Ch5 Stochastic Methods Dr. Bernard Chen Ph.D. University of Central Arkansas Spring 2011

Ch5 Stochastic Methods

Dr. Bernard Chen Ph.D.University of Central Arkansas

Spring 2011

Outline

Introduction Intro to Probability Baye’s Theory Naïve Baye’s Theory Application’s of the Stochastic

Methods

Introduction

Chapter 4 introduced heuristic search as an approach to problem solving in domains where A problem does not have an exact

solution or The full state space may be to

costly to calculate

Introduction

Important application domains for the use of the stochastic method are diagnostic reasoning where

cause/effect relationships are not always captured in a purely deterministic fashion

Gambling

Outline


Methods

Elements of Probability Theory Elementary Event

An elementary or atomic event is a happing or occurrence that cannot be made up of other events

Event E An event is a set of elementary events

Sample Space, S The set of all possible outcomes of an event E

Probability, p The probability of an event E in a sample space is

the ratio of the number of elements in E to the total number of possible outcomes

Elements of Probability Theory

For example, what is the probability that a 7 or an 11 are the result of the roll of two fair dice? Elementary Event: play two dice Event: Roll the dice Sample Space: each die has 6

outcomes, so the total set of outcomes of the two dice is 36

Elements of Probability Theory The number of combinations of the two dice

that can give an 7 is 1,6; 2,5; 3,4; 4,3; 5,2 and 6,1

So the probability of rolling a 7 is 6/36

The number of combinations of the two dice that can give an 11 is 5,6; 6,5

So the probability of rolling a 11 is 2/36

Thus, the probability to the answer is 6/36 + 2/36 =2/9

Probability Reasoning Suppose you are driving the

interstate highway and realize you are gradually slowing down because of increase traffic congestion

The you access to the state highway statistics and download the relevant statistical information

Probability Reasoning

In this situation, we have 3 parameter

Slowing down (S): T or F Whether or not there’s an accident (A): T or F Whether or not there’s a road construction (C): T of F

Probability Reasoning We may also present it in the

traditional Venn Diagram


Two events A and B are independent if and only if the probability of their both occurring is equal to the product o their occurring individually

P(A B) = P(A) * P(B)

Elements of Probability Theory Consider the situation where bit strings

of length 4 are randomly generated

We want to know whether the event of the bit sting containing an even number of 1s is independent of the event where the bit string ends with 0

We know the total space is 2^4 = 16

Elements of Probability Theory There are 8 bit strings of length four that

end with 0

There are 8 bit strings of length four that have even number of 1’s

The number of bit strings that have both an even number of 1s and end with 0 is 4:{1100, 1010, 0110, 0000}


P({even number of 1s} {end with 0})=p({even number of 1s}) * p({end with 0})

4/16=8/16*8/16


Finally, the conditional probability

p(d|s) = |d s|/|s|


Outline


Methods

Bayes’ Theorem

P(A), P(B) is the prior probability P(A|B) is the conditional probability of A,

given B. P(B|A) is the conditional probability of B,

given A.

Bayes’ Theorem Suppose there is a school with 60% boys and

40% girls as its students. The female students wear trousers (50%) or

skirts (50%) in equal numbers; the boys all wear trousers.

An observer sees a (random) student from a distance, and what the observer can see is that this student is wearing trousers.

What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem

Bayes’ Theorem P(B|A), or the probability of the student wearing

trousers given that the student is a girl. Since girls are as likely to wear skirts as trousers, this is 0.5.

P(A), or the probability that the student is a girl regardless of any other information, this probability equals 0.4.

P(B), or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since half of the girls and all of the boys are wearing trousers, this is 0.5×0.4 + 1.0×0.6 = 0.8.

