Representation and Learning in - PowerPoint Presentation

Slide1

Representation and Learning inDirected Mixed Graph Models

Ricardo Silva

Statistical Science/CSML, University

College London

ricardo@stats.ucl.ac.uk

Networks: Processes and Causality, Menorca 2012Slide2

Graphical ModelsGraphs provide a language for describing independence constraints

Applications to causal and probabilistic processes

The corresponding probabilistic models should obey the constraints encoded in the graph

Example: P(X1, X2, X3) is Markov with respect toif X1 is independent of X3 given X2 in P(  )

X1

X2

X3Slide3

Directed Graphical Models

X

1

X2

U

X3

X4

X

2

X

4

X

2

X

4

| X

3

X

2

X

4

| {X

3

, U}

...Slide4

Marginalization

X

1

X2

X3

U

X4

X

2

X

4

X

2

X

4

| X

3

X

2

X

4

| {X

3

, U}

...Slide5

Marginalization

No: X

1

X3 | X2

X1

X2

X3

X4

?

X1

X2

X3

X4

?

No: X

2

X

4

| X

3

X1

X2

X3

X4

?

OK, but not ideal

X

2

X

4

Slide6

The Acyclic Directed Mixed Graph (ADMG)

“Mixed” as in directed + bi-directed

“Directed” for obvious reasons

See also: chain graphs“Acyclic” for the usual reasonsIndependence model isClosed under marginalization (generalize DAGs)Different from chain graphs/undirected graphs

Analogous inference as in DAGs: m-separationX

1X

2X

3

X

4

(Richardson and Spirtes, 2002; Richardson, 2003)Slide7

Why do we care?Difficulty on computing scores or tests

Identifiability: theoretical issues and implications to optimization

Y

1

Y2

Y3

Y4

Y5

X

1

X

2

latent

observed

Y

1

Y

2

Y

3

Y

4

Y

5

X

1

X

3

Y

6

Y

6

X

2

Candidate I

Candidate IISlide8

Why do we care?

Set of

“target”

latent variables X (possibly none), and observations YSet of

“nuisance” latent variables XWith sparse structure implied over Y

Y1

Y2

Y

3

Y

4

Y

5

X

1

X

2

Y

6

X

3

X

4

X

5

X



XSlide9

Why do we care?

(

Bollen

, 1989)Slide10

The talk in a nutshell

The challenge:

How to specify families of distributions that respect the ADMG independence model, requires no explicit latent variable formulation

How NOT to do it: make everybody independent!Needed: rich families. How rich?Main results: A new construction that is fairly general, easy to use, and complements the state-of-the-art

Exploring this in structure learning problemsFirst, background: Current parameterizations, the good and bad issuesSlide11

The Gaussian bi-directed modelSlide12

The Gaussian bi-directed case

(Drton and Richardson, 2003)Slide13

Binary bi-directed case: the constrained Moebius

parameterization

(Drton and Richardson, 2008)Slide14

Binary bi-directed case:the constrained Moebius

parameterization

Disconnected sets

are marginally independent. Hence, define qA for connected sets onlyP(X1 = 0, X4

= 0) = P(X1 = 0)P(X4 = 0)

q14 = q1q4

However, notice there is a parameter q1234Slide15

Binary bi-directed case:the constrained Moebius

parameterization

The good:

this parameterization is complete. Every single binary bi-directed model can be represented with itThe bad: Moebius inverse is intractable, and number of connected sets can grow exponentially even for trees of low connectivity

...

...

...

...Slide16

The Cumulative Distribution Network (CDN) approach

Parameterizing cumulative distribution functions (CDFs) by a product of functions defined over subsets

Sufficient condition: each factor is a CDF itself

Independence model: the “same” as the bi-directed graph... but with extra constraints(Huang and Frey, 2008)

F(X1234) = F1(X12)F2(X

24)F3(X34)F4(X

13)X1 X

4

X

1

X

4

| X

2

etcSlide17

The Cumulative Distribution Network (CDN) approachWhich extra constraints?

