬁned by [8] Pr(zC ) S∈S Pr(zS ) C∈C Pr(z1:n ) = (5) where for each complete set B ⊆ G, we have Pr(zB ) given by (2). It follows that we have for any Πn ∈ PK Γ(θ) K! Pr(Πn ) = (K − n(Πn ))! C∈C Γ(θ) S∈S 2 K j=1 θ Γ(nj,C + K ) θ Γ(θ+nC )Γ( K )K K j=1 θ Γ(nj,S + K ) θ Γ(θ+nS )Γ( K )K (6) where for each complete set B ⊆ G, nj,B is the number of items associated to cluster j, j = 1, . . . , K in B and nB is the total number of items in B. Within each complete set B, the allocation variables deﬁne a partition distributed according to the Dirichlet-multinomial distribution. We now extend this approach to DP partition distributions; that is we derive a joint distribution over Πn such that the distribution of ΠB over each complete set B of the graph is given by (4) with θ > 0. Such a distribution satisﬁes the consistency condition over the separators as the restriction of any partition distributed according to (4) still follows (4) [7]. G Proposition. Let Pn be the set of partitions Πn ∈ Pn such that for each decomposition A, B, and any (i, j) ∈ A × B, i ↔ j ⇒ ∃k ∈ A ∩ B such that k ↔ i ↔ j. As K → ∞, the prior distribution G over partitions (6) is given for each Πn ∈ Pn by Pr(Πn ) = θn(Πn ) n(ΠC ) Γ(nj,C ) j=1 nC i=1 (θ+i−1) n(ΠS ) Γ(nj,S ) j=1 nS (θ+i−1) i=1 C∈C S∈S (7) where n(ΠB ) is the number of clusters in the complete set B. Proof. From (6), we have θ n(ΠC ) K(K − 1) . . . (K − n(Πn ) + 1) Pr(Πn ) = K C∈C n(ΠC )− S∈S n(ΠS ) C∈C θ n(ΠS ) S∈S n(ΠC ) θ Γ(nj,C + K ) j=1 nC (θ+i−1) i=1 n(ΠS ) θ Γ(nj,S + K ) j=1 nS (θ+i−1) i=1 Thus when K → ∞, we obtain (7) if n(Πn ) = C∈C n(ΠC ) − S∈S n(ΠS ) and 0 otherwise. We have n(Πn ) ≤ C∈C n(ΠC ) − S∈S n(ΠS ) for any Πn ∈ Pn and the subset of Pn verifying G n(Πn ) = C∈C n(ΠC ) − S∈S n(ΠS ) corresponds to the set Pn . Example. Let the notation i ∼ j (resp. i j) indicates an edge (resp. no edge) between two sites. Let n = 3 and G be the decomposable graph deﬁned by the relations 1 ∼ 2, 2 ∼ 3 and 1 3. G The set P3 is then equal to {{{1, 2, 3}}; {{1, 2}, {3}}; {{1}, {2, 3}}; {{1}, {2}, {3}}}. Note that G the partition {{1, 3}, {2}} does not belong to P3 . Indeed, as there is no edge between 1 and 3, they cannot be in the same cluster if 2 is in another cluster. The cliques are C1 = {1, 2} and C2 = {2, 3} Pr(ΠC1 ) Pr(ΠC2 ) hence we can and the separator is S2 = {2}. The distribution is given by Pr(Π3 ) = Pr(ΠS ) 2 check that we obtain Pr({1, 2, 3}) = (θ + 1)−2 , Pr({1, 2}, {3}) = Pr({1, 2}, {3}) = θ(θ + 1)−2 and Pr({1}, {2}, {3}) = θ2 (θ + 1)−2 . Let now deﬁne the full conditional distributions. Based on (7) the conditional assignment of an item k is proportional to the conditional over the cliques divided by the conditional over the separators. G Let denote G−k the undirected graph obtained by removing vertex k from G. Suppose that Πn ∈ Pn . G−k If Π−k ∈ Pn−1 , then do not change the value of item k. Otherwise, item k is assigned to cluster j / where j = 1, . . . , n(Π−k ) with probability proportional to {C∈C|n−k,j,C >0} n−k,j,C {S∈S|n−k,j,S >0} n−k,j,S (8) and to a new cluster with probability proportional to θ, where n−k,j,C is the number of items in the set C \ {k} belonging to cluster j. The updating process is illustrated by the Chinese wedding party process1 in Fig. 1. The results of this section can be extended to the Pitman-Yor process, and more generally to species sampling models. Example (continuing). Given Π−2 = {A1 = {1}, A2 = {3}}, we have −1 Pr( item 2 assigned to A1 = {1}| Π−2 ) = Pr( item 2 assigned to A2 = {3}| Π−2 ) = (θ + 2) −1 and Pr( item 2 assigned to new cluster A3 | Π−2 ) = θ (θ + 2) . Given Π−2 = {A1 = {1, 3}}, item 2 is assigned to A1 with probability 1. 1 Note that this representation describes the full conditionals while the CRP represents the sequential updat- ing. 3 (a) (b) (d) (c) (e) Figure 1: Chinese wedding party. Consider a group of n guests attending a wedding party. Each of the n guests may belong to one or several cliques, i.e. maximal groups of people such that everybody knows everybody. The belonging of each guest to the different cliques is represented by color patches on the ﬁgures, and the graphical representation of the relationship between the guests is represented by the graphical model (e). (a) Suppose that the guests are already seated such that two guests cannot be together at the same table is they are not part of the same clique, or if there does not exist a group of other guests such that they are related (“Any friend of yours is a friend of mine”). (b) The guest number k leaves his table and either (c) joins a table where there are guests from the same clique as him, with probability proportional to the product of the number of guests from each clique over the product of the number of guests belonging to several cliques on that table or (d) he joins a new table with probability proportional to θ. 2.3 Monte Carlo inference 2.3.1 MCMC algorithm Using the full conditionals, a single site Gibbs sampler can easily be designed to approximate the posterior distribution Pr(Πn |z1:n ). Given a partition Πn , an item k is taken out of the partition. If G−k Π−k ∈ Pn−1 , item k keeps the same