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共 2000 条

Proposed design for gR, a graphical models toolkit for R

Kevin P. Murphy

www.ai.mit.edu/∼murphyk

29 September 2003

Contents

1 Introduction 2

1.1 Comparison of proposed design with existing software . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Graphical model communities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Summary of proposed design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Representation 4

2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Class or structure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Kinds of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3 GUIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.4 Graph visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.5 File formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.6 Creating the graph via an API . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.7 Accessing the graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.8 Graph algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Local factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 CPDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Decision networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.4 Methods which local factors must implement . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Graphical Model class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 DAGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Chain graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Log-linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Factor graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.5 Decision networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.6 Frequentist models with tied parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.7 Generative 2D lattice MRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.8 Discriminative relational Markov networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.9 Structural/ relational uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.10 Dynamic Bayesian networks (DBNs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Inference 19

3.1 Types of inference engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Exact methods for models in which all hidden nodes are discrete . . . . . . . . . . . . . . . . 19

3.1.2 Exact methods for models in which some hidden nodes are not discrete . . . . . . . . . . . . 20

3.1.3 Deterministic approximations in which all hidden nodes are discrete . . . . . . . . . . . . . . 20

3.1.4 Deterministic approximations in which some hidden nodes are not discrete . . . . . . . . . . 21

3.1.5 Stochastic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.6 General comments on inference engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Things you can do with an inference engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 JPDs returned by query . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Applicability of inference engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 How inference engines are implemented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Internal potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Local factor methods needed by deterministic inference . . . . . . . . . . . . . . . . . . . . 24

3.4.3 Local factor methods needed by stochastic inference . . . . . . . . . . . . . . . . . . . . . . 24

3.4.4 How loopy belief propagation could be implemented . . . . . . . . . . . . . . . . . . . . . . 24

3.4.5 How junction tree inference could be implemented . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Online inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.1 High level online inference engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.2 The forwards and backwards operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.3 Representation of the belief state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.4 Possible implementation of the forwards and backwards operators . . . . . . . . . . . . . . . 28

4 Learning 28

4.1 Parameter learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Directed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.2 CPD methods needed by parameter learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.3 Undirected models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Structure Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Constraint-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Search-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Scaling up to large data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Decision making 33

6 Feasability of proposed design 34

7 Bibliography 35

1 Introduction

This document is an extension of the talk I gave at the gR meeting

in Aalborg on 19 September 2003. It outlines

a proposed design for gR, a graphical models library for R

. This design is similar to the design of BNT

, but is

much more general, in that it supports undirected models and chain graphs, and allows parameters to be represented

as random variables (Bayesian modeling).

1.1 Comparison of proposed design with existing software

There are a large number of software packages for graphical models in current use

, but none come close to the level of

generality, ﬂexibilityand openness that this design aspires to. Here I brieﬂy mention a few of the closest “competitors”:

• BNT probably comes closest, and implements about 60% of the functionality described here. The main func-

tionality it lacks is that BNT does not support Bayesian modeling, and only has limited support for undirected

models and no support for chain graphs.

• PNL

currently implements about half the functionalityof BNT (see discussion in Section 6). The main things it

lacks are inference for Gaussian models, decision networks, and good support for DBNs and structure learning.

http://www.r-project.org/gR

R is essentially an open-source version of S-plus. See www.r-project.org

Bayes Net Toolbox for Matlab. See www.ai.mit.edu/∼murphyk/Software/BNT/bnt.html

See www.ai.mit.edu/∼murphyk/Software/BNT/bnsoft.html for a comparison of existing software for graphical models.

Probabilistic Networks Library (Intel’s C++ version of BNT). See www.intel.com/research/mrl/pnl/

• BUGS

implements about 30% of the functionality described here. It handles Bayesian models, but not decision

networks, DBNs or structure learning. Also, it only has one kind of inference engine (Gibbs sampling). and is

not open source.

• GMTk

is designed for DBNs for speech recognition. It supports parameter estimation and structure learning,

but only has one kind of inference engine (junction tree), and does not support general graphical models. Also,

it is not currently open source.

• OpenBayes

was started by Richard Dybowski. This project never took off the ground, despite much discussion.