Meaning, event “X

1

 x1” is independent of “X3  x3” given “X2  x2”Clearly not true in general distributionsIf there is no natural order for variable values, encoding does matter

X1

X2

X3

F(X

123

)

=

F

1

(X

12

)F

2

(X

23

)Slide18

RelationshipCDN: the resulting PMF (usual CDF2PMF transform)

Moebius

: the resulting PMF is equivalent

Notice: qB = P(XB = 0) = P(X\B  1, XB  0)However, in a CDN, parameters further factorize over cliques

q

1234 = q12

q13q24q34Slide19

RelationshipCalculating likelihoods can be easily reduced to inference in factor graphs in a “pseudo-distribution”

Example: find the joint distribution of

X

1, X2, X3 belowX1

X2

X

3

reduces to

X

1

X

2

X

3

Z

1

Z

2

Z

3

P(

X

=

x

) = Slide20

RelationshipCDN

models are a strict subset of

marginal independence models

Binary case: Moebius should still be the approach of choice where only independence constraints are the targetE.g., jointly testing the implication of independence assumptionsBut...CDN models have a reasonable number of parameters, for small tree-widths any fitting criterion is tractable, and learning is trivially tractable anyway by marginal composite likelihood estimation (more on that later)Take-home message: a still flexible bi-directed graph model with no need for latent variables to make fitting tractableSlide21

The Mixed CDN model (MCDN)

How to construct a distribution Markov to this?

The binary ADMG parameterization by Richardson (2009) is complete, but with the same computational

difficulties

And how to easily extend it to non-Gaussian, infinite discrete cases, etc.?Slide22

Step 1: The high-level factorization

A

district

is a maximal set of vertices connected by bi-directed edgesFor an ADMG G with vertex set XV and districts {Di}, definewhere P() is a density/mass function and paG() are parent of the given set in GSlide23

Step 1: The high-level factorization

Also, assume that each

P

i( | ) is Markov with respect to subgraph Gi – the graph we obtain from the corresponding subsetWe can show the resulting distribution is Markovwith respect to the ADMG

X4

X1

X

4

X

1Slide24

Step 1: The high-level factorization

Despite the seemingly “cyclic” appearance, this factorization always gives a valid

P

() for any choice of Pi( | )P(X134) =

x2P(X1, x2 | X4

)P(X3, X4 | X1)

 P(X1 | X4)P(X3

, X4 | X1)

P(X

13

) =



x4

P(X

1

| x

4

)P(X

3

, x4 | X1

)

=

x4P(X1)P(X

3, x4 | X

1)  P(X1

)P(X3

| X1)Slide25

Step 2: Parameterizing Pi (barren case)

D

i

is a “barren” district is there is no directed edge within it

Barren

NOT BarrenSlide26

Step 2: Parameterizing Pi (barren case)

For a district

D

i with a clique set Ci (with respect bi-directed structure), start with a product of conditional CDFsEach factor FS(xS | xP) is a conditional CDF function, P(XS 

xS | XP = xP). (They have to be transformed back to PMFs/PDFs when writing the full likelihood function.)

On top of that, each FS(xS | xP) is defined to be Markov with respect to the corresponding G

iWe show that the corresponding product is Markov with respect to GiSlide27

Step 2a: A copula formulation of Pi

Implementing the local factor restriction could be potentially complicated, but the problem can be easily approached by adopting a

copula

formulationA copula function is just a CDF with uniform [0, 1] marginalsMain point: to provide a parameterization of a joint distribution that unties the parameters from the marginals from the remaining parameters of the jointSlide28

Step 2a: A copula formulation of Pi

Gaussian latent variable analogy:

X

1

X2

U

X1

= 1U

+

e

1

,

e

1

~ N(0,

v

1

)

X

2

= 

2U +

e2, e

2 ~ N(0, v

2)U ~ N(0,

1)

Marginal of

X1: N(0, 

12 +

v1)Covariance of X1

, X2:

1

2

Parameter sharingSlide29

Step 2a: A copula formulation of Pi

Copula idea: start from

then define

H(Ya, Yb) accordingly, where 0  Y*  1H(, ) will be a CDF with uniform [0, 1] marginalsFor any Fi

() of choice, Ui  Fi(Xi) gives an uniform [0, 1]We mix-and-match any marginals we want with any copula function we want

F(X1, X2) =

F( F1-1(F1 (

X1)), F2

-1

(

F

2

(

X

2

)))

H(Y

a

, Yb)

 F

( F1-1(

Ya), F

2-1(Yb

))Slide30

Step 2a: A copula formulation of Pi

The idea is to use a conditional marginal

F

i(Xi | pa(Xi)) within a copulaExampleCheck:X1

X2

X3

X4

U

2

(x

1

)



