Probabilistic Graphical Models

Transcription

1 Probabilisti Graphial Models Undireted Graphial Models Eri Xing Leture, Ot 7, 2005 Reading: MJ-Chap. 2,4, and KF-hap5 Review: independene properties of DAGs Defn: let I l (G) be the set of loal independene properties enoded by DAG G, namely: { X i NonDesendants(X ) i Parents(X ) i } Defn: A DAG G is an I-map (independene-map) of P if I l (G) I(P) A fully onneted DAG G is an I-map for any distribution, sine I l (G)= I(P) for any P. Defn: A DAG G is a minimal I-map for P if it is an I-map for P, and if the removal of even a single edge from G renders it not an I-map. A distribution may have several minimal I-maps Eah orresponding to a speifi node-ordering

2 Global Markov properties of DAGs X is d-separated (direted-separated) from Z given Y if we an't send a ball from any node in X to any node in Z using the "Bayes-ball" algorithm illustrated bellow: Defn: I(G)=all independene properties that orrespond to d- separation: I( G ) = { X Z Y ) : dsep ( X ; Z Y )} G D-separation is sound and omplete (Chap 3, Koller & Friedman) P-maps Defn: A DAG G is a perfet map (P-map) for a distribution P if I(P)=I(G). Thm: not every distribution has a perfet map as DAG. Pf by ounterexample. Suppose we have a model where A C {B,D}, and B D {A,C}. This annot be represented by any Bayes net. e.g., BN wrongly says B D A, BN2 wrongly says B D. D A B D B D A B C C A C BN BN2 MRF 2

3 Undireted graphial models X X 4 X 3 X 2 X 5 Pairwise (non-ausal) relationships Can write down model, and sore speifi onfigurations of the graph, but no expliit way to generate samples Contingeny onstrains on node onfigurations Canonial examples The grid model Naturally arises in image proessing, lattie physis, et. Eah node may represent a single "pixel", or an atom The states of adjaent or nearby nodes are "oupled" due to pattern ontinuity or eletro-magneti fore, et. Most likely joint-onfigurations usually orrespond to a "low-energy" state 3

4 Soial networks Ignoring the arrows, this is a "relational network" among people Protein interation networks 4

5 Modeling Go Information retrieval topi text image 5

6 Semantis of Undireted Graphs Let H be an undireted graph: B separates A and C if every path from a node in A to a node in C passes through a node in B: sep ( A; C B ) A probability distribution satisfies the global Markov property if for any disjoint A, B, C, suh that B separates A and C, A is independent of C given B: I( H ) = A C B ) :sep ( A; C B ) H { } H Undireted Graphial Models Defn: an undireted graphial model represents a distribution P(X,,X n ) defined by an undireted graph H, and a set of positive potential funtions ψ assoiated with liques of H, s.t. P ( x, K, xn ) = ψ ( x ) Z C where Z is known as the partition funtion: Z = x, K, x n C ψ ( x Also known as Markov Random Fields, Markov networks The potential funtion an be understood as an ontingeny funtion of its arguments assigning "pre-probabilisti" sore of their joint onfiguration. ) 6

7 Cliques For G={V,E}, a omplete subgraph (lique) is a subgraph G'={V' V,E' E} suh that nodes in V' are fully interonneted A (maximal) lique is a omplete subgraph s.t. any superset V" V' is not omplete. A sub-lique is a not-neessarily-maximal lique. A D B C Example: max-liques = {A,B,D}, {B,C,D}, sub-liques = {A,B}, {C,D}, all edges and singletons Example UGM using max liques A D B C P ( x, x2, x3, x4) = ψ ( x24) ψ ( x234) Z Z = x ψ ( x24) ψ ( x, x2, x3, x4 For disrete nodes, we an represent P(X :4 ) as two 3D tables instead of one 4D table 234 ) 7

8 Example UGM using subliques A D B P ( x, x2, x3, x4) = ψ ( x Z = ψ 2( x Z Z = 2 C ) ) ψ ( x 4, x2, x3, x4 x For disrete nodes, we an represent P(X :4 ) as 5 2D tables instead of one 4D table 4 ) ψ 23 ( x ψ (x 23 ) ) ψ 24 ( x 24 ) ψ 34 ( x 34 ) Interpretation of Clique Potentials X Y Z The model implies X Z Y. This independene statement implies (by definition) that the joint must fatorize as: p ( x, y, z ) = p( y ) p( x y ) p( z y ) p( x, y, z ) = p( x, y ) p( z y ) We an write this as:, but p( x, y, z ) = p( x y ) p( z, y ) annot have all potentials be marginals annot have all potentials be onditionals The positive lique potentials an only be thought of as general "ompatibility", "goodness" or "happiness" funtions over their variables, but not as probability distributions. 8

9 Exponential Form Constraining lique potentials to be positive ould be inonvenient (e.g., the interations between a pair of atoms an be either attrative or repulsive). We represent a lique potential ψ (x ) in an unonstrained form using a real-value "energy" funtion φ (x ): { φ ( x )} ψ ( x ) = exp For onveniene, we will all φ (x ) a potential when no onfusion arises from the ontext. This gives the joint a nie additive struture p( x) = exp φ ( x ) = exp x Z C Z where the sum in the exponent is alled the "free energy": In physis, this is alled the "Boltzmann distribution". In statistis, this is alled a log-linear model. H ( x) = φ ( x ) C { H ( )} Example: Boltzmann mahines A fully onneted graph with pairwise (edge) potentials on binary-valued nodes (for x i {, + } or xi { 0, } ) is alled a Boltzmann mahine P ( x, x2, x3, x4) = exp φ ( xi, x j ) Z = exp θ xi x j + αi xi + C Z i Hene the overall energy funtion has the form: T H ( x ) = ( x µ ) Θ ( x µ ) = ( x µ ) Θ( x µ ) i j 9

