Conditional Independence and Markov Properties

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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
              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 u
                    d       d u
                      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.
               A highly complex pedigree
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                 p314    p72 p288     p295 p318 p31 p293 p25 p296 p10 p73        p193 p81 p106 p354 p304 p357 p148     p337 p278 p171 p125 p368 p284       p170 p321    p9  p176 p262 p235 p260 p242    p243 p34 p3    p248 p95 p230 p228 p4    p348 p94 p7       p23 p272 p326 p338 p285 p325        p309 p76 p344 p323 p238 p180

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             p187 p551 p55 p96 p335p67 p552
               p311 p310                   p487p434 p486 p566
                                                 p305 p483   p481p312 p145p579
                                                                   p478             p581
                                                                              p139p580 p191p360p366p423 p527 p562 p532p424 p452 p520 p453 p459 p455 p521 p331 p328 p457p505 p477 p458p435
                                                                                                     p480 p528 p522     p430 p427 p194 p142 p437 p144 p535 p369 p454     p395 p467       p557p251 p465 p286p265
                                                                                                                                                                                               p173 p463       p538p433 p431
                                                                                                                                                                                                                     p141         p300 p269 p429 p264 p504 p250
                                                                                                                                                                                                                            p24 p253 p525 p428 p229 p419 p169 p448p417p494p495 p418p261 p450 p492 p444p379 p466 p347
                                                                                                                                                                                                                                                                                     p493 p237 p443
                                                                                                                                                                                                                                                                            p266                          p381 p155 p482p588p351p196p289p339

           p553 p315 p910 p524 p907 p333 p889 p909 p913 p882 p548 p555 p620 p189 p940 p941 p929 p618 p619 p595 p363 p926 p384 p914 p421 p881 p377 p543 p596 p880 p432 p654 p906 p514 p510 p385 p506 p637 p927 p920 p930 p901 p256 p915 p621 p571 p886 p533 p502 p150 p885 p603 p623 p633 p605 p572 p593 p392 p439 p393 p599 p378 p473 p398 p684 p496 p399 p629 p627 p374 p576
              p554 p523 p573 p574 p912 p895 p156 p575 p190 p383 p938 p546 p549 p948 p946 p451 p388 p386 p902 p896 p425 p928 p585 p536 p440 p892 p888 p597 p364 p512 p714 p544 p138 p397 p905 p921 p622 p887 p479 p367 p934 p373 p916 p919 p540 p636 p884 p890 p656 p933 p168 p644 p688 p607 p387 p518 p517 p394 p146 p390 p686 p200 p602 p516 p771 p628 p770 p570 p630 p197 p769
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                     p701                               p682 p1077                                        p956
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                                                                                                                   p370 p970     p118 p611 p1123         p617p749 p711p733 p660 p612p662 p159 p1100 p657 p640 p1009 p641 p1027 p1026p1053 p976 p678p1025p715p974p965p130p408 p1044p409p1051 p382p403 p719p658p1012p718 p964p649 p721 p120 p671 p1079
                                                                                                                                                                p1013    p709   p706 p712 p1097 p1096p1007 p1039p1055p1071 p673 p1022p1073p977             p724                  p1040 p751     p601      p1083        p730 p380     p668 p402 p110 p132     p210

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               p163 p777 p1267 p789 p1455
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                                                                         p1068 p1084 p1454 p1593 p1087 p784 p1125
                                                   p786 p1371 p994                                                    p1331 p857     p732 p780 p1578 p1146                                                                                        p1419 p1094 p117
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                                                                                                                                                                                                                                                                                                                       p852         p972 p1438 p1165 p1222           p1591 p1224

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             p1394p128 p1389p12911292p1451p879 p1262p1326p1411p1255p1440p1245p1522p1311p1254p1473p1155p1248p1409p1566p1210p1347p1341p1427p126p1433p583p1285p1432p1430pp1196p1273 p1302p1310p1422p1239 p1306p860 p1113p859p1114p1235p811p1131p592 p1445p1423p1116p1156p1448 p861 p865 p1489p1472p1119p1215p806 p1552p1595p1525p219 p1382p115 p1514p1528p1487
              p1512p1393p121 p863 p1328p1287p1415p773 p1463p1246p1452p1410p1524p1523p1405p1413p1345p1348p1401p1283p1279p1399p1276p1233p1431p1309p1468 p1323p1307p818 820p1237p1320p1313p814 p1315p1316 p1426p1350p813 p1236 p1530p1517 p1537p1516p1444p1118p1153p1160 p1234p1211p1178p868 p875 p1380p675 p804p1377p1167p1385p1594p809 p1383p1527p1526
               p1286p1388p1390p1293p1257p1449p1403 p1264p1192p1395p1250p1252p1253p1325p1150p1249p1400p1251p1324p1498p1340p1342p1406p1346p1397 p1429p1398p1284p823 p819
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                                                                                                                                                                          p822 p1318p167 p1298p812 p1314p1278p1241p1240 p1238p1213p847 p1117p1120p1518p1424p877 p1476p872 p1214p1556p1564p1470p1461p808p807 p1381 p842 p1164p15971596p1379p1602p1604


Family relationship of 1641 members of Greenland Eskimo
               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)
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.

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
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}.

Define further the conditional independence relation ‡U as
                                ∗        ∗           ∗
    A ‡U B | C ⇐⇒ ∀u∗ : U uC = {U uC }A × {U uC }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
     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 u
                      d       d u
                        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 u
                     d       d u
                       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 u
                    d       d u
                      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),

where A are complete subsets of G or, equivalently, if
                     f (x) =         ˜
                                     ψc (x),

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
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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