# A Tutorial On Learning With

AI and Robotics

Nov 7, 2013 (4 years and 6 months ago)

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Haimonti Dutta , Department Of
Computer And Information
Science

1

David HeckerMann

A Tutorial On Learning With

Bayesian Networks

Haimonti Dutta , Department Of
Computer And Information
Science

2

Outline

Introduction

Bayesian Interpretation of probability and review methods

Bayesian Networks and Construction from prior knowledge

Algorithms for probabilistic inference

Learning probabilities and structure in a bayesian network

Relationships between Bayesian Network techniques and
methods for supervised and unsupervised learning

Conclusion

Haimonti Dutta , Department Of
Computer And Information
Science

3

Introduction

A bayesian network is a graphical model for
probabilistic relationships among a set of
variables

Haimonti Dutta , Department Of
Computer And Information
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4

What do Bayesian Networks and Bayesian
Methods have to offer ?

Handling of Incomplete Data Sets

Facilitating the combination of domain knowledge
and data

Efficient and principled approach for avoiding the
over fitting of data

Haimonti Dutta , Department Of
Computer And Information
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5

The Bayesian Approach to

Probability and Statistics

Bayesian Probability : the degree of belief in that event

Classical Probability : true or physical probability of an event

Haimonti Dutta , Department Of
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Some Criticisms of Bayesian Probability

Why degrees of belief satisfy the rules of
probability

On what scale should probabilities be measured?

What probabilites are to be assigned to beliefs
that are not in extremes?

Haimonti Dutta , Department Of
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7

Researchers have suggested different sets of
properties that are satisfied by the degrees of
belief

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8

Scaling Problem

The probability wheel : a tool for assessing

probabilities

What is the probability that the fortune wheel stops

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Probability assessment

An evident problem : SENSITIVITY

How can we say that the probability of an event is 0.601
and not .599 ?

Another problem : ACCURACY

Methods for improving accuracy are available in
decision analysis techniques

Haimonti Dutta , Department Of
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10

Learning with Data

Thumbtack problem

When tossed it can rest on either heads or tails

Tails

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Problem ………

From N observations we want to determine the
probability of heads on the N+1 th toss.

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Two Approaches

Classical Approach :

assert some physical probability of heads

(unknown)

Estimate this physical probability from N
observations

Use this estimate as probability for the heads
on the N+1 th toss.

Haimonti Dutta , Department Of
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The other approach

Bayesian Approach

Assert some physical probability

probability using the Bayesian probailities

Use the rules of probability to compute the
required probability

Haimonti Dutta , Department Of
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Some basic probability

formulas

Bayes theorem : the posterior probability for

given D and a background knowledge

:

p(

⽄Ⱐ

⤠㴠

⤠瀠⡄⼠

)

)

Where p(D/

⤽)⁰ 䐯D

⤠瀨

⤠搠

Note :

the possible true values of the physical probability

Haimonti Dutta , Department Of
Computer And Information
Science

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Likelihood function

How good is a particular value of

?

It depends on how likely it is capable of generating the observed
data

L (

:D ) = P( D/

)

Hence the likelihood of the sequence H, T,H,T ,T may be L (

:D )
=

. (1
-

).

. (1
-

). (1
-

).

Haimonti Dutta , Department Of
Computer And Information
Science

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Sufficient statistics

To compute the likelihood in the thumb tack
problem we only require h and t

(the number of
heads and the number of tails)

h and t are called sufficient statistics for the
binomial distribution

A sufficient statistic is a function that summarizes
from the data , the relevant information for the
likelihood

Haimonti Dutta , Department Of
Computer And Information
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Finally

……….

We average over the possible values of

determine the probability that the N+1 th toss of
the thumb tack will come up heads

P(X

⤠㴠=

⽄Ⱐ

⤠d

n+1

The above value is also referred to as the
Expectation of

w楴栠牥獰散琠瑯t瑨攠摩獴物扵瑩潮

⽄/

)

Haimonti Dutta , Department Of
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To remember

We need a method to assess the prior distribution
for

.

䄠捯浭潮⁡灰牯A捨⁵獵慬ay 慤a灴敤p楳⁡獳s浥m
that the distribution is a beta distribution.

Haimonti Dutta , Department Of
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Maximum Likelihood Estimation

MLE principle

:

We try to learn the parameters that maximize the
likelihood function

It is one of the most commonly used estimators in
statistics and is intuitively appealing

Haimonti Dutta , Department Of
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A graphical model that efficiently encodes the
joint probability distribution for a large set of
variables

What is a Bayesian Network ?

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Definition

A Bayesian Network for a set of variables

X = { X1,…….Xn} contains

network structure S encoding conditional

a set P of local probability distributions

The network structure S is a directed acyclic graph

And the nodes are in one to one correspondence
with the variables X.Lack of an arc denotes a
conditional independence.

Haimonti Dutta , Department Of
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Science

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Some conventions……….

Variables depicted as nodes

Arcs represent probabilistic dependence between
variables

Conditional probabilities encode the strength of
dependencies

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An Example

Detecting Credit
-

Card Fraud

Fraud

Age

Sex

Gas

Jewelry

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Correctly identify the goals of modeling

Identify many possible observations that may be relevant to
a problem

Determine what subset of those observations is worthwhile
to model

Organize the observations into variables having mutually
exclusive and collectively exhaustive states.

Finally we are to build a Directed A cyclic Graph that encodes
the assertions of conditional independence

Haimonti Dutta , Department Of
Computer And Information
Science

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A technique of constructing a

Bayesian Network

The approach is based on the following
observations :

People can often readily assert causal
relationships among the variables

Casual relations typically correspond to
assertions of conditional dependence

To construct a Bayesian Network we simply draw
arcs for a given set of variables from the cause
variables to their immediate effects.In the final
step we determine the local probability
distributions.