value. Otherwise, the item will be assigned to a cluster j, / j = 1, . . . , n(Π−k ), with probability proportional to p(z{k}∪Aj,−k ) × p(zAj,−k ) {C∈C|n−k,j,C >0} n−k,j,C {S∈S|n−k,j,S >0} n−k,j,S (9) and the item will be assigned to a new cluster with probability proportional to p(z{k} ) × θ. Similarly to [3], we can also deﬁne a procedure to sample from p(θ|n(Πn ) = k)). We assume that θ ∼ G(a, b) and use p auxiliary variables x1 , . . . , xp . The procedure is as follows. • For j = 1, . . . , p, sample xj |k, θ ∼ Beta(θ + nSj , nCj − nSj ) • Sample θ|k, x1:p ∼ G(a + k, b − j log xj ) 2.3.2 Sequential Monte Carlo We have so far only treated the case of an undirected decomposable graph G. We can formulate a sequential updating rule for the corresponding perfect directed version D of G. Indeed, let (a1 , . . . a|V | ) be a perfect ordering and pa(ak ) be the set of parents of ak which is by deﬁnition complete. Let Πk−1 = {A1,k−1 , . . . , An(Πk−1 ),k−1 } denote the partition of the ﬁrst k−1 vertices a1:k−1 and let nj,pa(ak ) be the number of elements with value j in the set pa(ak ), j = 1, . . . , n(Πk−1 ). Then the vertex ak joins the set j with probability nj,pa(ak ) / θ + cluster with probability θ/ θ + q q nq,pa(ak ) and creates a new nq,pa(ak ) . One can then design a particle ﬁlter/SMC method in a similar fashion as [4]. Consider a set of (i) (i) (i) (i) N N particles Πk−1 with weights wk−1 ∝ Pr(Πk−1 , z1:k−1 ) ( i=1 wk−1 = 1) that approximate (i) the posterior distribution Pr(Πk−1 |z1:k−1 ). For each particle i, there are n(Πk−1 ) + 1 possible 4 (i,j) allocations for component ak . We denote Πk the partition obtained by associating component ak (i,j) to cluster j. The weight associated to Πk is given by  nj,pa(ak ) (i)  if j = 1, . . . , n(Πk−1 ) θ+ q nq,pa(ak ) (i,j) (i) p(z{ak }∪Aj,k−1 ) wk−1 = wk−1 × (10) (i) θ  θ+ n p(zAj,k−1 ) if j = n(Πk−1 ) + 1 q q,pa(ak ) (i,j) Then we can perform a deterministic resampling step by keeping the N particles Πk with highest (i,j) (i) (i) weights wk−1 . Let Πk be the resampled particles and wk the associated normalized weights. 3 Prior distributions for inﬁnite binary matrices on decomposable graphs Assume we have n objects; each of these objects being associated to the vertex of a graph G. To K each object is associated a K-dimensional binary vector zn = (zn,1 , . . . , zn,K ) ∈ {0, 1} where zn,i = 1 if object n possesses feature i and zn,i = 0 otherwise. These vectors zt form a binary n × K matrix denoted Z1:n . We denote by ξ1:n the associated equivalence class of left-ordered matrices and let EK be the set of left-ordered matrices with at most K features. In the standard case where the graph G is complete, we review brieﬂy here two popular prior distributions on Z1:n , equivalently on ξ1:n : the Beta-Bernoulli model and the IBP [5]. We then extend these models to undirected decomposable graphs. This can be used for example to deﬁne a time-varying IBP as illustrated in Section 4. 3.1 Beta-Bernoulli and IBP distributions The Beta-Bernoulli distribution over the allocation Z1:n is K Pr(Z1:n ) = α + K )Γ(n − nj + 1) α Γ(n + 1 + K ) α K Γ(nj j=1 (11) where nj is the number of objects having feature j. It follows that Pr(ξ1:n ) = K K! 2n −1 h=0 α K Γ(nj α + K )Γ(n − nj + 1) α Γ(n + 1 + K ) Kh ! j=1 (12) where Kh is the number of features possessing the history h (see [5] for details). The nonparametric model is obtained by taking the limit when K → ∞ Pr(ξ1:n ) = αK K+ + 2n −1 h=1 Kh ! exp(−αHn ) where K + is the total number of features and Hn = 3.2 (n − nj )!(nj − 1)! n! j=1 n 1 k=1 k . (13) The IBP follows from (13). Markov combination of Beta-Bernoulli and IBP distributions Let G be a decomposable undirected graph, C = {C1 , . . . , Cp } a perfect ordering of the cliques and S = {S2 , . . . , Cp } the associated separators. As in the Dirichlet-multinomial case, it is easily seen that if for each clique C ∈ C, the marginal distribution is deﬁned by (11), then these distributions are consistent as they yield the same distribution (11) over the separators. Therefore, the unique Markov distribution over G with Beta-Bernoulli distribution over the cliques is deﬁned by [8] Pr(ZC ) S∈S Pr(ZS ) C∈C Pr(Z1:n ) = (14) where Pr(ZB ) given by (11) for each complete set B ⊆ G. The prior over ξ1:n is thus given, for ξ1:n ∈ EK , by Pr(ξ1:n ) = K! 2n −1 h=0 Kh ! α K α Γ(nj,C + K )Γ(nC −nj,C +1) α Γ(nC +1+ K ) α α Γ(nj,S + K )Γ(nS −nj,S +1) K K α j=1 Γ(nS +1+ K ) K j=1 C∈C S∈S 5 (15) where for each complete set B ⊆ G, nj,B is the number of items having feature j, j = 1, . . . , K in the set B and nB is the whole set of objects in set B. Taking the limit when K → ∞, we obtain after a few calculations Pr(ξ1:n ) = α + K[n] exp [−α ( C HnC − 2n −1 h=1 Kh ! HnS )] × C∈C + KC (nC −nj,C )!(nj,C −1)! j=1 nC ! S∈S S + KS (nS −nj,S )!