George F Luger

ARTIFICIAL INTELLIGENCE Structures and Strategies for Complex Problem Solving

STOCHASTIC METHODS

Luger: Artificial Intelligence, 5th edition. © Pearson Education Limited, 2005

5.0 Introduction

5.1 The Elements of Counting

5.2 Elements of Probability Theory

5.3 Applications of the Stochastic Methodology

5.4 Bayes’ Theorem

Presented By:

Misbah Arif, Kayras, Abid

5.0. IntroductionStochastic Methods- Next problem solving methodology after heuristics

"Stochastic" means being or having a random variable or “pertaining to chance” or probabilistic reasoning.

Definition “A stochastic process is one whose behavior is non-deterministic and is determined both by the process's predictable actions and by a random element (non predicatble actions)”

Applications of Stochastic Methodology

Diagnostic Reasoning Decides symptoms and their causes Example: Fever caused either by flu or an infection

Natural Language Understanding Supports understanding of language

Planning and Scheduling During planning, no deterministic sequence of operation is

guaranteed to succeed.

To solve problems, as in simulated annealing, stochastic neural networks, stochastic optimization, and genetic algorithms

http://en.wikipedia.org/wiki/Stochastic

5.1. Elements of Counting5.1.1. Addition Rules

|A|= no of elements in set A U= Universal Sets A’ =U-A A B⊆ (A is the subset of B) AUB (A union B) A B (A intersection B) |AUC |= |A|+|C|- | A C | |AUBUC| = ?

Elements of Counting--- cont5.1.1. Multiplication Rules –

If we have two sets of elements A and B of size a and b

respectively, then there are a*b unique ways of combining the

elements of sets together

Cartesian Product |A*B|=|A|*|B|

Permutations and Combinations P (n, r) = n! / (n − r)! C (n, r) = P (n, r)/ r! = n! / (n − r)! r!

5.2. Elements of ProbabilityTheory

ProbabilityProbability

the Likelihood of occurrence of a specified event, often represented as a number between 0 (never) and 1 (always)

a mathematic ratio of the number of times something will occur to the total number of possible outcomes

P(A) =   The Number Of Ways Event A Can Occur

The total number Of Possible Outcomes

Example of a Spinner..

Problem 1:   A spinner has 4 equal sectors colored yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner?

Outcomes:  The possible outcomes of this experiment are yellow, blue, green, and red.

Probabilities:  P(blue)  =  # of ways to land on blue total # of colors  

Can You Solve this??Problem 2:  

A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number???

Important Terms

ExampleProblem 3:  

What is the probability that a 7 or an 11 is the result of the roll of two fair dice ?

Solution:  

1. First determine the Sample Space and Event

2. Determine the atomic event i.e. combinations of two dice that can give 7

3. Determine their probability

3. Determine the probability of rolling 11 and then both 7 and 11

So…….Atomic event: (1,6), (2,5),(3,4),(4,3),(5,2),(6,1)

Event: 7

Sample Space: 36 (6 *6)

Probability : 6/36= 0.16 

•The probability of any event E from the sample space S is:

•The sum of the probabilities of all possible outcomes is 1

•The probability of the compliment of an event is

•The probability of the contradictory or false outcome of an event

Axioms

Explanation with Example

P(Blue) ¼= 0.25

P(Red) ¼= 0.25

P(Yellow) ¼= 0.25

P(Green) ¼= 0.25

P (S) 1

Overview of Previous LectureStochastic Process- non deterministic process

Stochastic Process are related to probabilistic reasoning

Probability- Likelihood of occurrence of a specified event

P(A) =   The Number Of Ways Event A Can Occur The total number Of Possible Outcomes

Sample Space- Set of all possible outcomes

Sum of Probabilities of all possible outcomes = 1

Today’s LectureMultiple Events

Mutually Exclusive and Non-Mutually Exclusive Events Independent and Dependent Events

Types of Probability Unconditional or Prior Probability Conditional or Posterior Probability

Bayes’ Thoerom

Probability of Multiple EventsFor multiple events, the answer combines the

probabilities for each event in 2 different ways

If two events have to occur together, generally an "and" is used. Statement 1: "I will only be happy today if I get email and win the lottery." The "and" means that both events are expected to happen together.

If both events do not necessarily have to occur together, an "or" may be used as in Statement 2, "I will be happy today if I win the lottery or have email."

Source: http://www.800score.com/guidec8bview1b.html

Multiple Events- cont..

These two types of probability are formulated as follows:

Probability of A and BP(A and B) = P(A) × P(B) {probability is smaller than the individual probabilities of either A or B}

Probability of A or BP(A or B) = P(A) + P(B) – P(A and B) {probability is greater than the individual probabilities of either A or B }

Example If a coin is tossed twice, what is the probability that on the

first toss the coin lands heads and on the second toss the coin lands tails?

If a coin is tossed twice what is the probability that it will land either heads both times or tails both times?