P

2

(

X

2



x

2

|

x

1)U3(x

4)

 P2(X

3  x

3 | x4)

P(X2  x

2,

X3 

x3 | x

1, x4) = H(U

2(x1),

U

3

(x

4

))

P

(X

2



x

2

|

x

1,

x

4

) = H

(U

2

(x

1

), 1) = H(

U

2

(

x

1

))

=

U

2

(x

1

) = P

2

(X

2



x

2

|

x

1

) Slide31

Step 2a: A copula formulation of Pi

Not done yet! We need this

Product of copulas is not a copula

However, results in the literature are helpful here. It can be shown that plugging in Ui1/d(i), instead of Ui will turn the product into a copulawhere d(i) is the number of bi-directed cliques containing Xi

Liebscher (2008)Slide32

Step 3: The non-barren case

What should we do in this case?

Barren

NOT BarrenSlide33

Step 3: The non-barren caseSlide34

Step 3: The non-barren caseSlide35

Parameter learningFor the purposes of illustration, assume a finite mixture of experts for the conditional marginals for continuous data

For discrete data, just use the standard CPT formulation found in Bayesian networksSlide36

Parameter learningCopulas: we use a bi-variate formulation only (so we take products “over edges” instead of “over cliques”).

In the experiments: Frank copulaSlide37

Parameter learningSuggestion: two-stage quasiBayesian learning

Analogous to other approaches in the copula literature

Fit marginal parameters using the posterior expected value of the parameter for each individual mixture of experts

Plug those in the model, then do MCMC on the copula parametersRelatively efficient, decent mixing even with random walk proposalsNothing stopping you from using a fully Bayesian approach, but mixing might be bad without some smarter proposalsNotice: needs constant CDF-to-PDF/PMF transformations!Slide38

ExperimentsSlide39

ExperimentsSlide40

The story so far

General toolbox for construction for ADMG models

Alternative estimators would be welcome:

Bayesian inference is still “doubly-intractable” (Murray et al., 2006), but district size might be small enough even if one has many variablesEither way, composite likelihood still simple. Combined with the Huang + Frey dynamic programming method, it could go a long wayHybrid Moebius/CDN parameterizations to be exploitedEmpirical applications in problems with extreme value issues, exploring non-independence constraints, relations to effect models in the potential outcome framework etc.Slide41

Back to: Learning Latent Structure

Difficulty on computing scores or tests

Identifiability: theoretical issues and implications to optimization

Y1

Y2

Y3

Y4

Y5

X

1

X

2

latent

observed

Y

1

Y

2

Y

3

Y

4

Y

5

X

1

X

3

Y

6

Y

6

X

2

Candidate I

Candidate IISlide42

Leveraging Domain StructureExploiting “main” factors

Y

7a

Y

7

b

Y7c

X7

Y

7

d

Y

7e

Y

12a

X

12

Y

12b

Y

12c

(NHS Staff Survey, 2009)Slide43

The “Structured Canonical Correlation” Structural Space

Set of pre-specified latent variables

X

, observations YEach Y in Y has a pre-specified single parent in

XSet of unknown latent variables X XEach Y in Y can have potentially infinite parents in X

“Canonical correlation” in the sense of modeling dependencies within a partition of observed variables

Y1

Y2

Y

3

Y

4

Y

5

X

1

X

2

Y

6

X

3

X

4

X

5

X



XSlide44

The “Structured Canonical Correlation”: Learning Task

Assume a

partition structure of

Y according to X is knownDefine the mixed graph projection of a graph over (X, Y) by a bi-directed edge Yi  Yj if they share a common ancestor in XPractical assumption: bi-directed substructure is sparseGoal: learn bi-directed structure (and parameters) so that one can estimate functionals of P(

X | Y)

Y1

Y2

Y3

Y

4

Y

5

X

1

X

2

Y

6Slide45

Parametric Formulation

X

~ N(0,

),  positive definiteIgnore possibility of causal/sparse structure in X for simplicityFor a fixed graph G, parameterize the conditional cumulative distribution function (CDF) of Y given X according to bi-directed structure:F(y | x