(nj,S −1)! j=1 nS ! + + + + if K[n] = C KC − S KS and 0 otherwise, where KB is the number of different features possessed by objects in B. G Let En be the subset of En such that for each decomposition A, B and any (u, v) ∈ A × B: {u and v possess feature j} ⇒ ∃k ∈ A ∩ B such that {k possesses feature j}. Let ξ−k be the left-ordered + matrix obtained by removing object k from ξn and K−k be the total number of different features in G−k + ξ−k . For each feature j = 1, . . . , K−k , if ξ−k ∈ En−1 then we have   b C∈C nj,C if i = 1 S∈C nj,S Pr(ξk,j = i) = (16)  b C∈C (nC −nj,C ) if i = 0 (nS −nj,S ) S∈C nS where b is the appropriate normalizing constant then the customer k tries Poisson α {S∈S|k∈S} nC {C∈C|k∈C} new dishes. We can easily generalize this construction to a directed version D of G using arguments similar to those presented in Section 2; see Section 4 for an application to time-varying matrix factorization. 4 4.1 Applications Sharing clusters among relative groups: An alternative to HDP Consider that we are given d groups with nj data yi,j in each group, i = 1, . . . , nj , j = 1, . . . , d. We consider latent cluster variables zi,j that deﬁne the partition of the data. We will use alternatively the notation θi,j = Uzi,j in the following. Hierarchical Dirichlet Process [12] (HDP) is a very popular model for sharing clusters among related groups. It is based on a hierarchy of DPs G0 ∼ DP (γ, H), Gj |G0 ∼ DP (α, G0 ) j = 1, . . . d θi,j |Gj ∼ Gj , yi,j |θi,j ∼ f (θi,j ) i = 1, . . . , nj . Under conjugacy assumptions, G0 , Gj and U can be integrated out and we can approximate the marginal posterior of (zi,j ) given y = (yi,j ) with Gibbs sampling using the Chinese restaurant franchise to sample from the full conditional p(zi,j |z−{i,j} , y). Using the graph formulation deﬁned in Section 2, we propose an alternative to HDP. Let θ0,1 , . . . , θ0,N be N auxiliary variables belonging to what we call group 0. We deﬁne each clique Cj (j = 1, . . . , d) to be composed of elements from group j and elements from group 0. This deﬁnes a decomposable graphical model whose separator is given by the elements of group 0. We can rewrite the model in a way quite similar to HDP G0 ∼ DP (α, H), θ0,i |G0 ∼ G0 i = 1, ..., N α α Gj |θ0,1 , . . . , θ0,N ∼ DP (α + N, α+N H + α+N θi,j |Gj ∼ Gj , yi,j |θi,j ∼ f (θi,j ) i = 1, . . . , nj N i=1 δθ0,i ) j = 1, . . . d, N For any subset A and j = k ∈ {1, . . . , p} we have corr(Gj (A), Gk (A)) = α+N . Again, under conjugacy conditions, we can integrate out G0 , Gj and U and approximate the marginal posterior distribution over the partition using the Chinese wedding party process deﬁned in Section 2. Note that for latent variables zi,j , j = 1, . . . , d, associated to data, this is the usual CRP update. As in HDP, multiple layers can be added to the model. Figures 2 (a) and (b) resp. give the graphical DP alternative to HDP and 2-layer HDP. 6 z0 root z0 root corpora docs z1 z2 z1 z2 z3 z1,1 z1,2 z2,1 z2,2 z2,3 docs (a) Graphical DP alternative to HDP (b) Graphical DP alternative to 2-layer HDP Figure 2: Hierarchical Graphs of dependency with (a) one layer and (b) two layers of hierarchy. If N = 0, then Gj ∼ DP (α, H) for all j and this is equivalent to setting γ → ∞ in HDP. If N → ∞ then Gj = G0 for all j, G0 ∼ DP (α, H). This is equivalent to setting α → ∞ in the HDP. One interesting feature of the model is that, contrary to HDP, the marginal distribution of Gj at any layer of the tree is DP (α, H). As a consequence, the total number of clusters scales logarithmically (as in the usual DP) with the size of each group, whereas it scales doubly logarithmically in HDP. Contrary to HDP, there are at most N clusters shared between different groups. Our model is in that sense reminiscent of [9] where only a limited number of clusters can be shared. Note however that contrary to [9] we have a simple CRP-like process. The proposed methodology can be straightforwardly extended to the inﬁnite HMM [12]. The main issue of the proposed model is the setting of the number N of auxiliary parameters. Another issue is that to achieve high correlation, we need a large number of auxiliary variables. Nonetheless, the computational time used to sample from auxiliary variables is negligible compared to the time used for latent variables associated to data. Moreover, it can be easily parallelized. The model proposed offers a far richer framework and ensures that at each level of the tree, the marginal distribution of the partition is given by a DP partition model. 4.2 Time-varying matrix factorization Let X1:n be an observed matrix of dimension n × D. We want to ﬁnd a representation of this matrix in terms of two latent matrices Z1:n of dimension n × K and Y of dimension K × D. Here Z1:n 2 is a binary matrix whereas Y is a matrix of latent features. By assuming that Y ∼ N 0, σY IK×D and 2 X1:n = Z1:n Y + σX εn where εn ∼ N 0, σX In×D , we obtain p(X1:n |Z1:n ) ∝ −D/2 2 2 + Z+T Z+ + σX /σY IKn 1:n 1:n + (n−Kn )D σX exp − + Kn D σY 2 2 + where Σ−1 = I − Z+ Z+T Z+ + σX /σY IKn n 1:n 1:n 1:n −1 1 T −1 2 tr X1:n Σn X1:n 2σX (17) + Z+T , Kn the number of non-zero columns of 1:n + Z1:n and Z+ is the ﬁrst Kn columns of Z1:n . To avoid having to set K, [5, 14] assume that Z1:n 1:n follows an IBP. The resulting posterior distribution p(Z1:n |X1:n ) can be estimated through MCMC [5] or SMC [14]. We consider here a different model where the object Xt is assumed to arrive at time index t and