Haimonti Dutta , Department Of
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Problems

Steps are often intermingled in practice

Judgments of conditional independence and /or
cause and effect can influence problem
formulation

Assessments in probability may lead to changes
in the network structure

Haimonti Dutta , Department Of
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Bayesian inference

On construction of a Bayesian network we need to determine
the various probabilities of interest from the model

Observed data

Query

Computation of a probability of interest given a model is probabilistic
inference

x1

x2

x[m]

x[m+1]

Haimonti Dutta , Department Of
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Science

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Learning Probabilities in a Bayesian

Network

Problem

: Using data to update the probabilities of a
given network structure

Thumbtack problem

: We do not learn the
probability of the heads , we update the posterior
distribution for the variable that represents the

The problem restated

:Given a random sample D
compute the posterior probability .

Haimonti Dutta , Department Of
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29

Assumptions to compute the posterior

probability

There is no missing data in the random sample D.

Parameters are independent .

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But……

Data may be missing and then how do

we proceed ?????????

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Obvious concerns….

Why was the data missing?

Missing values

Hidden variables

Is the absence of an observation
dependent on the actual states of the
variables?

We deal with the missing data that are
independent of the state

Haimonti Dutta , Department Of
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Science

32

Incomplete data (contd)

Observations reveal that for any interesting set of
local likelihoods and priors the exact
computation of the posterior distribution will be
intractable.

We require approximation for incomplete data

Haimonti Dutta , Department Of
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Science

33

The various methods of approximations for

Incomplete Data

Monte Carlo Sampling methods

Gaussian Approximation

MAP and Ml Approximations and EM algorithm

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Gibb’s Sampling

The steps involved :

Start :

Choose an initial state for each of the variables in X at
random

Iterate :

Unassign the current state of X1.

Compute the probability of this state given that of n
-
1
variables.

Repeat this procedure for all X creating a new sample of X

After “ burn in “ phase the possible configuration of X will
be sampled with probability p(x).

Haimonti Dutta , Department Of
Computer And Information
Science

35

Problem in Monte Carlo method

Intractable when the sample size is large

Gaussian Approximation

Idea : Large amounts of data can be approximated
to a multivariate Gaussian Distribution.

Haimonti Dutta , Department Of
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36

Criteria for Model Selection

Some criterion must be used to determine the
degree to which a network structure fits the prior
knowledge and data

Some such criteria include

Relative posterior probability

Local criteria

Haimonti Dutta , Department Of
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Science

37

Relative posterior probability

A criteria for model selection is the logarithm of the
relative posterior probability given as follows :

Log p(D /Sh) = log p(Sh) + log p(D /Sh)

log prior log marginal

likelihood

Haimonti Dutta , Department Of
Computer And Information
Science

38

Local Criteria

An Example :

A Bayesian network structure for medical diagnosis

Ailment

Finding 1

Finding 2

Finding n

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Priors

To compute the relative posterior probability

We assess the

Structure priors p(Sh)

Parameter priors p(

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Priors on network parameters

Key concepts :

Independence Equivalence

Distribution Equivalence

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Illustration of independent equivalence

Independence assertion : X and Z are conditionally
independent given Y

X

Y

Z

X

Y

Z

X

Y

Z

Haimonti Dutta , Department Of
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42

Priors on structures

Various methods….

Assumption that every hypothesis is equally
likely ( usually for convenience)

Variables can be ordered and presence or
absence of arcs are mutually independent

Use of prior networks

Imaginary data from domain experts

Haimonti Dutta , Department Of
Computer And Information
Science

43

Benefits of learning structures

Efficient learning
---

more accurate models with
less data

Compare P(A) and P(B) versus P(A,B) former
requires less data

Discover structural properties of the domain

Helps to order events that occur sequentially and
in sensitivity analysis and inference

Predict effect of the actions

Haimonti Dutta , Department Of
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Science

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Search Methods

Problem : We are to find the best network from the
set of all networks in which each node has no
more than k parents

Search techniques :

Greedy Search

Greedy Search with restarts

Best first Search

Monte Carlo Methods

Haimonti Dutta , Department Of
Computer And Information
Science

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Bayesian Networks for Supervised and

Unsupervised learning

Supervised learning

: A natural representation in
which to encode prior knowledge

Unsupervised learning

:

Apply the learning technique to select a model with no hidden
variables

Look for sets of mutually dependent variables in the model

Create a new model with a hidden variable

Score new models possibly finding one better than the original.

Haimonti Dutta , Department Of
Computer And Information
Science

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What is all this good for anyway????????

Implementations in real life :

It is used in the Microsoft products(Microsoft
Office)

Medical applications and Biostatistics (BUGS)

In NASA Autoclass projectfor data analysis

Collaborative filtering (Microsoft

MSBN)

Fraud Detection (ATT)

Speech recognition (UC , Berkeley )

Haimonti Dutta , Department Of
Computer And Information
Science

47

Limitations Of Bayesian Networks

Typically require initial knowledge of many
probabilities…quality and extent of prior
knowledge play an important role

Unanticipated probability of an event is not taken
care of.

Haimonti Dutta , Department Of
Computer And Information
Science

48

Conclusion

Inducer

Bayesian Network

Data +prior

knowledge

Haimonti Dutta , Department Of
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49

Cross fertilization with other techniques?

For e.g with decision trees, R trees and neural networks

Improvements in search techniques using the classical
search methods ?

Application in some other areas as estimation of population
death rate and birth rate, financial applications ?