graphical models - oxford statisticssteffen/teaching/grad/graphicalmodels.pdf · google gives 7 420...

Genesis and historyExamples

Markov theoryComplex models

References

Graphical Models

Steffen Lauritzen, University of Oxford

Graduate Lectures Hilary Term 2011

January 27, 2011

Steffen Lauritzen, University of Oxford Graphical Models



References

I Precursors originate mostly from Physics (Gibbs, 1902),Genetics (Wright, 1921, 1934), and Economics (Wold, 1954);

I Early graphical models in statistics include covarianceselection models (Dempster, 1972) and log-linear models(Haberman, 1974);

I Papers setting the scene include Darroch et al. (1980),Wermuth and Lauritzen (1983), and Lauritzen and Wermuth(1989).

I Subject took off after Pearl (1988) and Lauritzen andSpiegelhalter (1988), and in particular after Whittaker (1990)and Lauritzen (1996).

I Developments now prolific and it is largely impossible to keeptrack. Google gives 7 420 000 hits.




References

A directed graphical model

Directed graphical model (Bayesian network) showing relationsbetween risk factors, diseases, and symptoms.




References

A pedigree

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




References

A large pedigree

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p1327 p1324p1325p1326

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Family relationship of 1641 members of Greenland Eskimopopulation.




References

Conditional IndependenceUndirected graphsDirected acyclic graphsMoralization

Random variables X and Y are conditionally independent given therandom 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:

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).




References


A distribution P is said to factorize w.r.t. and undirected graph ifits joint density f can be written as

f (x) = Z−1∏A∈A

φA(xA), (1)

where A are complete subsets of the graph.

Here x = (xv , v ∈ V ), xA = (xv , v ∈ A) so φA only depends theA-coordinates of x .

The factorization is matched by a global Markov property, ie thatA⊥⊥B | S if S separates A from B in G, written as A⊥G B | S(Hammersley and Clifford, 1971).




References


Factorization example

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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).




References


Global Markov property

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To find conditional independence relations, one should look forseparating 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}.




References


Pairwise and local Markov properties

G = (V ,E ) simple undirected graph; A distribution P satisfies

(P) the pairwise Markov property if

α 6∼ β ⇒ α⊥⊥P β |V \ {α, β};

(L) the local Markov property if

∀α ∈ V : α⊥⊥P V \ cl(α) | bd(α);




References


Pairwise Markov property

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Any non-adjacent pair of random variables are conditionallyindependent given the remaning.For example, 1⊥⊥ 5 | {2, 3, 4, 6, 7} and 4⊥⊥ 6 | {1, 2, 3, 5, 7}.




References


Local Markov property

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Every variable is conditionally independent of the remaining, givenits neighbours.For example, 5⊥⊥{1, 4} | {2, 3, 6, 7} and 7⊥⊥{1, 2, 3} | {4, 5, 6}.




References


Let (F) denote the property that f factorizes w.r.t. G and let (G),(L) and (P) denote the Markov properties as defined. Then itholds that

(F)⇒ (G)⇒ (L)⇒ (P).

All reverse implications are false in general.

If f (x) > 0 for all x it further holds that

(P)⇒ (F)

so then(F) ⇐⇒ (G) ⇐⇒ (L) ⇐⇒ (P)

(Lauritzen, 1996, Chap. 3).




References


A probability distribution P over X = XV factorizes over a DAG Dif its density or probability mass function f has the form

f (x) =∏v∈V

fv (xv | xpa(v)).

A well-known example is a Markov chain:

X1 X2 X3 X4 X5

s s ss s- - - - - sXn

with Xi+1⊥⊥ (X1, . . . ,Xi−1) |Xi for i = 3, . . . , n.




References


Example of DAG factorization

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1 5 7

2 4

u uu u u

u u

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The above graph corresponds to the factorization

f (x) = f (x1)f (x2 | x1)f (x3 | x1)f (x4 | x2)

× f (x5 | x2, x3)f (x6 | x3, x5)f (x7 | x4, x5, x6).




References


Local directed Markov property

A distribution P satisfies the local Markov property (L) w.r.t. adirected acyclic graph D if

∀α ∈ V : α⊥⊥P {nd(α) \ pa(α)} | pa(α).

Here nd(α) are the non-descendants of α.




References


Local directed Markov property

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For example, the local Markov property says4⊥⊥{1, 3, 5, 6} | 2,5⊥⊥{1, 4} | {2, 3}3⊥⊥{2, 4} | 1.