we want a prior distribution on Z1:n ensuring that objects close in time are more likely to possess similar features. To achieve this, we consider the simple directed graphical model D of Fig. 3 where the site numbering corresponds to a time index in that case and a perfect numbering of D is (1, 2, . . .). The set of parents pa(t) is composed of the r preceding sites {{t − r}, . . . , {t − 1}}. The time-varying IBP to sample from p(Z1:n ) associated to this directed graph follows from (16) and proceeds as follows. At time t = 1 + new new • Sample K1 ∼Poisson(α), set z1,i = 1 for i = 1, ..., K1 and set K1 = Knew . At times t = 2, . . . , r n + new ∼Poisson( α ). • For k = 1, . . . Kt , sample zt,k ∼ Ber( 1:t−1,k ) and Kt t t 7 ? ? - t−r - t−r+1 - . . . - t−1 - t - t+1 6 6 Figure 3: Directed graph. At times t = r + 1, . . . , n n + α new ∼Poisson( r+1 ). • For k = 1, . . . Kt , sample zt,k ∼ Ber( t−r:t−1,k ) and Kt r+1 + Here Kt is the total number of features appearing from time max(1, t − r) to t − 1 and nt−r:t−1,k the restriction of n1:t−1 to the r last customers. Using (17) and the prior distribution of Z1:n which can be sampled using the time-varying IBP described above, we can easily design an SMC method to sample from p(Z1:n |X1:n ). We do not detail it here. Note that contrary to [14], our algorithm does not require inverting a matrix whose dimension grows linearly with the size of the data but only a matrix of dimension r × r. In order to illustrate the model and SMC algorithm, we create 200 6 × 6 images using a ground truth Y consisting of 4 different 6 × 6 latent images. The 200 × 4 binary matrix was generated from Pr(zt,k = 1) = πt,k , where πt = ( .6 .5 0 0 ) if t = 1, . . . , 30, πt = ( .4 .8 .4 0 ) if t = 31, . . . , 50 and πt = ( 0 .3 .6 .6 ) if t = 51, . . . , 200. The order of the model is set to r = 50. The feature occurences Z1:n and true features Y and their estimates are represented in Figure 4. Two spurious features are detected by the model (features 2 and 5 on Fig. 3(c)) but quickly discarded (Fig. 4(d)). The algorithm is able to correctly estimate the varying prior occurences of the features over time. Feature1 Feature2 Feature1 Feature2 Feature3 20 20 40 40 60 60 Feature4 80 100 Feature4 Feature5 Feature6 Time Feature3 Time 80 100 120 120 140 140 160 160 180 200 180 1 2 3 200 4 Feature (a) 1 2 3 4 5 6 Feature (b) (c) (d) Figure 4: (a) True features, (b) True features occurences, (c) MAP estimate ZM AP and (d) associated E[Y|ZM AP ] t=20 t=50 t=20 t=50 t=100 t=200 t=100 t=200 (a) (b) Figure 5: (a) E[Xt |πt , Y] and (b) E[Xt |X1:t−1 ] at t = 20, 50, 100, 200. 5 Related work and Discussion The ﬁxed-lag version of the time-varying DP of Caron et al. [1] is a special case of the proposed model when G is given by Fig. 3. The bivariate DP of Walker and Muliere [13] is also a special case when G has only two cliques. In this paper, we have assumed that the structure of the graph was known beforehand and we have shown that many ﬂexible models arise from this framework. It would be interesting in the future to investigate the case where the graphical structure is unknown and must be estimated from the data. Acknowledgment The authors thank the reviewers for their comments that helped to improve the writing of the paper. 8 References [1] F. Caron, M. Davy, and A. Doucet. Generalized Polya urn for time-varying Dirichlet process mixtures. In Uncertainty in Artiﬁcial Intelligence, 2007. [2] A.P. Dawid and S.L. Lauritzen. Hyper Markov laws in the statistical analysis of decomposable graphical models. The Annals of Statistics, 21:1272–1317, 1993. [3] M.D. Escobar and M. West. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90:577–588, 1995. [4] P. Fearnhead. Particle ﬁlters for mixture models with an unknown number of components. Statistics and Computing, 14:11–21, 2004. [5] T.L. Grifﬁths and Z. Ghahramani. Inﬁnite latent feature models and the Indian buffet process. In Advances in Neural Information Processing Systems, 2006. [6] D. Heinz. Building hyper dirichlet processes for graphical models. Electonic Journal of Statistics, 3:290–315, 2009. [7] J.F.C. Kingman. Random partitions in population genetics. Proceedings of the Royal Society of London, 361:1–20, 1978. [8] S.L. Lauritzen. Graphical Models. Oxford University Press, 1996. [9] P. M¨ ller, F. Quintana, and G. Rosner. A method for combining inference across related nonu parametric Bayesian models. Journal of the Royal Statistical Society B, 66:735–749, 2004. [10] R.M. Neal. Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9:249–265, 2000. [11] J. Pitman. Exchangeable and partially exchangeable random partitions. Probability theory and related ﬁelds, 102:145–158, 1995. [12] Y.W. Teh, M.I. Jordan, M.J. Beal, and D.M. Blei. Hierarchical Dirichlet processes. Journal of the American Statistical Association, 101:1566–1581, 2006. [13] S. Walker and P. Muliere. A bivariate Dirichlet process. Statistics and Probability Letters, 64:1–7, 2003. [14] F. Wood and T.L. Grifﬁths. Particle ﬁltering for nonparametric Bayesian matrix factorization. In Advances in Neural Information Processing Systems, 2007. 9