10 Example: Ising (spin-glass) models Nodes are arranged in a regular topology (often a regular paking grid) and onneted only to their geometri neighbors. Same as sparse Boltzmann mahine, where Θ 0 iff i,j are neighbors. e.g., nodes are pixels, potential funtion enourages nearby pixels to have similar intensities. Potts model: multi-state Ising model. Example: multivariate Gaussian Distribution A Gaussian distribution an be represented by a fully onneted graph with pairwise (edge) potentials over ontinuous nodes. The overall energy has the form T H ( x ) = ( x µ ) Θ ( x µ ) = ( x µ ) Θ( x µ ) i where µ is the mean and Θ is the inverse ovariane (preision) matrix. j Also known as Gaussian graphial model (GGM), same as Boltzmann mahine exept x i R 0

11 Sparse preision vs. sparse ovariane in GGM Σ 6 = Σ = X Σ 5 = 0 X X5 X nbrs ( ) or nbrs ( 5) X5 Σ5 = 0 Independene properties of UGM Let us return to the question of what kinds of distributions an be represented by undireted graphs (ignoring the details of the partiular parameterization). Defn: the global Markov properties of a UG H are I( H ) = { X Z Y ) : sep ( X ; Z Y )} H Y X Z Is this definition sound and omplete?

12 Soundness and ompleteness of global Markov property Defn: An UG H is an I-map for a distribution P if I(H) I(P), i.e., P entails I(H). Defn: P is a Gibbs distribution over H if it an be represented as P ( x, K, xn ) = ψ ( x ) Z C Thm (soundness): If P is a Gibbs distribution over H, then H is an I-map of P. Thm (ompleteness): If sep H (X; Z Y), then X P Z Y in some P that fatorizes over H. Loal and global Markov properties For direted graphs, we defined I-maps in terms of loal Markov properties, and derived global independene. For undireted graphs, we defined I-maps in terms of global Markov properties, and will now derive loal independene. Defn: The pairwise Markov independenies assoiated with UG H = (V;E) are I l { X Y V \{ X, Y }:{ X, Y E } ( H ) = } e.g., X X5 { X2, X3, X4}

13 Loal Markov properties A distribution has the loal Markov property w.r.t. a graph H=(V,E) if the onditional distribution of variable given its neighbors is independent of the remaining nodes I l { X V \ ( X N ( X )) N ( X )) V} ( H ) = X H H : Theorem (Hammersley-Clifford): If the distribution is stritly positive and satisfies the loal Markov property, then it fatorizes with respet to the graph. N H (X) is also alled the Markov blanket of X. Relationship between loal and global Markov properties Thm If P = I l (H) then P = I p (H). Thm If P = I(H) then P = I l (H). Thm If P > 0 and P = I p (H), then P = I(H). Pf sketh: p(a,b,d)=p(a,d)p(b,d) and d separate b from {a,} p(a,b,d)p( d)=p(a,d)p(b,d)p( d)=p(a, d)p(b d) Corollary 5.5.6: If P > 0, then I l = I p = I. If x: P(x) = 0, then we an onstrut an example (using deterministi potentials) where I p I or I l I. Given more info Given less info Given more info 3

14 I-maps for undireted graphs Defn: A Markov network H is a minimal I-map for P if it is an I- map, and if the removal of any edge from H renders it not an I-map. How an we onstrut a minimal I-map from a positive distribution P? Pairwise method: add edges between all pairs X,Y s.t. P ( X Y V \{ X, Y }) Loal method: add edges between X and all Y MB P (X), where MB P (X) is the minimal set of nodes U s.t. ( X V \{ X }\ U Y ) P Thm 5.5./2: both methods indue the unique minimal I-map. If x s.t. P(x) = 0, then we an onstrut an example where either method fails to indue an I-map. Perfet maps Defn: A Markov network H is a perfet map for P if for any X; Y;Z we have that ( X Z Y ) seph ( X ; Z Y ) P = Thm: not every distribution has a perfet map as UGM. Pf by ounterexample. No undireted network an apture all and only the independenies enoded in a v-struture X Z Y. 4

15 The expressive power of UGM Can we always onvert direted undireted? No. W X Y X Y Z MRF No direted model an represent these and only these independenies. X Y {W, Z} W Z {X, Y} Z BN No undireted model an represent these and only these independenies. X Y Converting Bayes nets to Markov nets Defn: A Markov net H is an I-map for a Bayes net G if I(H) I(G). We an onstrut a minimal I-map for a BN by finding the minimal Markov blanket for eah node. We need to blok all ative paths oming into node X, from parents, hildren, and o-parents; so onnet them all to X. 5

16 Moralization The moral graph H(G) of a DAG is onstruted by adding undireted edges between any pair of disonneted ("unmarried") nodes X,Y that are parents of a hild Z, and then dropping all remaining arrows. To turn a BN into a MRF, We assign eah CPD to one of the lique potentials that ontains it. 6