References


A distribution satisfies the global Markov property w.r.t. D if

A⊥D B |S ⇒ A⊥⊥B |S .

Here ⊥D is d-separation, which is somewhat subtle.

It is always true for a DAG that

(F) ⇐⇒ (G) ⇐⇒ (L)

(Pearl, 1986; Geiger and Pearl, 1990; Lauritzen et al., 1990).




References


Separation in DAGs

A trail τ from vertex α to vertex β in a DAG D is blocked by S ifit contains a vertex γ ∈ τ such that

I either γ ∈ S and edges of τ do not meet head-to-head at γ, or

I γ and all its descendants are not in S , and edges of τ meethead-to-head at γ.

A trail that is not blocked is active. Two subsets A and B ofvertices are d-separated by S if all trails from A to B are blockedby S . We write A⊥D B | S .




References


Separation by example

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1 5 7

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For S = {5}, the trail (4, 2, 5, 3, 6) is active, whereas the trails(4, 2, 5, 6) and (4, 7, 6) are blocked.For S = {3, 5}, they are all blocked.




References


Returning to example

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1 5 7

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Hence 4⊥D 6 | 3, 5, but it is not true that 4⊥D 6 | 5 nor that4⊥D 6.




References


The moral graph Dm of a DAG D is obtained by adding undirectededges between unmarried parents and subsequently droppingdirections, as in the example below:

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References


Undirected factorizations

If P factorizes w.r.t. D, it factorizes w.r.t. the moralised graph Dm.

This is seen directly from the factorization:

f (x) =∏v∈V

f (xv | xpa(v)) =∏v∈V

ψ{v}∪pa(v)(x),

since {v} ∪ pa(v) are all complete in Dm.

Hence if P satisfies any of the directed Markov properties w.r.t. D,it satisfies all Markov properties for Dm.




References


Alternative equivalent separation

To resolve query involving three sets A, B, S :

1. Reduce to subgraph induced by ancestral set DAn(A∪B∪S) ofA ∪ B ∪ S ;

2. Moralize to form (DAn(A∪B∪S))m ;

It then holds that A⊥D B | S if and only if S separates A from Bin this undirected graph.

Proof in Lauritzen (1996) needs to allow self-intersecting paths tobe correct.




References


Forming ancestral set

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The subgraph induced by all ancestors of nodes involved in thequery 4⊥D 6 | 3, 5?




References


Adding links between unmarried parents

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Adding an undirected edge between 2 and 3 with common child 5in the subgraph induced by all ancestors of nodes involved in thequery 4⊥D 6 | 3, 5?




References


Dropping directions

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Since {3, 5} separates 4 from 6 in this graph, we can conclude that4⊥D 6 | 3, 5




References

Bayesian inference using Gibbs sampling

A particular successful development is associated with BUGS,(Gilks et al., 1994) (WinBUGS, OpenBUGS).

I enables a Bayesian analyst to focus on substantive modellingwhereas the technical model specification and computationalside is taken care of automatically,

I exploiting modularity, factorization, and MCMC methodology,including the Gibbs and Metropolis–Hastings sampler.

I Conforming with Bayesian paradigm, parameters andobservations are explicitly represented in model as nodes ingraph, all being observables;




References


Linear regression

for(i IN 1 : N)

sigma

taubetaalpha

mu[i]

Y[i]

Linear regression as a full Bayesian graphical model.




References


Linear regression

model

{

for( i in 1 : N ) {

Y[i] ~ dnorm(mu[i],tau)

mu[i] <- alpha + beta * (x[i] - xbar)

}

tau ~ dgamma(0.001,0.001) sigma <- 1 / sqrt(tau)

alpha ~ dnorm(0.0,1.0E-6)

beta ~ dnorm(0.0,1.0E-6)

}




References


Data and BUGS model for pumps

The number of failures Xi is assumed to follow a Poissondistribution with parameter θi ti , i = 1, . . . , 10where θi is the failure rate for pump i and ti is the length ofoperation time of the pump (in 1000s of hours). The data areshown below.

Pump 1 2 3 4 5 6 7 8 9 10ti 94.5 15.7 62.9 126 5.24 31.4 1.05 1.05 2.01 10.5xi 5 1 5 14 3 19 1 1 4 22

A gamma prior distribution is adopted for the failure rates:θi ∼ Γ(α, β), i = 1, . . . , 10




References


Gamma model for pumpdata

for(i IN 1 : N)

betaalpha

t[i]theta[i]

lambda[i]

x[i]

Failure of 10 power plant pumps.




