probabilistic graphical modelsesucar/clases-mgp/notes/ch1-introduction.pdf · development of...

Uncertainty

A brief history

BasicprobabilisticmodelsExample

Probabilisticgraphicalmodels

Applications

Overview

References

Probabilistic Graphical Models:Principles and Applications

Chapter 1: INTRODUCTION

L. Enrique Sucar, INAOE

(L E Sucar: PGM) 1 / 42

Uncertainty

A brief history



Applications

Overview

References

Outline

1 Uncertainty

2 A brief history

3 Basic probabilistic modelsExample

4 Probabilistic graphical models

5 Applications

6 Overview

7 References


Uncertainty

A brief history



Applications

Overview

References

Uncertainty

Intelligent Agents

• Intelligent agents, natural or artificial, have to select acourse of actions among many possibilities

• They make decisions based on the information they canobtain from their environment, their previous knowledgeand their objectives

• Information and knowledge is incomplete or unreliable,and the results of their decisions are not certain, theyhave to make decisions under uncertainty: a medicaldoctor in an emergency, an autonomous vehicle thatdetects what might be an obstacle in its way, a financialagent needs to select the best investment

• One of the goals of artificial intelligence is to developsystems that can reason and make decisions underuncertainty


Uncertainty

A brief history



Applications

Overview

References

Uncertainty

Effects of Uncertainty

• Early artificial intelligence systems were based onclassical logic, in which knowledge can be representedas a set of logic clauses or rules. These systems havetwo important properties:

• modularity, each piece of knowledge can be usedindependently to arrive to conclusions

• monotonicity, knowledge always increasesmonotonically: any deduced fact is maintained even ifnew facts are known by the system

• If we have uncertainty these two properties are not truein general, which makes a system that has to reasonunder uncertainty more complex


Uncertainty

A brief history



Applications

Overview

References

A brief history

Beginnings (1950’s and 60’s)

• Artificial intelligence (AI) researchers focused on solvingproblems such as theorem proving, games like chess,and the “blocks world” planning domain, which do notinvolve uncertainty

• The symbolic paradigm dominated AI in the beginnings


Uncertainty

A brief history



Applications

Overview

References

A brief history

Ad hoc techniques (1970’s)

• The development of expert systems for realisticapplications such as medicine and mining, required thedevelopment of uncertainty management approaches,ad hoc techniques were developed for specific expertsystems:

• MYCIN’s certainty factors• Prospector’s pseudo-probabilities

• Later it was shown that these techniques had a set ofimplicit assumptions

• Alternative theories were proposed to manageuncertainty: fuzzy logic and the Dempster-Shafer theory


Uncertainty

A brief history



Applications

Overview

References

A brief history

Resurgence of probability (1980’s)

• Probability theory was used in some initial expertsystems, however it was later discarded as itsapplication in naive ways implies a high computationalcomplexity

• New developments, in particular Bayesian networks,make it possible to build complex probabilistic systemsin an efficient manner


Uncertainty

A brief history



Applications

Overview

References

A brief history

Diverse formalisms (1990’s)

• Bayesian networks continued and were consolidatedwith the development of efficient inference and learningalgorithms

• Other techniques such as fuzzy and non-monotoniclogics were considered as alternatives


Uncertainty

A brief history



Applications

Overview

References

A brief history

Probabilistic graphical models (2000’s)

• Several techniques based on probability and graphicalrepresentations were consolidated as powerful methodsfor representing, reasoning and making decisions underuncertainty

• Bayesian networks, Markov networks, influencediagrams and Markov decision processes, amongothers


Uncertainty

A brief history



Applications

Overview

References

Basic probabilistic models

The basic approach

• If we apply probability in a naive way to complexproblems, we are soon deterred by computationalcomplexity

• We can model a problem using a naive probabilisticapproach based on a flat representation; and then wecan use this representation to answer someprobabilistic queries

• This will help to understand the limitations of the basicapproach, motivating the development of probabilisticgraphical models


Uncertainty

A brief history



Applications

Overview

References


Model Formulation

• Many problems can be formulated as set of variables,X1,X2, ...Xn such that we know the values for some ofthese variables and the others are unknown. In medicaldiagnosis, the variables might represent certainsymptoms and the associated diseases

• Let us consider that the set of possible values is finite;for example, X = {x1, x2, ..., xm} might represent the mpossible diseases in a medical domain

• Each value of a random variable will have a certainprobability associated in a context, we can represent adomain as:

1 A set of random variables, X1,X2, ...Xn.2 A joint probability distribution associated to these

variables, P(X1,X2, ...Xn).


