Conditional Independence and Markov Properties

Transcription

1 Conditional Independence and Markov Properties Lecture 1 Saint Flour Summerschool, July 5, 2006 Steffen L. Lauritzen, University of Oxford

2 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.

3 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 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.

4 Graphical models 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.

5 A directed graphical model Directed model showing relations between risk factors, diseases, and symptoms.

6 A pedigree Graphical model for a pedigree from study of Werner s syndrome. Each node is itself a graphical model.

7 A highly complex pedigree p1379 p1380 p1381 p1382 p1383 p1384 p1385 p1386 p1387 p1129 p1128 p1133 p1134 p1135 p1136 p1137 p1536 p1535 p1131 p1408 p1407 p1406 p1405 p1404 p1126 p1396 p1395 p1394 p1393 p1392 p1391p1390 p1389 p1388 p1127 p1426 p1425 p1424 p1423 p1130 p1565p1564 p1530 p1132 p1073 p1457 p1456 p1455 p1454 p1078 p1349 p1079 p1226 p1225 p1224 p1223 p1222p1221 p1220 p1076 p1242 p1241p1240 p1239 p1238 p1237 p1561 p1236 p1575 p1235 p1499 p1234 p1480 p1511 p1534 p1233 p1219 p1074 p1376 p1375 p1374 p1373 p1372 p1371 p1370 p1369 p1368 p1077 p1232 p1231 p1230 p1229 p1228p1227 p1072 p1438p1437 p1436 p1435 p1434 p1439 p1080 p1362 p1361 p1360 p1190 p1449 p1450 p1451 p1452 p1453 p1187 p1427 p1428 p1429 p1430 p1431 p1432 p1433 p1184 p1189 p1188 p1186 p1185 p1075 p927 p928 p929 p331 p544 p543 p542 p328 p1585 p657 p659 p660 p1556 p1555 p1557 p767 p1514 p1568 p765 p875 p876 p877 p766 p661 p662 p745 p842 p1367 p1366 p1365 p1364 p1363 p746 p747 p658 p506 p744 p742 p743 p512 p712 p709 p711 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p1031 p1041 p903 p904p906 p1039 p1038 p1037 p1036 p1035 p905 p263 p198 p26 p440 p441 p647 p646 p442 p231 p228 p446 p730 p513 p1211 p1212 p1570p1574 p1210 p1213 p1214 p1215 p1216p1217 p519 p518 p517 p516 p515 p514 p445 p600 p599 p748 p601 p602 p444 p1042 p1147 p1397 p1398 p1399 p1141 p1142 p1498 p1497 p1496 p1466 p1143 p1506 p1524p1523 p1522 p1144 p1541 p1505p1504 p1478 p1474 p1145 p1567 p1544 p1146 p1043 p892 p443 p447 p227 p269 p960 p1218 p961 p962 p268 p632 p751 p1420 p1419 p1418 p1417 p1421 p752 p633 p634 p264 p620 p1593 p757 p756 p796 p795 p794 p753 p854 p848 p754 p755 p619 p618 p267 p266 p582 p265 p226 p966 p968 p1284 p1283 p1282p1281 p1280 p1279 p1278 p1277 p1276 p1275 p1274 p1273 p1285 p969 p970 p971 p972 p973 p1209 p1208 p1207 p1206 p1205 p1204 p1203 p1566 p1202 p1470 p1471 p1472 p1200 p1198 p1540 p1539 p1538 p1537 p1199 p1516 p1517 p1518 p1519 p1201 p967 p888 p229 p230 p5 p6 p907 p908p909 p910 p911 p912 p560 p562 p570p569 p561 p563 p106 p340 p181 p439 p438 p1550 p1551 p1552 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8 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).

9 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).

10 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 are all connected. y + z f(y, z) > 0, Alternatively it is sufficient if the same condition is satisfied with X replacing Y. Is there a simple necessary and sufficient condition?

11 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.

12 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;

13 (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.

14 Probabilistic semigraphoids V finite set, X = (X v, v V ) random variables. For A V, let X A = (X v, v A). Let X v denote state space of X v. Similarly x A = (x v, v A) X A = v A X v. Abbreviate: A B S X A X B X S. 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.

15 Second order conditional independence Sets of random variables A and B are partially uncorrelated for fixed C if their residuals after linear regression on X C are uncorrelated: Cov{X A E (X A X C ), X B E (X B X C )} = 0, in other words, if the partial correlations are zero We then write A 2 B C. ρ AB C = 0. 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.

16 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.

17 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;

18 (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).

19 Variation independence Let U X = v V X v and define for S V the S-section U u S of U as U u S = {uv \S : u S = u S, u U}. Define further the conditional independence relation U as A U B C u C : U u C = {U u C }A {U u C }B 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.

20 Markov properties for semigraphoids G = (V, E) simple undirected graph; σ relation. Say σ satisfies (semi)graphoid (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.

21 Pairwise Markov property 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}.

22 Local Markov property 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}.

23 Global Markov property 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}.

24 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.

25 (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).

26 (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).

27 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.

28 Factorisation and Markov properties For a V, ψ a (x) is a function depending on x a only, i.e. x a = y a = ψ a (x) = ψ a (y). We can then write ψ a (x) = ψ a (x a ) 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) = a A ψ a (x), where A are complete subsets of G or, equivalently, if f(x) = c C ψ c (x), where C are the cliques of G.

29 Factorization example 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) = ψ 12 (x 1, x 2 )ψ 13 (x 1, x 3 )ψ 24 (x 2, x 4 )ψ 25 (x 2, x 5 ) ψ 356 (x 3, x 5, x 6 )ψ 47 (x 4, x 7 )ψ 567 (x 5, x 6, x 7 ).

30 Factorisation of the multivariate Gaussian Consider a multivariate Gaussian random vector X = N V (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 = k αβ = 0 i.e. if the concentration matrix has zero entries for non-adjacent vertices.

31 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 and further: (F) = (G) 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).