)  P(Y  y | X = x)   Pi(

Yi  yi | X[i

] = x[i])Each set Yi

forms a bi-directed clique in G, X[i

]

being the corresponding parents in

X

of the set

Y

i

We

assume here each

Y

is binary for simplicitySlide46

Parametric FormulationIn order to calculate the likelihood function, one should convert from the (conditional) CDF to the probability mass function (PMF)

P(

y

, x) = {F(y | x)} P(x)F(y | x) represents a difference operator. As we discussed before, for p-dimensional binary (unconditional) F(y

) this boils down toSlide47

Learning with Marginal Likelihoods

For

X

j parent of Yi in X:LetMarginal likelihood:Pick graph Gm that maximizes the marginal likelihood (maximizing also with respect to  and ), where  parameterizes local conditional CDFs Fi(y

i | x[i])Slide48

Computational ConsiderationsIntractable, of course

Including possible large tree-width of bi-directed component

First option: marginal bivariate composite likelihood

G

m

+/-

is the space of graphs that differ from

Gm by at most one bi-directed edge

Integrates



ij

and

X

1:N

with a crude quadrature methodSlide49

Beyond Pairwise ModelsWanted: to include terms that account for more than pairwise interactions

Gets expensive really fast

An indirect compromise:

Still only pairwise terms just like PCLHowever, integrate ij not over the prior, but over some posterior that depends on more than on Yi1:N, Yj1:N:Key idea: collect evidence from p(ij | Y

S1:N), {i, j}  S, plug it into the expected log of marginal likelihood . This corresponds to bounding each term of the log-composite likelihood score with different distributions for 

ij:Slide50

Beyond Pairwise ModelsNew score function

S

k

: observed children of Xk in XNotice: multiple copies of likelihood for ij when Yi and Yj have the same latent parentUse this function to optimize parameters {, } (but not necessarily structure)Slide51

Learning with Marginal LikelihoodsIllustration: for each pair of

latents

Xi, Xj, do

i

j

q

ij

(



ij

)  p(



ij

|

Y

S

, , )



ijk

Compute by

Laplace

approximation

and dynamic

programming

q

ij

(



ij

)

Defines

~

… +

E

q

(



ijk

)

[log P(

Y

ij1a

,

Y

ij1b,



ijk

|

, )] + …

Y

ij1a

Y

ij1b

Marginalize

and add term

~Slide52

Algorithm 2

q

mn

comes from conditioning on all variables that share a parent with Yi and YjIn practice, we use PCL when optimizing structureEM issues with discrete optimization: model without edge has an advantage, sometimesbad saddlepointSlide53

Experiments: Synthetic Data20 networks of 4 latent variables with 4 children per latent variable

Average number of bi-directed edges: ~18

Evaluation criteria:

Mean-squared error of estimate of slope  for each observed variableEdge omission error (false negatives)Edge commission error (false positives)Comparison against “single-shot” learningFit model without bi-directed edges, add edge Yi  Yj if implied pairwise distribution P(Yi, Y

j) doesn’t fit the dataEssentially a single iteration of Algorithm 1Slide54

Experiments: Synthetic DataQuantify results by taking the difference

between number of times Algorithm

2

does better than Algorithm 1 and 0 (“single-shot” learning)The number of times where the difference is positive with the corresponding p-values for a Wilcoxon signed rank test (stars indicate numbers less than 0.05)Slide55

Experiments: NHS DataFit model with 9 factors and 50 variables on the NHS data, using questionnaire as the partition structure

100,000 points in training set, about 40 edges discovered

Evaluation:

Test contribution of bi-directed edge dependencies to P(X | Y): compare against model without bi-directed edgesComparison by predictive ability: find embedding for each X(d) given Y(d) by maximizing Test on independent 50,000 points by evaluating how well we can predict other 11 answers based on latent representation using logistic regressionSlide56

Experiments: NHS DataMCCA: mixed graph structured canonical correlation model

SCCA: null model (without bi-directed edges)

Table contains AUC scores for each of the 11 binary prediction problems using estimated

X as covariates:Slide57

ConclusionMarginal composite likelihood and mixed graph models are a good match

Still requires some choices of approximations for posteriors over parameters, and numerical methods for integration

Future work:

Theoretical properties of the alternative marginal composite likelihood estimatorIdentifiability issuesReduction on the number of evaluations of qmnNon-binary dataWhich families could avoid multiple passes over data?