representing the proportion of days on which playis yes. A similar calculation
for the outcome noleads toThis indicates that for the new day,nois more likely than yes—four times more
likely. The numbers can be turned into probabilities by normalizing them so
that they sum to 1:This simple and intuitive method is based on Bayes’s rule of conditional prob-
ability. Bayes’s rule says that if you have a hypothesis Hand evidence Ethat bears
on that hypothesis, thenWe use the notation that Pr[A] denotes the probability of an event Aand that
Pr[A|B] denotes the probability ofAconditional on another event B.The
hypothesis His that playwill be, say,yes,and Pr[H|E] is going to turn out to be
20.5%, just as determined previously. The evidence Eis the particular combi-
nation of attribute values for the new day,outlook=sunny, temperature=cool,
humidity=high, and windy=true. Let’s call these four pieces of evidence E 1 ,E 2 ,
E 3 , and E 4 , respectively. Assuming that these pieces of evidence are independent
(given the class), their combined probability is obtained by multiplying the
probabilities:Don’t worry about the denominator: we will ignore it and eliminate it in the
final normalizing step when we make the probabilities ofyesand nosum to 1,
just as we did previously. The Pr[yes] at the end is the probability of a yes
outcome without knowing any of the evidence E,that is, without knowing any-
thing about the particular day referenced—it’s called the prior probabilityof the
hypothesis H. In this case, it’s just 9/14, because 9 of the 14 training examples
had a yesvalue for play. Substituting the fractions in Table 4.2 for the appro-
priate evidence probabilities leads toPr
Pryes E
E[]=¥¥¥¥
[]29 39 39 39 914
,PrPr Pr Pr Pr Pr
Pryes EE yes E yes E yes E yes yes
E[]=[]¥ []¥ []¥ []¥ []
[]12 3 4.PrPr Pr
PrHEEH H
E[]=[][]
[].Probability of no=
+=0 0206
0 0053 0 020679 5.
...%.Probability of yes=
+=0 0053
0 0053 0 020620 5.
...%,likelihood of no=¥¥¥¥ =3 5 1 5 4 5 3 5 5 14 0 0206..90 CHAPTER 4| ALGORITHMS: THE BASIC METHODS