Mutually Exclusive EventsTwo or more events are said to be mutually

exclusive if the occurrence of any one of them means the others will not occur (That is, we cannot have 2 events occurring at the same time)

For example, throwing a die once can yield a 5 or 6, but not both, in the same toss.

Thus if E1 and E2 are mutually exclusive events, then P(E1 and E2) = 0.

Example Male students and Female Students

If E1 and E2 are mutually exclusive events: P(E1 or E2) = P(E1) + P(E2)

http://www.tpub.com/math2/89.htm

Non-Mutually Exclusive Two or more events are said to be non- mutually

exclusive if the occurrence of any one of them means the others will also occur (That is, we can have 2 events occurring at the same time)

An example for non-mutually exclusive events could be:

E1 = students in the swimming team E2 = students in the debating team

If E1 and E2 are not mutually exclusive events: P(E1 or E2) = P(E1) + P(E2) − P(E1 and E2) Or P(E1 U E2) = P(E1) U P(E2) − P(E1 ∩ E2)

ExampleProblem 1: It is known that the probability of obtaining zero defectives in a

sample of 40 items is 0.34 whilst the probability of obtaining 1 defective item in the sample is 0.46. What is the probability of obtaining not more than 1 defective item in a sample?

Problem 2: The probability that a student passes Mathematics is 2/3 and the probability that he passes English is 4/9 . If the probability that he will pass at least one subject is 4/5 , What is the probability that he will pass both subjects?

http://www.intmath.com/Counting-probability/9_Mutually-exclusive-events.php

• Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other.

•For example, the probability of flipping a coin twice and the coin landing on heads the second time is not affected by (i.e. is independent of) whether the first coin flip turned up heads or tails.

•P(A and B) = P(A) × P(B)

Independent Events

What is the difference between Independent and Mutually Exclusive Events?Mutually Exclusive- If A occurs, B cannot

occur

Independent Events- Outcome of A doesn't affect outcome of B

If result of P(A) * P (B) = 0, then A and B are mutually exclusive and independent events

http://mathforum.org/library/drmath/view/69825.html

Dependent events• If the outcome of one event affects the outcome of another, then the

events are said to be Dependent Events.

Real-world Connections for Dependent Events If we wake up late, we will be late to office. If it rains, we use an umbrella.

For dependent events P(A and B) = P(A) × P(B\A)

Trick is to figure out ahead of time if the events are independent or dependent, and then use the formula

http://mathforum.org/library/drmath/view/56494.html

Can You Tell?? Which of the following are dependent events

1. Getting an even number in the first roll of a number cube and getting an even number in the second roll

2. Getting an odd number on the number cube and spinning blue color on the spinner.

3. Fair die is tossed twice. Find the probability of getting a 4 or 5 on the first toss and a 1, 2, or 3 in the second toss

4. Getting a face card in the first draw from a deck of playing cards and getting a face card in the second draw. (The first card is not replaced.)

Choices:A. 2 B. 2 and 3C. 1 and 3 D. 4E. 3 and 4

SolutionCorrect Answer: D

Solution: Step 1: In (1), rolling a number cube two times are two independent events. Step 2: In (2), rolling an odd number and spinning blue color are two

independent events. Step 3: In (3), getting 1, 2, or 3 doesn’t depends on 4 or 5, so they are

independent events Step 4: In (4), since the first card is not replaced back, the probability of the

second draw depends on the first draw.

Step 5: So, the two events in (4) are dependent events.

ExampleProblem 4:  

For combination of 3 bits, determine whether event containing odd number of 1s is independent of and event of bit strings ends at 0?

Solution:  

1. Calculate atomic events of Event containing odd number of 1s and Atomic events of Event of bit strings ends at 0

2. Calculate atomic events of Event containing both odd number of 1s and bit strings ends at 0

ExampleProblem 5

A box contains 3 white marbles and 4 black marbles. What is the probability of picking 2 black marbles and 1 white marble in succession without replacement?

Solutionp (A)= Probability of picking 1st black marble: 4/7p (B\A) = On the second draw the probability of picking 2nd black marble: 3/6= ½p (C \ B and A) = On the third draw the probability of picking a white marble is: 3/5Therefore, the probability of drawing 2 black marbles and 1 white

marble is: p (A) * p (B\A) * p (C\B and A)= 4/7 * 1/2 * 3/5= 6/35

http://www.tpub.com/math2/88.htm

Types of Probability

Prior Probability or Unconditional Probability

Probabilities that worked out prior to have any new information about the expected outcome of events in a particular situation.

Prior probability of a person having a disease is the number of people with the disease divided by the total number of people in the domain

Symbolized by: p (event)

Types of Probability- cont..

Posterior or Conditional Probability

Probability of an event given some new evidence or information on that event

If E1 and E2 are two events, the probability that E2 occurs given that E1 has occurred is called the conditional probability and is denoted by P(E2|E1). where, P(E2|E1) of E2 given that E1 has occurred.

Let A denote the event `student is female' and let B denote the event `student is French'. In a class of 100 students suppose 60 are French, and suppose that 10 of the French students are females. Find the probability that if I pick a French student, it will be a girl, that is, find P(A|B).