• “Software Support for Bayesian Analysis Systems”. Various systems were described at this workshop held at

NIPS 2000

and at Interface 2001

. However, none are publicly available (except BNT).

• AutoBayes

[FS03, GFSB02]. The AutoBayes system uses theorem proving techniques to ﬁnd symbolic so-

lutions to certain graphical model inference/learning problems. For example, it can automatically reproduce

textbook examples of Bayesian analysis of conjugate-exponentialmodels in the fully observed data regime. Un-

fortunately, small changes to the model (e.g., non conjugate priors or latent variables/missing data) often render

it intractable to solve analytically. In such a case, it generates model-speciﬁc code to compute the answer using

numerical integration. (Clearly other approximations could be used.) It can also generate EM-based algorithms.

Although the system is in principle very powerful, in practice the ability of a general purpose theorem prover to

recognize which algorithm/schema to use is limited. Also, the system is not open source.

1.2 Graphical model communities

Graphical model technology is currently being developed in several different communities, each with their own termi-

nology and philosophical outlook. Ideally gR will be catholic enough in its approach to be of interest to researchers in

each of these communities. More importantly, gR should be useful to users of GM technology, who span an even wider

range of communities, such as genetics, forensic science, epidemiology, bio-informatics, fault diagnosis, geostatistics,

machine learning, data mining, computer vision, text processing, speech recognition, robotics, error-correcting codes,

etc.

I come from the UAI/NIPS/Matlab community, which is reﬂected in the design outlined in this document. The

design will no doubt need modifying to satisfy the needs/style of the statistics/R community. However, hopefully it

will prove possible to “tunnel through” (to quote Steffen Lauritzen) from the machine learning community to the stats

community (and vice versa!).

1.3 Summary of proposed design

At the highest level of abstraction, we can identify the following major topics:

• Representation - how do we specify the graph structure and the parametric forms of the local distributions?

• Non-parametric reasoning - computations on the graph structure

• Inference - state estimation

• Learning - parameter estimation and model selection

• Control - decision making

For representation, we need the following main classes:

• Graph

Bayesian inference Using Gibbs Sampling. See http://www.mrc-bsu.cam.ac.uk/bugs/.

Graphical Models Toolkit. See ssli.ee.washington.edu/∼bilmes/gmtk/

See www.ai.mit.edu/∼murphyk/OpenBayes/index.html

http://ase.arc.nasa.gov/nips2000/

http://www.ics.uci.edu/∼interfac/all-sessions.html#FBI5

See http://ase.arc.nasa.gov/docs/autobayes.html

• LocalFactor - this can be a CPD (conditional probability distribution), which sums to one, or a Potential, which

is an arbitrary positive function of its arguments.

• GraphicalModel - this is just a graph plus a list of local factors. We will also consider various special-purpose

graphical model classes.

For non-parametric reasoning, we need to provide a set of graph functions e.g., to assess (causal) identiﬁability

[Pea00].

Once we have speciﬁed the parameters of the model, we can do many more things with it, such as:

• Infer hidden variables given a single case of evidence.

• Estimate (learn) parameters given a set of data cases (model ﬁtting).

• Sample values (if it’s a generative, as opposed to discriminative, model).

• Make optimal decisions (if it’s a decision network).

Although these may be thought of as methods of the GM class, in fact there are many algorithms to perform each

of these tasks. Since these algorithms often have internal state (e.g., auxiliary data structures such as a junction tree,

or storing internal parameters, such as the number of samples to collect), it is better to think of these algorithms as

objects, which take the GM (which is just a passive data structure) as an argument. Hence we will deﬁne inference and

learning “engines” below. Learning must call inference as a subroutine (except in the important special case of fully

observed DAG models).

This suggests the following extra classes

• InfEngine - inference engine

• ParamLearnEngine - for ML/MAP parameter estimation

• StructLearnEngine - structure learning

• DecisionEngine - for decision networks (inﬂuence diagrams)

We will discuss all of these in detail below.

