1 Overview

During the last unit, our investigations of AI were rooted in logic; in particular, we studied “symbolic” AI,inwhich all statements were assignedbinary truth values. We nowembark uponastudy of what might be called “numeric” AI. Rather than associate discrete values with statements, we associate continuous values, or probabilities. This representational flexbility is intended to model uncertainty in an agent’s environment.

2 Introduction

Gambling was the primary force driving the early development of probability theory. As early as the 16th century, gamblers noticed that there are empirical laws which govern the frequencies of the various outcomes in a game of chance, even though the precise outcome cannot generally be predicted in advance. For example, Cardano noticed that for many simple games of chance, each outcome is realized in proportion to the reciprocal of the total number of outcomes. Cardano’s observation applies to rolling dice and flipping coins, for example.

Supposean experiment(such as spinning a red andblack roulette wheel) is repeated N times. Let #(A) denote the number of times the outcome A (e.g., “landed on red”) is observed. The ratio of #(A) to A is called the relative frequency of A. Empirically, when N is large, this relative frequency approximates some p∈ R: i.e.,

#(A)

≈ p

N

Since p depends on A, it is usually written P(A), anditis calledthe probability of A. Sometimes the symmetric nature of the experiments renders all outcomes are equally likely, asinthegames of chance studied by Cardano. In this case,

|A|

P(A)=

|Ω|

We write Ω to denote the sample space of possible outcomes, with ω ∈ Ω and A ⊆ Ω.

Example: Consider an experiment in which an unbiased coin is flipped twice in succession. In this experiment, the sample space Ω = {HH,HT,TH,TT}, with each outcome equally likely. If A is the event “at least one head,” then P(A)=3/4.

Similarly, consider an experiment in which an unbiased coin is flipped thrice in succession. In this experiment, the sample space Ω = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}, with each outcome equally likely. If B is the event “at most one tail,” then P(B)=1/2.

3 Axioms of Probability

The following three axioms characterize the probability function P :2^Ω → R:

1. P(Ω)=1
2. 0 ≤ P(A)≤ 1, for all events A ⊆ Ω
3. P(A∪B)= P(A)+P(B), for allevents A,B ⊆ Ω s.t. A∩B = ∅

The third axiom is often expressed as its inductive equivalent, namely

3^′ . given the finite set of disjoint events {A1,A2,...,An},

n



PAi = P(A1)+P(A2)+... + P(An)

i=1

4.1 Independence

A and B are independent events iff P(A |B)= P(A). If P(A |B)= P(A), then by Equation 4, P(B |A)= P(B). Moreover,

P(A∩B)= P(A |B)P(B)= P(A)P(B) (6)

Example: If A is the event “head on the first toss” and B is the event “head on the second toss,” then A = {HH,HT}, B = {HH,TH}, and A∩B = {HH}. A and B are independent events, since P(A ∩B) =1/4 =(1/2)(1/2) = P(A)P(B). On the other hand, if B is the event “at least one tail,” then B = {HT,TH,TT}, and the events A and B are not independent: P(A∩B)=1/4 but P(A)P(B)=(1/2)/(3/4) =3/8.

Exercise: Show that if A and B are independent events, then the pairs of events A and B^c , A^c and B, and A^c and B^c are all independent.

A and B are conditionally independent with respect to C iff P(A |B ∩C)= P(A |C). This definition also implies that P(B |A∩C)= P(B |C), since

P(A∩C |B)P(B)P(B |A∩C)=

P(A∩C) P(A |B ∩C)P(C |B)P(B)

=

P(A∩C) P(A |C)P(C |B)P(B)

=

P(A∩C) P(C |B)P(B)

=

P(C) = P(B |C)

Moreover, if A and B are conditionally independent of C, then

P(A∩B |C)= P(A |B ∩C)P(B |C)= P(A |C)P(B |C) (7)

Exercise: Show that if A and B are independent events, then A and B are conditionally independent of C for all events C.

5.1 Expectation and Variance

The expected value E[X]of a random variable X is defined as:

E[X]= p(xi)xi i

The expected value is also called the mean, in which case it is denoted µX , or µ, whenever the random variable over which it is defined is clear from context.

For example, if I is a random variable with range {0,1,...,n}, and if I is uniformly distributed (i.e., P(I = i)=1/(n +1) for all 0 ≤ i ≤ n), then

nn

� 11 � 1 n(n +1) n

E[I]= i = i ==

n +1 n +1 n +1 2 2

i=0 i=0

Exercise: Compute E[X]and E[Y]in our running example. The following properties hold of expectation:

• If X and Y are independent random variables, then E[XY]= E[X]E[Y].
• Linearity of expectation: i.e., E[X + Y]= E[X]+E[Y].
• For arbitrary constant c ∈ R, E[cX]= cE[X].

Exercise: Prove the stated properties of expectation.

Expectation suffices to predict an average data point. The expected value of the two sequences 0,0,... and −1,1,−1,1,... both equal zero, however. Variance captures the variability in a series of data points.

Given random variable X, the variance var(X)= E[(X −µX )²]:

var(X)= E[(X −µX )²]

2

= E[X² −2µX X + µ ]

X

2

= E[X²]−2µX E[X]+µ

X

= E[X²]−2E[X]E[X]+(E[X])² = E[X²]−(E[X])²

Variance is oftendenoted σ² , or simply σ², whenever X is clear from context. The value σ is called

X

the standard deviation. If I is as above, a random variable with range {0,1,...,n}and p(i)is uniformly distributed, then

n

� 11 n(2n +1)

E[I²]= i² = (0² +1² +2² + ... + n ²)=

n +1 n +16

i=0

Thus,

��₂

n(2n +1) nn(n +2)

var(I)= E[I²]−(E[I])² = − =

6 2 12

The following properties hold of variance:

• For arbitrary constant a ∈ R, var(aX)= a² var(X).
• For arbitrary constant c ∈ R, var(X + c)= var(X).

Exercise: Prove the stated properties of variance. Exercise: Let Z represent the number of heads after three successive coin tosses. Given p(Z)as

follows, compute E[Z]and var(Z).

Z

0 1 2 3

p(Z) 1/8 3/8 3/8 1/8

5.2 Correlation and Covariance

Correlation and covariance provide numerical measurements of the strength of the relationship between two random variables.

Given random variables X and Y with respective means µX and µY , the covariance cov(X,Y)= E[(X −µX )(Y −µY )]:

cov(X,Y)= E[(X −µX )(Y −µY )] = E[(XY −µX Y −µY X −µxµY )] = E[XY]−µX E[Y]−µY E[X]−µxµY )] = E[XY]−µX µY −µY µX −µxµY )] = E[XY]−µX µY

The correlation coefficient ρXY is defined in terms of covariance and standard deviation as follows:

cov(X,Y)

ρXY = (8)

σX σY

The following properties hold of covariance and correlation:

• For independent random variables X and Y, cov(X,Y)= ρXY =0.
• For arbitrary random variables X and Y, var(X + Y)= var(X)+var(Y)+2cov(X,Y).