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 ca Abstract Over recent years Dirichlet processes and the associated Chinese restaurant process (CRP) have found many applications in clustering while the Indian buffet process (IBP) is increasingly used to describe latent feature models. [sent-5, score-0.151]

2 We propose here extensions of these models where the dependency between samples is given by a known decomposable graph. [sent-7, score-0.295]

3 We consider here the case where the data we are interested in are ‘locally’ dependent; these dependencies being represented by a known graph G where each data point/object is associated to a vertex. [sent-10, score-0.15]

4 For example, in the context of clustering, we might want to propose a prior distribution on partitions enforcing that data which are ‘close’ in the graph are more likely to be in the same cluster. [sent-14, score-0.203]

5 Similarly, in the context of latent feature models, we might be interested in a prior distribution on features enforcing that data which are ‘close’ in the graph are more likely to possess similar features. [sent-15, score-0.323]

6 The ‘standard’ CRP and IBP correspond to the case where the graph G is complete; that is it is fully connected. [sent-16, score-0.103]

7 In this paper, we generalize the CRP and IBP to decomposable graphs. [sent-17, score-0.295]

8 Each clique of the graph follows marginally a CRP or an IBP process and explicit expressions for the joint prior distribution on the graph is available. [sent-19, score-0.347]

9 In Section 2, we review the popular Dirichlet multinomial allocation model and the Dirichlet Process (DP) partition distribution. [sent-22, score-0.329]

10 We propose an extension of these two models to decomposable graphical models. [sent-23, score-0.366]

11 In Section 3 we discuss nonparametric latent feature models, reviewing brieﬂy the construction in [5] and extending it to decomposable graphs. [sent-24, score-0.399]