Uncertainty

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Probabilistic queries (1)

Marginal probabilities: the probability of one ofthe variables taking a certain value. This canbe obtained by summing over all the othervariables of the joint probability distribution. Inother words, P(Xi) =

∑∀X 6=Xi

P(X1,X2, ...Xn).This is known as marginalization.Conditional probabilities: the conditionalprobability of Xi given that we know Xj isP(Xi | Xj) = P(Xi ,Xj)/P(Xj),P(Xj) 6= 0.P(Xi ,Xj) and P(Xj) can be obtained viamarginalization, and from them we can obtainconditional probabilities.


Uncertainty

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Probabilistic queries (2)

Total Abduction or MPE: given that a subset (E)of variables is known, abduction consists infinding the values of the rest of variables (J)that maximize their conditional probability giventhe evidence, maxP(J | E). That isArgX∈J [maxP(X1,X2, ...Xn)/P(X ∈ E)].Partial abduction or MAP: in this case there are3 subsets of variables: the evidence, E , thequery variables that we want to maximize, J,and the rest of the variables, K , such that wewant to maximize P(J | E). This is obtained bymarginalizing over K and maximizing over J,that isArgX∈J [max

∑X∈K P(X1,X2, ...Xn)/P(X ∈ E)].


Uncertainty

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Basic probabilistic models Example

golf example

• In this problem we have 5 variables: outlook,temperature, humidity, windy, play.

• All variables are discrete so they can take a value froma finite set of values, for instance outlook could besunny, overcast or rain


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A sample data set for the golf example

Outlook Temperature Humidity Windy Playsunny high high false nosunny high high true no

overcast high high false yesrain medium high false yesrain low normal false yesrain low normal true no

overcast low normal true yessunny medium high false nosunny low normal false yesrain medium normal false yes

sunny medium normal true yesovercast medium high true yesovercast high normal false yes

rain medium high true no


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Reduced Example

Simplify the example using only two variables, Outlook andTemperature:

Table: Joint probability distribution for Outlook and Temperature.

Temp.Outlook H M LS 0.143 0.143 0.071O 0.143 0.071 0.071R 0 0.214 0.143


Uncertainty

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Marginal probabilities

• If we sum per row (marginalizing Temperature) then weobtain the marginal probabilities for Outlook,P(Outlook) = [0.357,0.286,0.357]

• If we sum per column we obtain the marginalprobabilities for Temperature,P(Temperature) = [0.286,0.428,0.286]

• From these distributions, we obtain that the mostprobable Temperature is M and the most probablevalues for Outlook are S and R.


Uncertainty

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Conditional Probabilities

P(Temp. | Outlook = R) = P(Temp.∧Outlook = R)/P(Outlook = R)

= [0,0.6,0.4]

P(Outlook | Temp. = L) = P(Outlook ∧ Temp. = L)/P(Temp. = L)

= [0.25,0.25,0.5]

• The most probable Temperature given that the Outlookis Rain is Medium, and the most probable Outlook giventhat the Temperature is Low is Rain

• The most probable combination of Outlook andTemperature is {Rain,Medium}, which in this case canbe obtained directly from the joint probability table


Uncertainty

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Limitations

• The basic approach becomes impractical for complexproblems with many variables, as the size of the tableand the direct computation of marginal and conditionalprobabilities grow exponentially with the number ofvariables in the model.

• Another disadvantage is that to have good estimates forthe joint probabilities from data, we will require a verylarge database if there are many variables in the model

• The joint probability table does not say much about theproblem to a human; so this approach also hascognitive limitations.


Uncertainty

A brief history



Applications

Overview

References

Probabilistic graphical models

Probabilistic Graphical Models (PGMs)

• Provide a framework for managing uncertainty basedon probability theory in a computationally efficient way

• Consider only those independence relations that arevalid for a certain problem, and include these in theprobabilistic model to reduce complexity in terms ofmemory requirements and also computational time

• Represent the dependence and independence relationsbetween a set of variables using graphs


Uncertainty

A brief history



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Representation

• A Probabilistic Graphical Model is a compactrepresentation of a joint probability distribution, fromwhich we can obtain marginal and conditionalprobabilities. It has several advantages over a flatrepresentation:

• It is generally much more compact (space).• It is generally much more efficient (time).• It is easier to understand and communicate.• It is easier to learn form data or to construct based on

expert knowledge.


Uncertainty

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Specification

A PGM is specified by two aspects: (i) a graph, G(V ,E), thatdefines the structure of the model; and (ii) a set of localfunctions, f (Yi), that define the parameters, where Yi is asubset of X . The joint probability is obtained by the productof the local functions:

P(X1,X2, . . . ,XN) = KM∏

i=1

f (Yi) (1)


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Inference and learning

Inference: obtain the marginal or conditional probabilitiesof any subset of variables Z given any othersubset Y .