Bayes’ Theorem

Outline


Methods

Naïve Bayesian Classifier: Training Dataset

Class:C1:buys_computer = ‘yes’C2:buys_computer = ‘no’

Data sample X = (age <=30,Income = medium,Student = yesCredit_rating = Fair)

age income studentcredit_ratingbuys_computer<=30 high no fair no<=30 high no excellent no31…40 high no fair yes>40 medium no fair yes>40 low yes fair yes>40 low yes excellent no31…40 low yes excellent yes<=30 medium no fair no<=30 low yes fair yes>40 medium yes fair yes<=30 medium yes excellent yes31…40 medium no excellent yes31…40 high yes fair yes>40 medium no excellent no

Bayesian Theorem: Basics Let X be a data sample

Let H be a hypothesis (our prediction) that X belongs to class C

Classification is to determine P(H|X), the probability that the hypothesis holds given the observed data sample X

Example: customer X will buy a computer given that know the customer’s age and income

Naïve Bayesian Classifier: Training Dataset

Class:C1:buys_computer = ‘yes’C2:buys_computer = ‘no’

Data sample X = (age <=30,Income = medium,Student = yesCredit_rating = Fair)

age income studentcredit_ratingbuys_computer<=30 high no fair no<=30 high no excellent no31…40 high no fair yes>40 medium no fair yes>40 low yes fair yes>40 low yes excellent no31…40 low yes excellent yes<=30 medium no fair no<=30 low yes fair yes>40 medium yes fair yes<=30 medium yes excellent yes31…40 medium no excellent yes31…40 high yes fair yes>40 medium no excellent no

Naïve Bayesian Classifier: An Example P(Ci): P(buys_computer = “yes”) = 9/14 = 0.643 P(buys_computer = “no”) = 5/14= 0.357

Compute P(X|Ci) for each class P(age = “<=30” | buys_computer = “yes”) = 2/9 = 0.222 P(age = “<= 30” | buys_computer = “no”) = 3/5 = 0.6 P(income = “medium” | buys_computer = “yes”) = 4/9 =

0.444 P(income = “medium” | buys_computer = “no”) = 2/5 = 0.4 P(student = “yes” | buys_computer = “yes) = 6/9 = 0.667 P(student = “yes” | buys_computer = “no”) = 1/5 = 0.2 P(credit_rating = “fair” | buys_computer = “yes”) = 6/9 =

0.667 P(credit_rating = “fair” | buys_computer = “no”) = 2/5 = 0.4

Naïve Bayesian Classifier: An Example X = (age <= 30 , income = medium, student = yes,

credit_rating = fair)

P(X|Ci) :

P(X|buys_computer = “yes”) = 0.222 x 0.444 x 0.667 x 0.667 = 0.044 P(X|buys_computer = “no”) = 0.6 x 0.4 x 0.2 x 0.4 = 0.019

P(X|Ci)*P(Ci) : P(X|buys_computer = “yes”) * P(buys_computer = “yes”) = 0.028

P(X|buys_computer = “no”) * P(buys_computer = “no”) = 0.007

Therefore, X belongs to class (“buys_computer = yes”)

Towards Naïve Bayesian Classifier This can be derived from Bayes’

theorem

Since P(X) is constant for all classes, only

needs to be maximized

)()()|(

)|(X

XX

PiCPiCP

iCP

)()|()|( iCPiCPiCP XX

Naïve Bayesian Classifier: An Example

Test on the following example:

X = (age > 30, Income = Low, Student = yes Credit_rating = Excellent)

Outline


Methods

Tomato You say [t ow m ey t ow] and I say [t ow

m aa t ow]

Probabilistic finite machine A finite state machine where the next state

function is a probability distribution over the full set of states of the machine

Probabilistic finite state acceptor An acceptor, whene one or more states are

indicates as the start state and one or more as the accept states

So how is “Tomato” pronounced

A probabilistic finite state acceptor for the pronunciation of “tomato”, adapted from Jurafsky and Martin (2000).

Natural Language Processing IN the second example, we consider

the phoneme recognition problem, Often called decoding

Suppose a phoneme recognition algorithm has identified the phone ni (as in “knee”) that occurs just after the recognized word I

Natural Language Processing

We want to associate ni with either a word or the first part of the word

Then we need Switchboard Corpora, which is 1.4M word collection of telephone conversation, to assist us.



We next apply a form of Naïve Baye’s theorem to analysis the phone ni following I

Documents

Ch5 Stochastic Methods Dr. Bernard Chen Ph.D. University of Central Arkansas Spring 2011