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An Introduction to Bayesian Networks and their Contemporary Applic... http://www.niedermayer.ca/papers/bayesian

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1 of 12 17/10/2005 09:51

An Introduction to Bayesian Networks and their

Contemporary Applications

Daryle Niedermayer, I.S.P., PMP, B.Sc., B.A., M.Div.

December 1, 1998

Table of Contents

Abstract1.

Introduction

2.

An Introduction to Bayesian Inference

Bayes Theoremi.

Bayes Theorem Applied

ii.

3.

Bayesian Networks

Introductioni.

Definition

ii.

Bayesian Networks Illustrated

iii.

4.

Algorithmic Implications of Bayesian Networks

5.

Practical Uses for Bayesian Networks

AutoClassi.

Introduction of Search Heuristics

ii.

Lumiere

iii.

6.

Limitations of Bayesian Networks

7.

Conclusion

8.

References

9.

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Abstract

Bayesian Networks are becoming an increasingly important area for research and application in the entire

field of Artificial Intelligence. This paper explores the nature and implications for Bayesian Networks

beginning with an overview and comparison of inferential statistics and Bayes' Theorem. The nature,

relevance and applicability of Bayesian Network theory for issues of advanced computability forms the

core of the current discussion. A number of current applications using Bayesian networks is examined.

The paper concludes with a brief discussion of the appropriateness and limitations of Bayesian Networks

for human-computer interaction and automated learning.

An Introduction to Bayesian Networks and their Contemporary Applic... http://www.niedermayer.ca/papers/bayesian

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2 of 12 17/10/2005 09:51

Back to the Table of Contents

Introduction

Inferential statistics is a branch of statistics that attempts to make valid predictions based on only a sample

of all possible observations[1]

. For example, imagine a bag of 10,000 marbles. Some are black and some

white, but of which the exact proportion of these colours is unknown. It is unnecessary to count all the

marbles in order to make some statement about this proportion. A randomly acquired sample of 1,000

marbles may be sufficient to make an inference about the proportion of black and white marbles in the

entire population. If 40% of our sample are white, then we may be able to infer that about 40% of the

population are also white.

To the layperson, this process seems rather straight forward. In fact, it might seem that there is no need to

even acquire a sample of 1,000 marbles. A sample of 100 or even 10 marbles might do.

This is assumption is not necessarily correct. As the sample size becomes smaller, the potential for error

grows. For this reason, inferential statistics has developed numerous techniques for stating the level of

confidence that can be placed on these inferences.

If we took ten samples of 100 marbles each, we might find the following results:

Table 1: Relative proportions of 10 samples from a

population of 10,000

Sample Number

Number of White

Marbles

Number of Black Marbles

1

40 60

2

35 65

3

47 53

4

50 50

5

31 69

6

25 75

7

36 64

8

20 80

9

45 55

10

55 45

We are then in a position to calculate the "Standard Deviation" of these samples:

(eq. 1)[2]

where x

2

is the sum of the squares so that the equation is expanded to:

(eq. 2)

and n is the number of samples. In our example, the mean number of White marbles is

.

We might be tempted to say that about 40% of the marbles are white, but we are unable to argue that point

with any degree of certainty. Using equation 2 above, we determine that the Standard Deviation is 11.15.

We must then determine the "Sample Error of the Mean" (where s=[sigma]):

An Introduction to Bayesian Networks and their Contemporary Applic... http://www.niedermayer.ca/papers/bayesian

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3 of 12 17/10/2005 09:51

(eq. 4)

The confidence we can put on our hypothesis that u=40 of the marbles are white is found using a standard

statistical test called the "z-test":

(eq. 5)

Using a z-test table [3]

and our resulting z-value of -.4532, we find that 32% of the area of the normal

curve would fall below this "z" value. In other words, in 32% of samples given

, would be less than

38.4. Since the distribution is two-sided or "two-tailed" (i.e. the sample average could also be greater than

the population average), we would also expect

to greater than ((u- )+u=) 41.6 in another 32% of cases.

In summary, if we expect 40% of all marbles in the bag to be white, then a series of ten samples with only

38.4% of marbles being white would be expected in (100-64%=) 36% of the time. Clearly, the confidence

we can place in our conclusion is not as good as it was on first glance. This lack of confidence is due to

the high variability among the samples. If we took more samples or larger samples, our confidence in our

conclusion might increase.

Back to the Table of Contents

An Introduction to Bayesian Inference

Classical inferential models do not permit the introduction of prior knowledge into the calculations. For

the rigours of the scientific method, this is an appropriate response to prevent the introduction of

extraneous data that might skew the experimental results. However, there are times when the use of prior

knowledge would be a useful contribution to the evaluation process.

Assume a situation where an investor is considering purchasing some sort of exclusive franchise for a

given geographic territory. Her business plan suggests that she must achieve 25% of market saturation for

the enterprise to be profitable. Using some of her investment funds, she hires a polling company to

conduct a randomized survey. The results conclude that from a random sample of 20 consumers, 25% of

the population would indeed be prepared to purchase her services. Is this sufficient evidence to proceed

with the investment?

If this is all the investor has to go on, she could find herself on her break-even point and could just as

easily turn a loss instead of a profit. She may not have enough confidence in this survey or her plan to

proceed.

Fortunately, the franchising company has a wealth of experience in exploiting new markets. Their results

show that in 20% of cases, new franchises only achieve a 25% market saturation, while in 40% of cases,

new franchises achieve a 30% market saturation. The entire table of their findings appears below:

Table 2: Percent of New Franchises achieving a given Market Saturation

Market Saturation (Proportion) =p Percent of Franchises (Relative Frequency)

0.10 0.05

0.15 0.05

0.20 0.20

0.25 0.20

0.30 0.40

0.35 0.10

Total = 1.00

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