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Conditional Independence and Markov Properties Lecture 1 Saint Flour Summerschool, July 5, 2006 Steﬀen L. Lauritzen, University of Oxford Overview of lectures 1. Conditional independence and Markov properties 2. More on Markov properties 3. Graph decompositions and junction trees 4. Probability propagation and similar algorithms 5. Log-linear and Gaussian graphical models 6. Conjugate prior families for graphical models 7. Hyper Markov laws 8. Structure learning and Bayes factors 9. More on structure learning. Conditional independence The notion of conditional independence is fundamental for graphical models. For three random variables X, Y and Z we denote this as ⊥ X ⊥ Y | Z and graphically as u u u X Z Y If the random variables have density w.r.t. a product measure µ, the conditional independence is reﬂected in the relation f (x, y, z)f (z) = f (x, z)f (y, z), where f is a generic symbol for the densities involved. Graphical models 2 4 u u 1 d 5 d 7 u d u d d u d d d d u d d u d 3 6 For several variables, complex systems of conditional independence can be described by undirected graphs. Then a set of variables A is conditionally independent of set B, given the values of a set of variables C if C separates A from B. A directed graphical model Directed model showing relations between risk factors, diseases, and symptoms. A pedigree Graphical model for a pedigree from study of Werner’s syndrome. Each node is itself a graphical model. 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relationship of 1641 members of Greenland Eskimo population. Conditional independence Random variables X and Y are conditionally independent given the random variable Z if L(X | Y, Z) = L(X | Z). ⊥ ⊥ We then write X ⊥ Y | Z (or X ⊥ P Y | Z) Intuitively: Knowing Z renders Y irrelevant for predicting X. Factorisation of densities w.r.t. product measure: ⊥ X ⊥ Y |Z ⇐⇒ f (x, y, z)f (z) = f (x, z)f (y, z) ⇐⇒ ∃a, b : f (x, y, z) = a(x, z)b(y, z). Fundamental properties For random variables X, Y , Z, and W it holds ⊥ ⊥ (C1) if X ⊥ Y | Z then Y ⊥ X | Z; ⊥ ⊥ (C2) if X ⊥ Y | Z and U = g(Y ), then X ⊥ U | Z; ⊥ ⊥ (C3) if X ⊥ Y | Z and U = g(Y ), then X ⊥ Y | (Z, U ); ⊥ ⊥ (C4) if X ⊥ Y | Z and X ⊥ W | (Y, Z), then ⊥ X ⊥ (Y, W ) | Z; If density w.r.t. product measure f (x, y, z) > 0 also ⊥ ⊥ ⊥ (C5) if X ⊥ Y | Z and X ⊥ Z | Y then X ⊥ (Y, Z). Additional note on (C5) f (x, y, z) > 0 is not necessary for (C5). Enough e.g. that f (y, z) > 0 for all (y, z) or f (x, z) > 0 for all . In discrete and ﬁnite case it is even enough that the bipartite graphs G+ = (Y ∪ Z, E+ ) deﬁned by y ∼+ z ⇐⇒ f (y, z) > 0, are all connected. Alternatively it is suﬃcient if the same condition is satisﬁed with X replacing Y . Is there a simple necessary and suﬃcient condition? Graphoid axioms Ternary relation ⊥σ among subsets of a ﬁnite set V is graphoid if for all disjoint subsets A, B, C, and D of V : (S1) if A ⊥σ B | C then B ⊥σ A | C; (S2) if A ⊥σ B | C and D ⊆ B, then A ⊥σ D | C; (S3) if A ⊥σ B | C and D ⊆ B, then A ⊥σ B | (C ∪ D); (S4) if A ⊥σ B | C and A ⊥σ D | (B ∪ C), then A ⊥σ (B ∪ D) | C; (S5) if A ⊥σ B | (C ∪ D) and A ⊥σ C | (B ∪ D) then A ⊥σ (B ∪ C) | D. Semigraphoid if only (S1)–(S4) holds. Irrelevance Conditional independence can be seen as encoding irrelevance in a fundamental way. With the interpretation: Knowing C, A is irrelevant for learning B, (S1)–(S4) translate to: (I1) If, knowing C, learning A is irrelevant for learning B, then B is irrelevant for learning A; (I2) If, knowing C, learning A is irrelevant for learning B, then A is irrelevant for learning any part D