Chapter Dr. Bernard Chen Ph.D. University of Central Arkansas Spring 2009.
Ch5 Stochastic Methods Dr. Bernard Chen Ph.D. University of Central Arkansas Spring 2011.
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Transcript of 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
Introduction Intro to Probability Baye’s Theory Naïve Baye’s Theory Application’s of the Stochastic
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
Elements of Probability Theory
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}
Elements of Probability Theory
P({even number of 1s} {end with 0})=p({even number of 1s}) * p({end with 0})
4/16=8/16*8/16
Outline
Introduction Intro to Probability Baye’s Theory Naïve Baye’s Theory Application’s of the Stochastic
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.
Outline
Introduction Intro to Probability Baye’s Theory Naïve Baye’s Theory Application’s of the Stochastic
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
Introduction Intro to Probability Baye’s Theory Naïve Baye’s Theory Application’s of the Stochastic
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.