Probabilistic reasoning

October 9, 2013 Probabilistic Reasoning - K.Iniya CSE 1

The Conditional Independence relationship
among the Variables can greatly reduce the
number of Probabilities we specified in order
to define the Full Joint distribution.

It introduces the Data Structure called a
Bayesian Network to represent the
dependencies among variables and to concise
specification of any Full Joint distribution.

A Bayesian Network is a
Directed Graph in which each node is
annotated with Quantitative Probability
information

The Full Specification is as follows:
A set of Random Variables makes up the
nodes of the network. Variable may be
Discrete or Continuous.
A set of directed links or arrows connects
pairs of nodes. if there is an arrow from node
X to node Y, X is said to be parent of Y.

Each node Xi has a conditional probability
distribution p(Xi |Parents(Xi)) than Quantifies
the effect of the parents on the node.
The graph has no directed cycles (directed,
DAG, or acyclic graph).

The Set of Nodes and Links
specifies the Conditional independences
relationships that hold in the domain.
The intuitive meaning of an
arrow in a properly constructed network is
usually that X has a direct influence on Y.

Once the Topology of the
Bayesian Network is laid out, we need only
specify a conditional probability distribution
for each variable given it’s parents.

Consisting the Variables,
i)Toothache ii)Cavity iii)Catch and iv)Weather.
Here,
Draw the Bayesian Network for the
above Variable

1) Weather is independent of the other
variables.
2) Toothache and Catch are conditionally
independent, given Cavity.

This relationships are represents in Bayesian
Network
Weather Cavity
CatchToothache

Weather Cavity
CatchToothache
Conditionally
independence

Probabilistic Reasoning - K.Iniya CSE
Weather Cavity
CatchToothache
Absence
of a link
October 9, 2013 13

Probabilistic Reasoning - K.Iniya CSEOctober 9, 2013 14
Weather Cavity
CatchToothache
No direct causal
relationship

Weather Cavity
CatchToothache
Independent
with other
variables

Consider the variables
i)Dog ii)Venus iii)Pet iv)Human v)Planet
vi)Mercury vii) Cat
BAYESIAN NETWORK?

Dog
Pet
Cat
Human
Mercury
Planet
Venus

A Burglar Alarm is installed at Home.
It’s fairly reliable at detecting a burglary and
It also responds for minor Earthquakes.
You have two neighbors John and Mary who
have promised to call you at work when they
hear the alarm
Draw the Bayesian Network

Burglary
Alarm
John calls Mary calls
Earthquake

John always calls when hears the alarm,
but sometimes confuses the telephone ringing
with the alarm and calls.

Mary likes rather loud Music and sometimes
misses the alarm.

The Burglary and Earthquake directly affect
the probability of the alarm going’s off.
But John and Mary call depends only on
the alarm.

The degree of approximation can be improved if
we introduce the additional relevant information.
CPT Stands for Conditional Probability Table.
Each row in a CPT contains the conditional
probability of each node for a conditioning case.
A Conditioning Case is just possible combination
of values for the parent nodes(a miniature atomic
event) .

Burglary Earthquake
Alarm
John calls Mary calls

Probabilistic Reasoning - K.Iniya CSEOctober 9, 2013 25
Burglary Earthquake
Alarm
John
calls
Mary
calls
P(Burglary)
0.001
P(Earthquake)
0.002
Alarm
t
f
P(John)
0.90
0.05
Alarm
t
f
P(Mary)
0.70
0.01
B E P(Alarm)
t t
t f
f t
f f
0.95
0.94
0.29
0.001

For Boolean Variables, once you know the
probability of a true value is p.
the probability of a false value is 1-p.
A Table for a Boolean variable with k Boolean
parents contains 2 independently specifiable
probabilities.
A node with no parents has only one row
representing prior probabilities.
k

Probabilistic reasoning

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