bayesian decision theory

Institute of Systems and RoboticsISR – Coimbra

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Bayesian Approaches Bayesian Approaches 11

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Retrospective

Bayesian Multimodal Perception by J. F. Fereira

Bayes' theorem - Bayes rule

Knowledge of past behavior and stateform prediction of current state

Non-Gaussian likelihood functions

Multimodal Sensing in human perception


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Retrospective

distribution of object position unknown => flat

Noise in each modality is independent

bimodal posterior distribution = product of the unimodal distributions

Simplification: Probability distributions are Gaussian


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1. Introduction to Pattern Recognition

Example:

“Sorting incoming Fish on a conveyor according to species using optical sensing”


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Selecting length feature

Example: Fish Classifier

Selecting lightness feature


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Example: Fish Classifier

Selecting two features and defining a simple straight line as decision boundary

Best performance but complicated classifier – will not perform well with novel patterns

Search for the optimal tradeoff between performance on the training set and simplicity


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post-processing

classification

feature extraction

segmentation

sensing

input

decision

Invariant FeaturesTranslation

RotationScale

OcclusionProjective Distortion

RateDeformation

Feature Selection

NoiseMissing Features

Error RateRisk

ContextMultiple Classifiers

Pattern Recognition System


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collect data

choose features

choose model

train classifier

evaluate classifier

end

start

Prior KnowledgeOverfitting

Design Cycle


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2. Continouos Features

State of nature Finite set of c states of nature (‘categories’) {1, … , c}

Prior P(j)If the state of nature is finite:

Decision rule (for c =2):Decide 1 if P(1) > P(2); otherwise decide 2


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Feature vector x : x d the feature spacex is (for d=1) a continuous random variable x

Class(State)-conditional probability density function: p(x| j) expresses the distribution of x depending on the state of nature


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Bayes formula(Posterior)

Evidence

Bayes Decision rule (for c =2):Decide 1 if P(1 | x) > P(2 | x) ; otherwise decide 2

Bayes Decision rule (expressed in terms of Priors):Decide 1 if p(x|1)P(1) > p(x|2)P(2) ; otherwise decide 2


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Conditional Risk

We can minimize our expected loss by selecting the action that minimizes the conditional risk.

This Bayes decision procedure provides the optimal performance

Two-Category Classification

Bayes Risk

Decide 1 if (21-11)P(1 | x) > (12-22)P(2 | x) ; otherwise decide 2


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Discrete Features

Probabilities rather than probability densities.

Bayes decision ruleTo minimize the overall risk, select

the action I for which R(i|x) is minimum

Feature vector x can assume m discrete values

Po

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nceR

isk


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Discrete FeaturesExample: Independent Binary Features

2 category problemFeature vector x = {x1, …, xd}T where xi = {0;1}


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Bayesian Belief Networks

Represents knowledge about a distribution.

Knowledge: Statistical Dependencies – Causal Relations among the

component variables

Knowledge from e.g. structural information

Graphical representation: Bayesian Belief Nets

node variableP(a)link

parents(of C)

children(of E)


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Bayesian Belief NetworksApplying Bayes rule to determine the probability of any configuration of variables in the joint distribution.

P(a1) P(a2)

0.739 0.261

P(c1|ak) P(c2|ak)

a1 0.3 0.7

a2 0.6 0.4 =1

=1

=1

Discrete Case:Discrete number of possible values A (e.g. 2: a={a1, a2} andcontinues-valued probabilities

Conditional Probability Table


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Bayesian Belief NetworksDetermining the probabilities of the variablesP(a) P(b|a) P(c|b) P(d|c)

A B C D

independance

Summing the full joint distribution P(a,b,c,d) over all variables other than d

E.g.: Probability distribution over d1, d2, … at D

simple split

P(b)P(c)P(d)

simple interpretation

Probability of a particular value of D


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Bayesian Belief NetworksGive the values of some variables (evidence e)

… and search to determine some particular configuration of other variables x


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Bayesian Belief NetworksExample: Belief Network for Fish

As usual: Compute P(x1 salmon) and P(x2 sea bass) Decide for the minimum expected classification error

Ex.2 Classify the fish:Known: Fish is light (c1) and caught in the south Atlantic (b2).Unknown: Time of year (a), thickness (d)