References


BUGS program for pumps

With suitable priors the program becomes

model

{

for (i in 1 : N) {

theta[i] ~ dgamma(alpha, beta)

lambda[i] <- theta[i] * t[i]

x[i] ~ dpois(lambda[i])

}

alpha ~ dexp(1)

beta ~ dgamma(0.1, 1.0)

}




References


Growth of rats

for(j IN 1 : T)for(i IN 1 : N)

sigma

tau.cx[j]

Y[i, j]

mu[i, j]

beta[i]alpha[i]

beta.taubeta.calpha0alpha.calpha.tau

Growth of 30 young rats.




References


Finding full conditionals for Gibbs sampler

Inference in Bayesian complex graphical models as above uses theGibbs sampler.

For a DAG the densities of full conditional distributions are:

f (xi | xV \i ) ∝∏v∈V

f (xv | xpa(v))

∝ f (xi | xpa(i))∏

v∈ch(i)

f (xv | xpa(v))

= f (xi | xbl(i)),

x where bl(i) is the Markov blanket of node i :

bl(i) = pa(i) ∪ ch(i) ∪{∪v∈ch(i) pa(v) \ {i}

}.




References


Markov blanket

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Markov blanket of 6 is bl(6) = {3, 5, 7, 4}.




References


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The Markov blanket is just the neighbours of in the moral graph:

bl(v) = nem(v) so bl(6) = {3, 5, 7, 4} and bl(3) = {1, 5, 6, 2}.The DAG is used for modular specification of the model, and themoral graph for local computation.




References


I Is a huge conceptual extension of so-called Bayesianhierarchical models;

I distinction prior/likelihood and parameter/random variableless well defined;

I If founder nodes in network are considered fixed and unknown,no reason not to consider models in Fisherian paradigm.




References

Darroch, J. N., S. L. Lauritzen, and T. P. Speed (1980). Markovfields and log-linear interaction models for contingency tables.The Annals of Statistics 8, 522–539.

Dempster, A. P. (1972). Covariance selection. Biometrics 28,157–175.

Geiger, D. and J. Pearl (1990). On the logic of causal models. InR. D. Shachter, T. S. Levitt, L. N. Kanal, and J. F. Lemmer(Eds.), Uncertainty in Artificial Intelligence 4, pp. 3–14.Amsterdam, The Netherlands: North-Holland.

Gibbs, W. (1902). Elementary Principles of Statistical Mechanics.NewHaven, Connecticut: Yale University Press.

Gilks, W. R., A. Thomas, and D. J. Spiegelhalter (1994). BUGS: alanguage and program for complex Bayesian modelling. TheStatistician 43, 169–178.

Haberman, S. J. (1974). The Analysis of Frequency Data.Chicago, Illinois: University of Chicago Press.




References

Hammersley, J. M. and P. E. Clifford (1971). Markov fields onfinite graphs and lattices. Unpublished manuscript.

Lauritzen, S. L. (1996). Graphical Models. Oxford, UnitedKingdom: Clarendon Press.

Lauritzen, S. L., A. P. Dawid, B. N. Larsen, and H.-G. Leimer(1990). Independence properties of directed Markov fields.Networks 20, 491–505.

Lauritzen, S. L. and D. J. Spiegelhalter (1988). Localcomputations with probabilities on graphical structures and theirapplication to expert systems (with discussion). Journal of theRoyal Statistical Society, Series B 50, 157–224.

Lauritzen, S. L. and N. Wermuth (1989). Graphical models forassociations between variables, some of which are qualitativeand some quantitative. The Annals of Statistics 17, 31–57.

Pearl, J. (1986). A constraint–propagation approach toprobabilistic reasoning. In L. N. Kanal and J. F. Lemmer (Eds.),




References

Uncertainty in Artificial Intelligence, Amsterdam, TheNetherlands, pp. 357–370. North-Holland.

Pearl, J. (1988). Probabilistic Inference in Intelligent Systems. SanMateo, CA: Morgan Kaufmann Publishers.

Wermuth, N. and S. L. Lauritzen (1983). Graphical and recursivemodels for contingency tables. Biometrika 70, 537–552.

Whittaker, J. (1990). Graphical Models in Applied MultivariateStatistics. Chichester, United Kingdom: John Wiley and Sons.

Wold, H. O. A. (1954). Causality and econometrics.Econometrica 22, 162–177.

Wright, S. (1921). Correlation and causation. Journal ofAgricultural Research 20, 557–585.

Wright, S. (1934). The method of path coefficients. The Annals ofMathematical Statistics 5, 161–215.


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