Learning: given a set of data values for X (that can beincomplete) estimate the structure (graph) andparameters (local functions) of the model.


Uncertainty

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Classification of PGMs

1 Directed or Undirected – Undirected graphs representsymmetric relations, while directed graphs representrelations in which the direction is important

2 Static or Dynamic – Defines if the model represents aset of variables at a certain point in time (static) oracross different times (dynamic)

3 Probabilistic or Decisional – Probabilistic models onlyinclude random variables, while decisional models alsoinclude decision and utility variables


Uncertainty

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Common PGMs

Type Dir/Undir Static/Dyn Prob./DecBayesian Classifiers D/U S PMarkov Chains D D PHidden Markov Models D D PMarkov Random Fields U S PBayesian Networks D S PDynamic Bayesian Networks D D PInfluence Diagrams D S DMarkov Decision Processes D D DPartially Observable MDPs D D D


Uncertainty

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Representation, Inference and Learning

• The representation is the basic property of each model,and it defines which entities constitute it and how theseare related.

• Inference consists in answering different probabilisticqueries based on the model and some evidence

• To construct these models there are basically twoalternatives: to build it “by hand” with the aid of domainexperts or to induce the model from data.


Uncertainty

A brief history



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General Paradigm

• Separate the inference and learning techniques fromthe model –the reasoning mechanisms are general andcan be applied to different models


Uncertainty

A brief history



Applications

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Applications

Application Examples

• Medical diagnosis and decision making.• Mobile robot localization, navigation and planning.• Diagnosis for complex industrial equipment such as

turbines and power plants.• User modeling for adaptive interfaces and intelligent

tutors.• Speech recognition and natural language processing.• Pollution modeling and prediction.• Reliability analysis of complex processes.• Modeling the evolution of viruses.• Object recognition in computer vision.• Information retrieval.• Energy markets.


Uncertainty

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Applications

Bayesian classifiers - skin detection


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HMMs - gesture recognition


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MRFs - image segmentation


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BNs - ozone prediction


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TNBNs - HIV mutation prediction


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Decision net - airport location


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MDP - robot motion planning


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PRMs - student modeling


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Causal BN - attention deficit model


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Overview

Overview (1)

• Part I provides the mathematical foundations:probability theory and graph theory

• Part II covers the probabilistic models that only haverandom variables

• Bayesian classifiers• Markov chains and hidden Markov models• Markov random fields• Bayesian networks• Dynamic Bayesian networks and temporal networks


Uncertainty

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Overview

Overview (2)

• Part III presents those models that consider decisionsand utilities: Decision Diagrams and Markov DecisionProcesses

• Part IV considers alternative paradigms that can bethought as extensions to the traditional probabilisticgraphical models:

• relational probabilistic models,• causal graphical models


Uncertainty

A brief history



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References

Additional Reading – Other books onPGMs

Bessiere, Mazer, E., P., Ahuactzin, J.M., Mekhnacha, K.:Bayesian Programming. CRC Press, Boca Raton,Florida (2014)

Daphne Koller, D., Friedman. N.: Probabilistic GraphicalModels: Principles and Techniques. MIT Press.Cambridge (2009)

Lauritzen, S.L.: Graphical Models. Oxford UniversityPress, Oxford(1996)


Uncertainty

A brief history



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Additional Reading – Books on BayesianNetworks

Darwiche, A.: Modeling and Reasoning with BayesianNetworks. Cambridge University Press, New York (2009)

Jensen, F.V.: Bayesian Networks and Decision Graphs.Springer-Verlag, New York (2001)

Neapolitan, R. E.: Probabilistic Reasoning in ExpertSystems. John Wiley & Sons, New York (1990)

Pearl, J.: Probabilistic Reasoning in Intelligent Systems:Networks of Plausible Inference. Morgan Kaufmann,San Francisco (1988)


Uncertainty

A brief history



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References

Additional Reading – Alternative Models

Duda, R.O., Hart, P.A., Nilsson, N.L.: Subjective BayesianMethods for Rule-Based Inference Systems. In: Proc. of theNat. Comp. Conf., vol. 45, pp. 1075–1082 (1976)

Heckerman, D.: Probabilistic Interpretations for MYCIN’sCertainty Factors. In: Proc. of the First Conf. on Uncertainty inArtificial Intelligence, pp. 9–20 (1985)

Shafer, G.: A Mathematical Theory of Evidence. PrincetonUniversity Press, Princeton (1976)

Shortliffe, E.H., Buchanan B.G.: A Model of InexactReasoning in Medicine. Mathematical Biosciencies, 23,351–379 (1975)

Zadeh, L.A.: Knowledge Representation in Fuzzy Logic. IEEETrans. on Knowledge and Data Engr., 1(1), 89–100 (1989)


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