Supervised Learning [1] Artificial Intelligence 2015-2016

Artificial Intelligence

Supervised Learning [2] Artificial Intelligence 2015-2016

Types of learning problems

{D1, D2, ..., Dn}

{D1, D2, ..., Dn}

{Y1, Y2, ..., Yn}

P

{D1, D2, ..., Dn}

P

{D1, D2, ..., Dn}

Xi ai ri

ai = (Di)

v(< r1, r2, ..., rn >)

Supervised Learning [3] Artificial Intelligence 2015-2016

Events and observations

P

W X = x1

X = x2

X = xn

...

Y = y1

Y = y2

Y = ym ...

W

Supervised Learning [4] Artificial Intelligence 2015-2016

Observations and Independence

{X, Y}

n D1 ={X1, Y1}, D2 ={X2, Y2}, ..., Dn ={Xn, Yn}

{X1, X2, ... , Xn}

<Xi Xj> i ≠ j

P(Xi) = P(Xi), i ≠ j

Supervised Learning [5] Artificial Intelligence 2015-2016

Maximum Likelihood Estimation (MLE) P(X)

P(X) P(X)

D = {D1, D2, ... , Dn}

P(X)

P(D | )

D

{D1, D2, ... , Dn}

)|,,,()|()|( 21 nDDDPDPDL

i

in DPDPDPDPDP )|()|()|()|()|( 21

)|(maxarg* DLML

i

i

i

i DPDPDLD )|(log)|(log)|(log)|(

)|(maxarg* DML

Supervised Learning [6] Artificial Intelligence 2015-2016

Example: coin tossing (Bernoulli Trials)

X X = 1 X = 0

P(X = 1) = , P(X = 0) = 1 < D1, D2, ... , Dn >

D = {D1={X1 = 1}, D2={X2 = 1}, D2={X3 = 0} ...}

P

i

i

i

ii XPXPXPDPD )|(log)|(log)|}({log)|(log)|(

i

i

i

i

i

XXXXD ii ]0[)1(log]1[log)1(log)|(

]0[]1[

( where NX=1 is the number of Xi = 1 in the sequence D )

)1(

01

XX NN

01

1*0

XX

XML

NN

N

vXif

vXifvX

i

i

i0

1][where:

)1(loglog 01 XX NN

]0[]1[ )1()|( XXXP

Supervised Learning [7] Artificial Intelligence 2015-2016

Anti-spam filter

D = {D1 ={Y1 = 1, X11 = 1, X12 = 1, ..., X1n = 0}, D1 ={Y2 = 0, X21 = 0, X22 = 1, ..., X2n = 1}, ...}

kjiik kYjXPkYP )|(,)(

m

kjikmkjikmkjik DPDPDPDL }),{|(}),{|}({)|()|},({

i

kjimmimim

m

kmm

m

kjimmimimkmm

m

kjikmmimimkjikmm

m

kjikimimmm

yYxXPyYP

yYxXPyYP

yYxXPyYP

xXyYP

}){,|(}){|(

}){,|}({}){|(

}),{,|}({}),{|(

}),{|},({

(<Xi Xj, Y>)

X1

Y

X2 ... Xn

n

i

ii YXPYPXYP1

)|()(}){,(

Supervised Learning [8] Artificial Intelligence 2015-2016

Anti-spam filter

P

j k

kYjX

kjikjikjimi

k

kY

kkk

ikYjXP

kYP

]][[

][

}){,|(

}){|(

m i

kjimmimim

m

kmmkjik yYxXPyYPD }){,|(log}){|(log)|},({

m i j k

kjimim

m

k

k

mkjik kYjXkYD log]][[log][)|},({

X1

Y

X2 ... Xn

n

i

ii YXPYPXYP1

)|()(}){,(

Supervised Learning [9] Artificial Intelligence 2015-2016

Anti-spam filter

m i j k

kjimim

m

k

k

mkjik kYjXkYD log]][[log][)|},({

)1(log][)|}({* k

k

m

k

k

mk kYD

D

k

kY

k

kY

k

k

kYk

k

k

m

m

k

NNN

N

kY

11

0

][

*

*

D

kYk

N

N *

D k

k

X1

Y

X2 ... Xn

n

i

ii YXPYPXYP1

)|()(}){,(

Supervised Learning [10] Artificial Intelligence 2015-2016

Anti-spam filter

i k j

kjiki

m i j k

kjimimkji kYjXD )1(log]][[)|}({*

kY

kYjX

kjiN

Ni

,*

kY

j

kYjX

j ik

kYjX

j

kji

ik

kYjX

kji

kji

ik

kji

m

mim

kji

NNN

N

kYjX

i

i

i

,

,

,*

*

11

0

]][[

i j k

X1

Y

X2 ... Xn

n

i

ii YXPYPXYP1

)|()(}){,(

m i j k

kjimim

m

k

k

mkjik kYjXkYD log]][[log][)|},({

Supervised Learning [11] Artificial Intelligence 2015-2016

Learning CPTs for a graphical model

T F

A S

L

R

T P(T)

0

1

T F A P(A | T,F)

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

F P(F)

0

1 F S P(S | F)

0 0

0 1

1 0

1 1

A L P(L | A)

0 0

0 1

1 0

1 1

L R P(R | L)

0 0

0 1

1 0

1 1

0

0,0

F

FS

N

ND

F

N

N 0

D

F

N

N 1D

T

N

N 0

D

T

N

N 1

0

0,1

F

FS

N

N

1

1,0

F

FS

N

N

1

1,1

F

FS

N

N

0

0,0

A

AL

N

N

0

0,1

A

AL

N

N

1

1,0

A

AL

N

N

1

1,1

A

AL

N

N

Supervised Learning [12] Artificial Intelligence 2015-2016

Bayesian learning

D

D

)()|(

)()|(

)(

)()|()|(

PDP

PDP

DP

PDPDP

)()|( PDP

)(P

Supervised Learning [13] Artificial Intelligence 2015-2016

Beta distribution n > 0

> 0

> 0

)!1(:)( nn

)!1(

)!1()!1(

)(

)()(:),(B

),(B

)1(:),;(Beta

11

xxx

Beta(x;1,1) Beta(x;2,2) Beta(x;10,10) Beta(x;2,3)

2

1

x

(*) In a finitary setting

Supervised Learning [14] Artificial Intelligence 2015-2016

Conjugate prior distributions

]0[]1[)1()|(

ii XX

iDP

DD

i

ii DPDPDP )1()|()|}({)|(

),(B

)1()1(),;(Beta)1()()|(

11

PP

PP

PP

DDDDPDP

),;(Beta),(B

),(B

),(B

)1(11

PDPD

PP

PDPD

PP

PDPD

),;(Beta)()|( PDPDPDP

2

1),;(Betamaxarg*

PDPD

PDPDPDMAP

2 PP

)(P )|( DP

)()|( PDP )(P

𝛼𝐷 𝛽𝐷

𝛼𝑃 𝛽𝑃

Supervised Learning [15] Artificial Intelligence 2015-2016

Anti-spam filter

D

2

1,*

kYkjikji

kYjXkji

kjiN

Ni

i j k

2

1*

Dkk

kYkk

N

N

k

)()|(maxarg* PDPMAP

kjikjikk ,,,

X1

Y

X2 ... Xn

n

i

ii YXPYPXYP1

)|()(}){,(