12 2 Prior distributions for partitions on decomposable graphs Assume we have n observations. [sent-26, score-0.433]

13 When performing clustering, we associate to each of this observation an allocation variable zi ∈ [K] = {1, . [sent-27, score-0.138]

14 , n(Πn ), where ∪j Aj = [n] and n(Πn ) is the number of subsets for partition Πn . [sent-41, score-0.149]

15 We also denote by Pn be the set of all partitions of [n] and let nj , j = 1, . [sent-42, score-0.287]

16 Each allocation variable zi is associated to a vertex/site of an undirected graph G, which is assumed to be known. [sent-46, score-0.357]

17 In the standard case where the graph G is complete, we ﬁrst review brieﬂy here two popular prior distributions on z1:n , equivalently on Πn . [sent-47, score-0.173]

18 We then extend these models to undirected decomposable graphs; see [2, 8] for an introduction to decomposable graphs. [sent-48, score-0.659]

19 1 Dirichlet multinomial allocation model and DP partition distribution Assume for the time being that K is ﬁnite. [sent-52, score-0.329]

20 When the graph is complete, a popular choice for the allocation variables is to consider a Dirichlet multinomial allocation model [11] θ θ , . [sent-53, score-0.386]

21 Integrating out π, we obtain the following Dirichlet multinomial prior distribution π ∼ D( Pr(z1:n ) = K j=1 Γ(θ) Γ(nj + θ K) (2) θ Γ(θ + n)Γ( K )K and then, using the straightforward equality Pr(Πn ) = PK where PK = {Πn ∈ Pn |n(Πn ) ≤ K}, we obtain K! [sent-57, score-0.112]

22 Γ(θ + n)Γ( K )n(Πn ) (3) DP may be seen as a generalization of the Dirichlet multinomial model when the number of components K → ∞; see for example [10]. [sent-62, score-0.077]

23 In this case the distribution over the partition Πn of [n] is given by [11] n(Π ) θn(Πn ) j=1n Γ(nj ) . [sent-63, score-0.149]

24 , An(Π−k ),−k } be the partition induced by removing item k to Πn and nj,−k be the size of cluster j for j = 1, . [sent-67, score-0.412]

25 It follows from (4) that an item k is assigned to an existing cluster j, j = 1, . [sent-71, score-0.313]

26 , n(Π−k ), with probability proportional to nj,−k / (n − 1 + θ) and forms a new cluster with probability θ/ (n − 1 + θ). [sent-74, score-0.155]

27 We now extend the Dirichlet multinomial allocation and the DP partition distribution models to decomposable graphs. [sent-76, score-0.624]

28 2 Markov combination of Dirichlet multinomial and DP partition distributions Let G be a decomposable undirected graph, C = {C1 , . [sent-78, score-0.625]

29 , Cp } a perfect ordering of the cliques and S = {S2 , . [sent-81, score-0.21]

30 It can be easily checked that if the marginal distribution of zC for each clique C ∈ C is deﬁned by (2) then these distributions are consistent as they yield the same distribution (2) over the separators. [sent-85, score-0.177]

31 Therefore, the unique Markov distribution over G with Dirichlet multinomial distribution over the cliques is deﬁned by [8] Pr(zC ) S∈S Pr(zS ) C∈C Pr(z1:n ) = (5) where for each complete set B ⊆ G, we have Pr(zB ) given by (2). [sent-86, score-0.303]

32 C∈C Γ(θ) S∈S 2 K j=1 θ Γ(nj,C + K ) θ Γ(θ+nC )Γ( K )K K j=1 θ Γ(nj,S + K ) θ Γ(θ+nS )Γ( K )K (6) where for each complete set B ⊆ G, nj,B is the number of items associated to cluster j, j = 1, . [sent-89, score-0.252]

33 Within each complete set B, the allocation variables deﬁne a partition distributed according to the Dirichlet-multinomial distribution. [sent-93, score-0.315]

34 We now extend this approach to DP partition distributions; that is we derive a joint distribution over Πn such that the distribution of ΠB over each complete set B of the graph is given by (4) with θ > 0. [sent-94, score-0.315]

35 Such a distribution satisﬁes the consistency condition over the separators as the restriction of any partition distributed according to (4) still follows (4) [7]. [sent-95, score-0.149]

36 As K → ∞, the prior distribution G over partitions (6) is given for each Πn ∈ Pn by Pr(Πn ) = θn(Πn ) n(ΠC ) Γ(nj,C ) j=1 nC i=1 (θ+i−1) n(ΠS ) Γ(nj,S ) j=1 nS (θ+i−1) i=1 C∈C S∈S (7) where n(ΠB ) is the number of clusters in the complete set B. [sent-98, score-0.211]

37 Let n = 3 and G be the decomposable graph deﬁned by the relations 1 ∼ 2, 2 ∼ 3 and 1 3. [sent-109, score-0.398]

38 Note that G the partition {{1, 3}, {2}} does not belong to P3 . [sent-111, score-0.149]

39 Indeed, as there is no edge between 1 and 3, they cannot be in the same cluster if 2 is in another cluster. [sent-112, score-0.096]