of B; (I3) If, knowing C, learning A is irrelevant for learning B, it remains irrelevant having learnt any part D of B; (I4) If, knowing C, learning A is irrelevant for learning B and, having also learnt A, D remains irrelevant for learning B, then both of A and D are irrelevant for learning B. The property (S5) is slightly more subtle and not generally obvious. Also the symmetry (C1) is a special property of probabilistic conditional independence, rather than of general irrelevance, where (I1) could appear dubious. Probabilistic semigraphoids V ﬁnite set, X = (Xv , v ∈ V ) random variables. For A ⊆ V , let XA = (Xv , v ∈ A). Let Xv denote state space of Xv . Similarly xA = (xv , v ∈ A) ∈ XA = ×v∈A Xv . ⊥ ⊥ Abbreviate: A ⊥ B | S ⇐⇒ XA ⊥ XB | XS . Then basic properties of conditional independence imply: ⊥ The relation ⊥ on subsets of V is a semigraphoid. ⊥ If f (x) > 0 for all x, ⊥ is also a graphoid. Not all (semi)graphoids are probabilistically representable. Second order conditional independence Sets of random variables A and B are partially uncorrelated for ﬁxed C if their residuals after linear regression on XC are uncorrelated: Cov{XA − E∗ (XA | XC ), XB − E∗ (XB | XC )} = 0, in other words, if the partial correlations are zero ρAB·C = 0. We then write A ⊥2 B | C. Also ⊥2 satisﬁes the semigraphoid axioms (S1) -(S4) and the graphoid axioms if there is no non-trivial linear relation between the variables in V . Separation in undirected graphs Let G = (V, E) be ﬁnite and simple undirected graph (no self-loops, no multiple edges). For subsets A, B, S of V , let A ⊥G B | S denote that S separates A from B in G, i.e. that all paths from A to B intersect S. Fact: The relation ⊥G on subsets of V is a graphoid. This fact is the reason for choosing the name ‘graphoid’ for such separation relations. Geometric Orthogonality As another fundamental example, consider geometric orthogonality in Euclidean vector spaces or Hilbert spaces. Let L, M , and N be linear subspaces of a Hilbert space H and deﬁne L ⊥ M | N ⇐⇒ (L N ) ⊥ (M N ), where L N = L ∩ N ⊥ . Then L and M are said to meet orthogonally in N . This has properties (O1) If L ⊥ M | N then M ⊥ L | N ; (O2) If L ⊥ M | N and U is a linear subspace of L, then U ⊥ M | N; (O3) If L ⊥ M | N and U is a linear subspace of M , then L ⊥ M | (N + U ); (O4) If L ⊥ M | N and L ⊥ R | (M + N ), then L ⊥ (M + R) | N . The analogue of (C5) does not hold in general; for example if M = N we may have L ⊥ M | N and L ⊥ N | M, but if L and M are not orthogonal then it is false that L ⊥ (M + N ). Variation independence Let U ⊆ X = ×v∈V Xv and deﬁne for S ⊆ V the S-section ∗ U uS of U as ∗ U uS = {uV \S : uS = u∗ , u ∈ U}. S Deﬁne further the conditional independence relation ‡U as ∗ ∗ ∗ A ‡U B | C ⇐⇒ ∀u∗ : U uC = {U uC }A × {U uC }B C i.e. if and only if the C-sections all have the form of a product space. The relation ‡U satisﬁes the semigraphoid axioms. In particular ‡U holds if U is the support of a probability measure satisfying the similar conditional independence restriction. Markov properties for semigraphoids G = (V, E) simple undirected graph; ⊥σ (semi)graphoid relation. Say ⊥σ satisﬁes (P) the pairwise Markov property if α ∼ β =⇒ α ⊥σ β | V \ {α, β}; (L) the local Markov property if ∀α ∈ V : α ⊥σ V \ cl(α) | bd(α); (G) the global Markov property if A ⊥G B | S =⇒ A ⊥σ B | S. Pairwise Markov property 2 4 u u 1 d 5 d 7 u d u d d u d d d d u d d u d 3 6 Any non-adjacent pair of random variables are conditionally independent given the remaning. ⊥ ⊥ For example, 1 ⊥ 5 | {2, 3, 4, 6, 7} and 4 ⊥ 6 | {1, 2, 3, 5, 7}. Local Markov property 2 4 u u 1 d 5 d 7 u