Report

Bayesian Networks

Submitted by:

Muahmmad Jamshaid Iqbal 14-ECT-03

Muzamil Ahmed 14-ECT-09

Subject:

Artificial Intelligence

Department of Electronics Engineering

U.E.T Taxila Sub Campus Chakwal

INTRODUCTION:

A Bayesian network is a directed graph, consisting of nodes that are connected to each other, each node represents a variable whose value can be discrete or continuous. Every node is assigned a quantitative probability value. These nodes are connected by arrows that shows us how the graph proceeds e.g. if there is an arrow from node A to node B then node A is parent and node B is child. The meaning of arrow is simply that A has direct influence on B. All nodes are connected in acyclic manner such as no loops are present in the graph and hence they are called directed acyclic graph or DAG. So, they Bayesian networks does not have feedback loops. Each node in Bayesian network has a conditional probability distribution that tells us about the effect of parent on the respective node.

If we have A = {A1, A2,…,An} nodes.

p(A) = ? p (Ai |pa(Ai))

Here p(A) is the joint probability of nodes A and pa(Ai) are parent nodes of Ai.

In the above equation we see that how the joint probability of A nodes depends on probability of individual nodes and their relative parent nodes.

If we have enough knowledge we can construct Bayesian network manually, but we can also construct it automatically from a large data set using some software.

Methodology:

The topology of the network consists of nodes and links between nodes. Once the topology being made we can find the conditional probability distribution on each node or variable according to the parent nodes. Then we can find the joint distribution for all variables.

Suppose we have four variables from outside world win lottery, rain, wet ground and slip. Now, we make a DAG from these variables and see how they are connected.

1847850145415Win Lottery

00Win Lottery

2847975302895894715331470396240012700Slip

00Slip

left12700Rain

00Rain

184721512700Wet Ground

00Wet Ground

In these four variables, we can see that win lottery has no connection with other three variables, so it is alone and not connected to other variables. Whereas, other three variables are connected such as, when it’s rain the ground wets and there is a chance of slipping on the wet ground.

Now, lets find out the probability.

p (L, R, G, S) = p(L), p(R), p(G|R), p(S|G)

In above equation, we can see from figure that lottery is different variable having no parent while rain is in connection but has no parent node according to our figure, so we write their individual probabilities in equation as p(L) and p(R). whereas, p(G|R) shows the probability distribution of wet ground according to rain as rain is its parent node. Similarly, p(S|G) shows the probability distribution of slip according to wet ground as wet ground is its parent node.

Let’s consider an example shown below. We have a new burglar alarm installed at home. It is reliable at detecting a burglary, but also responds on occasion to minor earthquakes. We also have two neighbors, John and Mary, who have promised to call you at work when they hear the alarm.

In below figure, we have also shown the CPT table with the figure.

P(E)

.002

P(B)

.001

4857759525Burglary

0Burglary

33909009525Earthquake

0Earthquake

1285875120650028479753111500

B E P(A)

T T

T F

F T

F F .95

.94

.29

.001

275272511245850016763991134110001933575218440Alarm

0Alarm

6191251723390John Calls

0John Calls

33623251713865Mary Calls

0Mary Calls

A P(J)

T

F .90

.05

A P(M)

T

F .70

.01

CPT table are the conditional probability tables having values of conditional probability for each entry. Here in the CPT table we have just shown the probability values when an event occurs, we can also find the probability values of not occurrence of an event by just calculating 1-P is equal to the required value. These two values will always equal to 1 because sum of probability of occurrence and not occurrence will always be equal to 1.

The probability equation for above figure is as following:

p(B,E,A,J,M) = p(B), p(E), p(A|B,E), p(J|A), p(M|A)

Here, B denotes Burglary, E denotes Earthquake, A denotes Alarm, J denotes John Calls and M denotes Mary Calls.

Now let’s find out the joint probability distribution for any case. Let’s assume we have a case in which alarm has sounded but the burglary has happened and earthquake hasn’t happened and both Mary and John didn’t call. It means B=T, E=F, A=T, J=F and M=F.

p(¬J, ¬M,A,B, ¬E) = p(¬J|A)p(¬M|A)p(A|B?¬E)p(B)p(¬E)

= 0.05×0.01×0.94×0.001×0.998

= 0.00000046906

Applications:

Bayesian network has many applications in health services, clinical researches, public health, symptoms findings and many other decisions making. There ae following applications:

Medical diagnosis systems

Manufacturing system diagnosis

Computer systems diagnosis

Network systems diagnosis

Helpdesk troubleshooting

Information retrieval

Customer modeling