40 The cliques are C1 = {1, 2} and C2 = {2, 3} Pr(ΠC1 ) Pr(ΠC2 ) hence we can and the separator is S2 = {2}. [sent-113, score-0.206]

41 Based on (7) the conditional assignment of an item k is proportional to the conditional over the cliques divided by the conditional over the separators. [sent-116, score-0.389]

42 G Let denote G−k the undirected graph obtained by removing vertex k from G. [sent-117, score-0.206]

43 G−k If Π−k ∈ Pn−1 , then do not change the value of item k. [sent-119, score-0.167]

44 Otherwise, item k is assigned to cluster j / where j = 1, . [sent-120, score-0.313]

45 , n(Π−k ) with probability proportional to {C∈C|n−k,j,C >0} n−k,j,C {S∈S|n−k,j,S >0} n−k,j,S (8) and to a new cluster with probability proportional to θ, where n−k,j,C is the number of items in the set C \ {k} belonging to cluster j. [sent-123, score-0.402]

46 The updating process is illustrated by the Chinese wedding party process1 in Fig. [sent-124, score-0.145]

47 Given Π−2 = {A1 = {1}, A2 = {3}}, we have −1 Pr( item 2 assigned to A1 = {1}| Π−2 ) = Pr( item 2 assigned to A2 = {3}| Π−2 ) = (θ + 2) −1 and Pr( item 2 assigned to new cluster A3 | Π−2 ) = θ (θ + 2) . [sent-128, score-0.747]

48 Given Π−2 = {A1 = {1, 3}}, item 2 is assigned to A1 with probability 1. [sent-129, score-0.217]

49 3 (a) (b) (d) (c) (e) Figure 1: Chinese wedding party. [sent-131, score-0.107]

50 Consider a group of n guests attending a wedding party. [sent-132, score-0.361]

51 Each of the n guests may belong to one or several cliques, i. [sent-133, score-0.21]

52 The belonging of each guest to the different cliques is represented by color patches on the ﬁgures, and the graphical representation of the relationship between the guests is represented by the graphical model (e). [sent-136, score-0.614]

53 (a) Suppose that the guests are already seated such that two guests cannot be together at the same table is they are not part of the same clique, or if there does not exist a group of other guests such that they are related (“Any friend of yours is a friend of mine”). [sent-137, score-0.768]

54 Given a partition Πn , an item k is taken out of the partition. [sent-143, score-0.316]

55 Otherwise, the item will be assigned to a cluster j, / j = 1, . [sent-145, score-0.313]

56 , n(Π−k ), with probability proportional to p(z{k}∪Aj,−k ) × p(zAj,−k ) {C∈C|n−k,j,C >0} n−k,j,C {S∈S|n−k,j,S >0} n−k,j,S (9) and the item will be assigned to a new cluster with probability proportional to p(z{k} ) × θ. [sent-148, score-0.431]

57 2 Sequential Monte Carlo We have so far only treated the case of an undirected decomposable graph G. [sent-160, score-0.467]

58 We can formulate a sequential updating rule for the corresponding perfect directed version D of G. [sent-161, score-0.109]

59 a|V | ) be a perfect ordering and pa(ak ) be the set of parents of ak which is by deﬁnition complete. [sent-165, score-0.255]

60 , An(Πk−1 ),k−1 } denote the partition of the ﬁrst k−1 vertices a1:k−1 and let nj,pa(ak ) be the number of elements with value j in the set pa(ak ), j = 1, . [sent-169, score-0.149]

61 Then the vertex ak joins the set j with probability nj,pa(ak ) / θ + cluster with probability θ/ θ + q q nq,pa(ak ) and creates a new nq,pa(ak ) . [sent-173, score-0.372]

62 For each particle i, there are n(Πk−1 ) + 1 possible 4 (i,j) allocations for component ak . [sent-176, score-0.225]

63 We denote Πk the partition obtained by associating component ak (i,j) to cluster j. [sent-177, score-0.417]

64 Let Πk be the resampled particles and wk the associated normalized weights. [sent-182, score-0.184]

65 3 Prior distributions for inﬁnite binary matrices on decomposable graphs Assume we have n objects; each of these objects being associated to the vertex of a graph G. [sent-183, score-0.591]

66 In the standard case where the graph G is complete, we review brieﬂy here two popular prior distributions on Z1:n , equivalently on ξ1:n : the Beta-Bernoulli model and the IBP [5]. [sent-190, score-0.173]

67 We then extend these models to undirected decomposable graphs. [sent-191, score-0.364]

68 1 Beta-Bernoulli and IBP distributions The Beta-Bernoulli distribution over the allocation Z1:n is K Pr(Z1:n ) = α + K )Γ(n − nj + 1) α Γ(n + 1 + K ) α K Γ(nj j=1 (11) where nj is the number of objects having feature j. [sent-194, score-0.658]

69 2n −1 h=0 α K Γ(nj α + K )Γ(n − nj + 1) α Γ(n + 1 + K ) Kh ! [sent-196, score-0.222]

70 Markov combination of Beta-Bernoulli and IBP distributions Let G be a decomposable undirected graph, C = {C1 , . [sent-205, score-0.399]