d u d d u d d d d u d d u d 3 6 Every variable is conditionally independent of the remaining, given its neighbours. ⊥ For example, 5 ⊥ {1, 4} | {2, 3, 6, 7} and ⊥ 7 ⊥ {1, 2, 3} | {4, 5, 6}. Global Markov property 2 4 u u 1 d 5 d 7 u d u d d u d d d d u d d u d 3 6 To ﬁnd conditional independence relations, one should look for separating sets, such as {2, 3}, {4, 5, 6}, or {2, 5, 6} ⊥ For example, it follows that 1 ⊥ 7 | {2, 5, 6} and ⊥ 2 ⊥ 6 | {3, 4, 5}. Structural relations among Markov properties For any semigraphoid it holds that (G) =⇒ (L) =⇒ (P) If ⊥σ satisﬁes graphoid axioms it further holds that (P) =⇒ (G) so that in the graphoid case (G) ⇐⇒ (L) ⇐⇒ (P). ⊥ The latter holds in particular for ⊥ , when f (x) > 0. (G) =⇒ (L) =⇒ (P) (G) implies (L) because bd(α) separates α from V \ cl(α). Assume (L). Then β ∈ V \ cl(α) because α ∼ β. Thus bd(α) ∪ ((V \ cl(α)) \ {β}) = V \ {α, β}, Hence by (L) and (S3) we get that α ⊥σ (V \ cl(α)) | V \ {α, β}. (S2) then gives α ⊥σ β | V \ {α, β} which is (P). (P) =⇒ (G) for graphoids Asuume (P) and A ⊥G B | S. We must show A ⊥σ B | S. Wlog assume A and B non-empty. Proof is reverse induction on n = |S|. If n = |V | − 2 then A and B are singletons and (P) yields A ⊥σ B | S directly. Assume |S| = n < |V | − 2 and conclusion established for |S| > n. First assume V = A ∪ B ∪ S. Then either A or B has at least two elements, say A. If α ∈ A then B ⊥G (A \ {α}) | (S ∪ {α}) and also α ⊥G B | (S ∪ A \ {α}) (as ⊥G is a semi-graphoid). Thus by the induction hypothesis (A \ {α}) ⊥σ B | (S ∪ {α}) and {α} ⊥σ B | (S ∪ A \ {α}). Now (S5) gives A ⊥σ B | S. For A ∪ B ∪ S ⊂ V we choose α ∈ V \ (A ∪ B ∪ S). Then A ⊥G B | (S ∪ {α}) and hence the induction hypothesis yields A ⊥σ B | (S ∪ {α}). Further, either A ∪ S separates B from {α} or B ∪ S separates A from {α}. Assuming the former gives α ⊥σ B | A ∪ S. Using (S5) we get (A ∪ {α}) ⊥σ B | S and from (S2) we derive that A ⊥σ B | S. The latter case is similar. Factorisation and Markov properties For a ⊆ V , ψa (x) is a function depending on xa only, i.e. xa = ya =⇒ ψa (x) = ψa (y). We can then write ψa (x) = ψa (xa ) without ambiguity. The distribution of X factorizes w.r.t. G or satisﬁes (F) if its density f w.r.t. product measure on X has the form f (x) = ψa (x), a∈A where A are complete subsets of G or, equivalently, if f (x) = ˜ ψc (x), c∈C where C are the cliques of G. Factorization example 2 4 s s 1 d 5 7 ds d s s d d s d s d d 3 6 The cliques of this graph are the maximal complete subsets {1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 5, 6}, {4, 7}, and {5, 6, 7}. A complete set is any subset of these sets. The graph above corresponds to a factorization as f (x) = ψ12 (x1 , x2 )ψ13 (x1 , x3 )ψ24 (x2 , x4 )ψ25 (x2 , x5 ) × ψ356 (x3 , x5 , x6 )ψ47 (x4 , x7 )ψ567 (x5 , x6 , x7 ). Factorisation of the multivariate Gaussian Consider a multivariate Gaussian random vector X = NV (ξ, Σ) with Σ regular so it has density f (x | ξ, Σ) = (2π)−|V |/2 (det K)1/2 e−(x−ξ) K(x−ξ)/2 , where K = Σ−1 is the concentration matrix of the distribution. Thus the Gaussian density factorizes w.r.t. G if and only if α ∼ β =⇒ kαβ = 0 i.e. if the concentration matrix has zero entries for non-adjacent vertices. Factorization theorem Consider a distribution with density f w.r.t. a product measure and let (G), (L) and (P) denote Markov properties ⊥ w.r.t. the semigraphoid relation ⊥ . It then holds that (F) =⇒ (G) and further: If f (x) > 0 for all x: (P) =⇒ (F). Thus in the case of positive density (but typically only then), all the properties coincide: (F) ⇐⇒ (G) ⇐⇒ (L) ⇐⇒ (P).

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