2 Representation

2.1 Graphs

2.1.1 Class or structure?

The graph class should presumably be compatible with the R graph package

, although this does not seem to support

mixed directed-undirected graphs. Also, I think it is better to represent graphs using a “lightweight” structure, such

as a C-style struct, rather than a “heavy” class with lots of unrelated methods, which the current R graph class suffers

from. Using a struct rather than a class means that functions (unlike methods) can be added independently by different

developers (see discussion in Section 6), and can be grouped/stored according to functionality, rather than forcing

them all to be centrally stored in the graph class. (Imagine adding all the functions in Figure 12 to the graph class

deﬁnition! Why should someone who never uses the junction tree algorithm have to see these functions?)

The graph should just represent the topology. Extra information about the nodes and arcs is usually speciﬁc to a

particular parameterization of a graph, and is better kept inside the GM object than inside the graph object. Functions

that need to know about node/edge types can just take extra arguments (we will see examples below).

“LocalFactor” is perhaps not a good name, since the term “factor” has another meaning in R. We need a word that describes the union of

potentials and CPDs. “LocalDistribution” is possible, except a potential is not (necessarily) a distribution. The term “factor” was introduced in

[KFL01] to mean a term in the product deﬁning the joint. “Local factor” is a compromise, to disambiguate from the R usage.

Bayesians do not distinguish between parameters and other latent variables, and hence for them, parameter learning (estimation) is just a special

case of inference.

Available at www.bioconductor.org

2.1.2 Kinds of graphs

It will be useful to distinguish different kinds of graphs, such as DAGs, undirected graphs, mixed directed-undirected,

bipartite, etc. These could be deﬁned as subclasses, but this seems quite “heavy”; I prefer to minimize inheritance and

subtyping. Instead, it will probably sufﬁce to add a ’type’ ﬁeld to the struct, which speciﬁes what kind of graph this is.

2.1.3 GUIs

There has been some discussion of GUIs for R

, but none of this pertains to graphs or graphical models. In the UAI

community, the GeNIe

package is considered one of the best GUIs, but it is only freely-available in binary form, and

is restricted to DAG models.

There are many things the GUI should be able to support, such as:

• Draw a graph. (We want to allow for different types of nodes and edges).

• Specify the local factors interactively. (Each local factor class should provide a method for creating/updating

itself via a GUI.)

• Specify queries (e.g., click on desired nodes).

• Specify evidence/ training data (e.g., from a ﬁle).

• Call inference and learning engines. (Each engine should provide a method for calling itself via a GUI.)

• Display results of query e.g., as histogram, Gaussian 2D ellipsoid, quantile plots, etc. (Each JPD should provide

a method for displaying itself, presumably using built-in R functions.)

• Display results of structure learning (see Section 2.1.4).

• Call arbitrary R functions.

This is clearly a large amount of work, and is probably not appropriate for the gR project to attempt to do all this

(in my opinion).

2.1.4 Graph visualization

The graph class should call Graphviz

to do automatic graph layout using the Rgraphiz package

. This is useful

for visualizing the results of structure learning. The system should be able to represent the conﬁdence that an arc is

present. See, for example, Nir Friedman’s pathway explorer for visualizing features of models learned from biological

data

. The conﬁdence that each arc is present is indicated by shading; nodes can be clicked on, to pull up relevant

pubMed citations/data, etc.

The GUI and the graph visualizer should be integrated, so one can create an initial model, ﬁt it to data (modifying

its structure), visualize the resulting ﬁtted model, edit the result (by manually changing some arcs), and repeating the

cycle.

2.1.5 File formats

Much time has been spent in the UAI community discussing ﬁle formats for Bayes nets

. The emphasis is on how

to represent the CPDs, rather than representing the graph (which is comparitively easy). However, all of the current

proposals are much too restrictive for our purposes, since they cannot handle continuous variables, Bayesian models,

undirected models, DBNs, etc. gR should probably just deﬁne its own XML-based format, and then provide converters

for other popular formats such as Hugin and BUGS if demand warrants.

See http://www.sciviews.org/

rgui/

www.sis.pitt.edu/∼genie/

www.research.att.com/sw/tools/graphviz/

Available at www.bioconductor.org

http://www.cs.huji.ac.il/labs/compbio/ismb01/Rose/index1.html

See e.g., www.ai.mit.edu/∼murphyk/OpenBayes/ﬁle

formats.html.

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