• Since 10 out of 100 students are both French and female, thenP(A and B) = 10/100• Also, 60 out of the 100 students are French, so P(B) = 60/100• So the required probability is:

Example of Conditional Probability

5.3. Applications of Stochastic Methodology

In this section, examples are there to use probability measures to reason about the interpretation of ambiguous information.

Probabilistic Finite State MachineProbabilistic Finite State Machine is 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

Probabilistic Finite State Machine is an acceptor, when one or more states are indicated as the start states or one or more as the accept states.

Example – Pronunciation of Tomato

Example- Hidden Markov Models

The Hidden Markov Model is a finite set of states, each of which is associated with a (generally multidimensional) probability distribution.

Transitions among the states are governed by a set of probabilities called transition probabilities. In a particular state an outcome or observation can be generated, according to the associated probability distribution. It is only the outcome, not the state visible to an external observer and therefore states are ``hidden'' to the outside; hence the name Hidden Markov Model.

5.4. Bayes’ Theorem

Definitions

Prior Probability

Unconditional probabilities of our hypothesis before we get any data or any NEW evidence. Simply speaking, it is the state of our knowledge before the data is observed

Posterior Probability

A conditional probability about our hypothesis (our state of knowledge) based on the new data

Likelihood

The conditional probability based on our observation data given that our hypothesis holds

Bayes’ Solution to a problem of “inverse probability”

The Theorem –

Relates cause & effect in such a way that by understanding the effect we can learn the probability of its causes.

Importance –

Important for determining the causes of diseases such as cancer.

Useful for determining the effects of some particular medication on that disease

Thomas Bayes (1702-1761)

Understanding Bayes’ TheoremUnderstanding Bayes’ TheoremMarie is getting married tomorrow, at an

outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie's wedding?

Solution: The sample space is defined by two mutually-exclusive events - it rains or it does not rain. Additionally, a third event occurs when the weatherman predicts rain. Notation for these events appears below:

Event A1. It rains on Marie's wedding.

Event A2. It does not rain on Marie's wedding

Event B. The weatherman predicts rain.

In Probabilistic Terms …We know the following:

P( A1 ) = 5/365 =0.0136985 [It rains 5 days out of the year.]

P( A2 ) = 360/365 = 0.9863014 [It does not rain 360 days out of the year.]

P( B | A1 ) = 0.9 [When it rains, the weatherman predicts rain 90% of the time.]

P( B | A2 ) = 0.1 [When it does not rain, the weatherman predicts rain 10% of the time.]

Cont..Cont..We want to know P( A1 | B ), the probability it will

rain on the day of Marie's wedding, given a forecast for rain by the weatherman. The answer can be determined from Bayes' theorem, as shown below.

Note the somewhat unintuitive result. Even when the weatherman predicts rain, it only rains only about 11% of the time. Despite the weatherman's gloomy prediction, there is a good chance that Marie will not get rained on at her wedding.

The general form of Bayes’ theorem where we assume the set of hypotheses H partition the evidence set E:

Luger: Artificial Intelligence, 5th edition. © Pearson Education Limited, 2005

22

Application of Bayes Theorem in Application of Bayes Theorem in AIAI

Basic rulesBasic rulesConditional probabilityConditional probability:

P(A|B)= P(AB) / P(B) if P(B)≠0Product ruleProduct rule:

P(AB) = P(A|B) P(B)Bayes’ RuleBayes’ Rule:

P(A|B)= P(B|A)P(A) / P(B)Why important??

Makes connection between Makes connection between diagnosticdiagnostic and and causalcausal probability. probability.

Non deterministic Algorithm Non deterministic Algorithm In computer science , a nondeterministic algorithm is an algorithm with one or more choice points where multiple different continuations are possible, without any specification of which one will be taken.

UseIn algorithm design, nondeterministic algorithms are often used as specifications. This is natural when the problem solved by the algorithm inherently allows multiple outcomes, or when there is a single outcome but there are multiple ways to get there and we simply don't care which of them is chosen.

What these cases have in common is that the nondeterministic algorithm always arrives at a valid solution, no matter which choices are made at the choice points encountered underway.

http://en.wikipedia.org/wiki/Nondeterministic_finite_state_machine

ExampleExampleExample 1: Shopping list

Consider a shopping list: a list of items to buy.

It can be interpreted in two ways:

•The instruction to buy all of those items, in any order. This is a (nondeterministic algorithm) •The instruction to buy all of those items, in the order given. This is a deterministic algorithm.

Monte Carlo MethodsMonte Carlo MethodsMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results.

Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer.

Monte Carlo methods tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm.

Monte Carlo Methods – cont..Monte Carlo Methods – cont..More broadly, Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business.

It is a widely successful method in risk analysis when compared with alternative methods or human intuition.

When Monte Carlo simulations have been applied in space exploration and oil exploration, actual observations of failures, cost overruns and schedule overruns are routinely better predicted by the simulations than by human intuition or alternative "soft" methods