71 , Cp } a perfect ordering of the cliques and S = {S2 , . [sent-208, score-0.21]

72 As in the Dirichlet-multinomial case, it is easily seen that if for each clique C ∈ C, the marginal distribution is deﬁned by (11), then these distributions are consistent as they yield the same distribution (11) over the separators. [sent-212, score-0.177]

73 Therefore, the unique Markov distribution over G with Beta-Bernoulli distribution over the cliques is deﬁned by [8] Pr(ZC ) S∈S Pr(ZS ) C∈C Pr(Z1:n ) = (14) where Pr(ZB ) given by (11) for each complete set B ⊆ G. [sent-213, score-0.226]

74 α K α Γ(nj,C + K )Γ(nC −nj,C +1) α Γ(nC +1+ K ) α α Γ(nj,S + K )Γ(nS −nj,S +1) K K α j=1 Γ(nS +1+ K ) K j=1 C∈C S∈S 5 (15) where for each complete set B ⊆ G, nj,B is the number of items having feature j, j = 1, . [sent-216, score-0.146]

75 + + + + if K[n] = C KC − S KS and 0 otherwise, where KB is the number of different features possessed by objects in B. [sent-227, score-0.076]

76 G Let En be the subset of En such that for each decomposition A, B and any (u, v) ∈ A × B: {u and v possess feature j} ⇒ ∃k ∈ A ∩ B such that {k possesses feature j}. [sent-228, score-0.15]

77 1 Applications Sharing clusters among relative groups: An alternative to HDP Consider that we are given d groups with nj data yi,j in each group, i = 1, . [sent-236, score-0.27]

78 We consider latent cluster variables zi,j that deﬁne the partition of the data. [sent-243, score-0.312]

79 Under conjugacy assumptions, G0 , Gj and U can be integrated out and we can approximate the marginal posterior of (zi,j ) given y = (yi,j ) with Gibbs sampling using the Chinese restaurant franchise to sample from the full conditional p(zi,j |z−{i,j} , y). [sent-253, score-0.071]

80 Using the graph formulation deﬁned in Section 2, we propose an alternative to HDP. [sent-254, score-0.103]

81 , θ0,N be N auxiliary variables belonging to what we call group 0. [sent-258, score-0.139]

82 , d) to be composed of elements from group j and elements from group 0. [sent-262, score-0.088]

83 This deﬁnes a decomposable graphical model whose separator is given by the elements of group 0. [sent-263, score-0.453]

84 Again, under conjugacy conditions, we can integrate out G0 , Gj and U and approximate the marginal posterior distribution over the partition using the Chinese wedding party process deﬁned in Section 2. [sent-280, score-0.365]

85 give the graphical DP alternative to HDP and 2-layer HDP. [sent-287, score-0.071]

86 6 z0 root z0 root corpora docs z1 z2 z1 z2 z3 z1,1 z1,2 z2,1 z2,2 z2,3 docs (a) Graphical DP alternative to HDP (b) Graphical DP alternative to 2-layer HDP Figure 2: Hierarchical Graphs of dependency with (a) one layer and (b) two layers of hierarchy. [sent-288, score-0.08]

87 One interesting feature of the model is that, contrary to HDP, the marginal distribution of Gj at any layer of the tree is DP (α, H). [sent-292, score-0.122]

88 As a consequence, the total number of clusters scales logarithmically (as in the usual DP) with the size of each group, whereas it scales doubly logarithmically in HDP. [sent-293, score-0.12]

89 Nonetheless, the computational time used to sample from auxiliary variables is negligible compared to the time used for latent variables associated to data. [sent-300, score-0.163]

90 The model proposed offers a far richer framework and ensures that at each level of the tree, the marginal distribution of the partition is given by a DP partition model. [sent-302, score-0.334]

91 We consider here a different model where the object Xt is assumed to arrive at time index t and we want a prior distribution on Z1:n ensuring that objects close in time are more likely to possess similar features. [sent-310, score-0.118]

92 To achieve this, we consider the simple directed graphical model D of Fig. [sent-311, score-0.133]

93 3 where the site numbering corresponds to a time index in that case and a perfect numbering of D is (1, 2, . [sent-312, score-0.176]

94 The time-varying IBP to sample from p(Z1:n ) associated to this directed graph follows from (16) and proceeds as follows. [sent-320, score-0.212]

95 The feature occurences Z1:n and true features Y and their estimates are represented in Figure 4. [sent-369, score-0.154]

96 The algorithm is able to correctly estimate the varying prior occurences of the features over time. [sent-373, score-0.152]

97 In this paper, we have assumed that the structure of the graph was known beforehand and we have shown that many ﬂexible models arise from this framework. [sent-379, score-0.103]

98 It would be interesting in the future to investigate the case where the graphical structure is unknown and must be estimated from the data. [sent-380, score-0.071]

99 Hyper Markov laws in the statistical analysis of decomposable graphical models. [sent-393, score-0.366]

100 Inﬁnite latent feature models and the Indian buffet process. [sent-409, score-0.104]

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tfidf for this paper:

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