Iowa State Universitycs573x/Notes/cs573week3... · 2010-02-07 · 2 ! # Department of Computer...

Department of Computer Science

Artificial Intelligence Research Laboratory

Copyright Vasant Honavar, 2010

11, 01,10, 00

c=A c=B

x 1 0 x

Tree 1 Tree 2 Data set

11, 01, 10, 00

P(X = H; Y = H) = 0.2. P(X= H; Y = T) = 0.4P(X = T; Y = H) = 0.3. P(X = T; Y = T) = 0.1H(X,Y)=-0.2log20.2+… 1.85P(X = H) = 0.6. H(X) 0.97P(Y = H) = 0.5. H(Y) = 1.0P(Y= H| X = H) = 0.2/0.6 = 0.333P(Y = T|X = H) = 1-0.333=0.667P(Y= H| X = T)= 0.3/0.4=0.75P(Y= T| X = T)=0.1/0.4=0.25 H(Y|X) 0.88

I(X,Y)=0?

I1 (t, d, l) +I2 (s, d, l) +I3 (t, b, l) -I4 (t, r, l) -I5 (s, b, l) -I6 (t, b, w) +I7 (t, d, w) +I8 (s, b, w) +

Height (tall, short)Hair (dark, blonde, red) Eye (blue, brown)

I1…I8

Height

I1 I3 I4 I6 I7 I2 I5 I8

Hair Hair

+ -Eye

P (h) h

P (D) D

P (h | D) h D

P (D | h) D h

hMAP = argmaxh

iHP(D | hi )P(hi )

= argmaxh

log2 P Dhi( ) + log2 P hi( )( )

= argminh

log2 P Dhi( ) log2 P hi( )( )

log2 P hi( ) = description length of hi under optimal encoding

of hypotheses

-log2P(D | hi ) = description length of data D given hi under optimal encoding based on P(D | hi )

MAP hypothesis learning implies Occam's razor!

D = {ei =(Xi , di)|Xi X di=c(Xi )}

i.i.d. samples imply

P D | h( ) = P ei | h( )i

= P Xi,di h( )i

= P di h,Xi( )i

P Xi | h( )

Assuming Xi are independent of h,

P D | h( ) = P(dii

| h,Xi )P Xi( )

P(di | h,Xi ) =1 if h(Xi ) = di;

P(di | h,Xi ) = 0 otherwise

P D( ) = P(D | hh H

)P(h) =1H

h VH, D

P Xi( )i

h VH, D

=VH ,D

HP Xi( )

h VH ,D, we have

P(h |D) =P Dh( )P h( )

P D( )=

P Xi( )i

VH ,D P Xi( )i

1VH ,D

and h VH ,D, we have P(h |D) = 0

EntropyRatio(S | A)Entropy(S | A)

SplitEntropy(S | A)

SplitEntropy(S | A)| Si || S |i=1

Values(A )log2

| Si || S |

Entropy Impurity(Z) =ZiZi=1

Classeslog2

Gini Impurity(Z) =ZiZ

Classes

Impurity(S | A) = Min

i Values(A )Impurity Si( ){ }

ASGain

ACost 1

ASGain2

Impurity(S) = ijij

Iowa State Universitycs573x/Notes/cs573week3... · 2010-02-07 · 2 ! # Department of Computer...

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