In this case D does not affect our results


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And if the dependency relation is unknown?

naïve Bayes – idiot Bayes

Features are conditionally independant

After normalization:


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Compound Bayesian Decision TheoryConsecutive ’s not statistically

independent=> exploit dependence

=> improved performance

Wait for n states to emerge and make all n decisions jointly

= compound decision problem

States of nature = ((1), … , (n))T

taking one of c values {1, … , c}

Prior P()for n states of nature

Feature matrix X : =(x1, …, xn)xi obtained when state of nature was i

n observations


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Compound Bayesian Decision TheoryDefine loss matrix for the

compound decision problem. Seek decision rule the minimizes the

compound risk (optimal procedure)

Assumption: Correct = no lossErrors = equally costly

=> simply calculate P(|X) for all and select for which P(.) is

maximum.

practice: calculate P(|X) is time expensive

assumption: xi depends only on (i) not on other x or

Conditional probability density function: p(X|) for X given the true set of

Po

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ensity


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Obrigado!


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Annex


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Book: Pattern Cl.

PrefaceCh 1: IntroductionCh 2: Bayesian Decision TheoryCh 3: Maximum Likelihood and Bayesian EstimationCh 4: Nonparametric TechniquesCh 5: Linear Discriminant FunctionsCh 6: Multilayer Neural NetworksCh 7: Stochastic MethodsCh 8: Nonmetric MethodsCh 9: Algorithm-Independent Machine LearningCh 10: Unsupervised Learning and ClusteringApp A: Mathematical Foundations


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2 Bug Algorithms 17

3 Configuration Space 39

4 Potential Functions 77

5 Roadmaps 107

6 Cell Decompositions 161

7 Sampling-Based Algorithms 197

8 Kalman Filtering 269

9 Bayesian Methods 30110 Robot Dynamics 349

11 Trajectory Planning 373

12 Nonholonomic and Underactuated Systems 401

Book: Principles of ...


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Book: Artificial ...

PrefacePart I Artificial Intelligence Part II Problem Solving Part III Knowledge and Reasoning Part IV Planning Part V Uncertain Knowledge and Reasoning Part VI Learning Part VII Communicating, Perceiving, and Acting Part VIII Conclusions


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Book: Bayesian ...

Preface xixPart I: Fundamentals of Bayesian Inference 11 Background 32 Single-parameter models 333 Introduction to multiparameter models 734 Large-sample inference and frequency properties of Bayesian inference 101Part II: Fundamentals of Bayesian Data Analysis 1155 Hierarchical models 1176 Model checking and improvement 1577 Modeling accounting for data collection 1978 Connections and challenges 2479 General advice 259Part III: Advanced Computation 27310 Overview of computation 27511 Posterior simulation 28312 Approximations based on posterior modes 31113 Special topics in computation 335Part IV: Regression Models 35114 Introduction to regression models 35315 Hierarchical linear models 38916 Generalized linear models 41517 Models for robust inference 44318 Mixture models 46319 Multivariate models 48120 Nonlinear models 49721 Models for missing data 51722 Decision analysis 541Appendixes 571


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Book: Classification ...

Preface. Foreword. 1. Introduction. 2. Detection and Classification. 3. Parameter Estimation. 4. State Estimation. 5. Supervised Learning. 6. Feature Extraction and Selection. 7. Unsupervised Learning. 8. State Estimation in Practice. 9. Worked Out Examples. Appendix


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Images


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2. Simple ExampleDesigning a simple classifier for gesture recognition.The observer tries to predict which gesture might be performed next.The sequence of gestures appears to be random.

Ten types of gestures:1. Big circle2. Small circle3. Vertical Line4. Horizontal Line5. Pointing North-West6. Pointing West7. Talk louder8. Talk more quiet9. Wave Bye-Bye10. I am hungry

State of nature Type of gesture (1 … 10)

We assume that there is some a priori probability (i.e. prior) P(1) that the next gesture is ‘Big Circle’, P(2) that the next gesture is ‘Small Circle’, etc.If the gesture lexicon is finite:


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Missing and noisy features

Missing Features:

Example: x1 is missingmeasured value of x2 is x^2mean x1 points to omega 3but omega2 better decision

bayesian decision theory

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