What s the probability that the next candy is lime? What is P(d - - PowerPoint PPT Presentation

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What s the probability that the next candy is lime? What is P(d i+1 | d 1 ,, d i ) = P(X | D )? We don t know which bag of candy we got, so we have to assume it could be any one of them: P(X | D ) = i P(X | D , h i )P(h i |

SLIDE 1

Whatʼs the probability that the next candy is lime?

CS440/ECE448: Intro AI

What is P(di+1 | d1 ,…, di ) = P(X | D)?

We donʼt know which bag of candy we got, so we have to assume it could be any one of them:

P(X | D) = ∑i P(X | D, hi )P(hi |D) = ∑i P(X | hi )P(hi |D)

SLIDE 2

Lecture 19  Learning graphical models

Prof. Julia Hockenmaier

juliahmr@illinois.edu

http://cs.illinois.edu/fa11/cs440
CS440/ECE448: Intro to Artificial Intelligence

SLIDE 3

The Burglary example

CS440/ECE448: Intro AI

Burglary Earthquake Alarm JohnCalls MaryCalls

B=t ¡ B=f ¡ .001 ¡ 0.999 ¡ E=t ¡ E=f ¡ .002 ¡ 0.998 ¡ B ¡ E ¡ A=t ¡ A=f ¡ t ¡ t ¡ .95 ¡ .05 ¡ t ¡ f ¡ .94 ¡ .06 ¡ f ¡ t ¡ .29 ¡ .71 ¡ f ¡ f ¡ .999 ¡ .001 ¡ A ¡ M=t ¡ M=f ¡ t ¡ .7 ¡ .3 ¡ f ¡ .01 ¡ .99 ¡ A ¡ J=t ¡ J=f ¡ t ¡ .9 ¡ .1 ¡ f ¡ .05 ¡ .95 ¡

What is the probability of a burglary if John and Mary call?

SLIDE 4

Learning Bayes Nets

SLIDE 5

How do we know the parameters

f a Bayes Net?

We want to estimate the parameters based

n data D.
Data = instantiations of some or all random

variables in the Bayes Net.

The data are our evidence.

SLIDE 6

Surprise Candy

There are two flavors of Surprise Candy: cherry and lime. Both have the same wrapper.

There are five different types of bags (which all

look the same) that Surprise Candy is sold in:

– h1: 100% cherry – h2: 75% cherry + 25% lime – h3: 50% cherry + 50% lime – h4: 25% cherry + 75% lime – h5: 100% lime 6

CS440/ECE448: Intro AI

SLIDE 7

Surprise Candy

You just bought a bag of Surprise Candy. Which kind of bag did you get?

There are five different hypotheses: h1-h5
You start eating your candy. This is your data

D1 = cherry, D2 = lime, …., DN= ….

What is the most likely hypothesis given your data

(evidence)?

CS440/ECE448: Intro AI

SLIDE 8

Conditional probability  refresher

CS440/ECE448: Intro AI

P(X |Y )P(Y ) = P(Y | X)P(X) P(X |Y ) = P(X,Y ) P(Y ) P(X |Y )P(Y ) = P(X,Y )

SLIDE 9

Bayes Rule

P(cause): prior probability of cause

P(cause | effect): posterior probability of cause. P(effect | cause): likelihood of effect

Prior ∝ posterior × likelihood

CS440/ECE448: Intro AI

P(cause | effect)! P(effect | cause)P(cause) P(cause | effect) = P(effect | cause)P(cause) P(effect)

SLIDE 10

Bayes Rule

P(h): prior probability of hypothesis

P(h | D): posterior probability of hypothesis. P(D | h): likelihood of data, given hypothesis

Prior ∝ posterior × likelihood

CS440/ECE448: Intro AI

P(h | D)! P(D | h)P(h) P(h | D) = P(D | h)P(h) P(D)

SLIDE 11

Bayes Rule

P(h): prior probability of hypothesis

P(h | D): posterior probability of hypothesis. P(D | h): likelihood of data, given hypothesis

CS440/ECE448: Intro AI

argmaxh P(h | D) = argmaxh P(D | h)P(h) P(D) = argmaxh P(D | h)P(h)

SLIDE 12

Bayesian learning

Use Bayes rule to calculate the probability of each hypothesis given the data.

How do we know the prior and the

likelihood?

CS440/ECE448: Intro AI

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

SLIDE 13

The prior P(h)

Sometimes we know P(h) in advance.

– Surprise Candy: (0.1, 0.2, 0.4, 0.2, 0.1)

Sometimes we have to make an assumption

(e.g. a uniform prior, when we donʼt know anything)

CS440/ECE448: Intro AI

SLIDE 14

The likelihood P(D|h)

We typically assume that each observation di is drawn “i.i.d.” - independently from the same (identical) distribution.

Therefore:
14

CS440/ECE448: Intro AI

P(D | h) = P(di | h)

i

!

SLIDE 15

The posterior P(h|D)

Assume weʼve seen 10 lime candies:

CS440/ECE448: Intro AI

0.2 0.4 0.6 0.8 1 2 4 6 8 10 Posterior probability of hypothesis Number of observations in d P(h1 | d) P(h2 | d) P(h3 | d) P(h4 | d) P(h5 | d)

SLIDE 16

Whatʼs the probability that the next candy is lime?

This probability will eventually (if we had an infinite amount of data) agree with the true hypothesis.

CS440/ECE448: Intro AI

0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 Probability that next candy is lime Number of observations in d

SLIDE 17

Bayes optimal prediction

We donʼt know which hypothesis is true,   so we marginalize them out:

P(X | D) = ∑i P(X | hi )P(hi |D)

This is guaranteed to converge to the true hypothesis.

CS440/ECE448: Intro AI

SLIDE 18

Maximum a-posteriori (MAP)

We assume the hypothesis with the maximum posterior probability     hMAP = argmaxh P(h|D) is true:

P(X | D) = P(X | hMAP )

CS440/ECE448: Intro AI

SLIDE 19

Maximum likelihood (ML)

We assume a uniform prior P(h). We then choose the hypothesis that assigns the highest likelihood to the data    hML = argmax h P(D|h) P(X | D) = P(X | hML ) This is commonly used in machine learning.

CS440/ECE448: Intro AI

SLIDE 20

Surprise candy again

Now the manufacturer has been bought up by another company.

Now we donʼt know the lime-cherry

proportions θ (= P(cherry)) anymore.

Can we estimate θ from data?

CS440/ECE448: Intro AI

flavor

cherry θ lime: 1-θ

SLIDE 21

Maximum likelihood learning

Given data D, we want to find the parameters that maximize P(D | θ).

We have a data set with N candies.

c candies are cherry. l = (N-c) candies are lime.

CS440/ECE448: Intro AI

SLIDE 22

Maximum likelihood learning

Out of N candies, c are cherry, (N-c) lime.

The likelihood of our data set:
22

CS440/ECE448: Intro AI

P(d |!) = P(d j |

j=1 N

!

!) = ! c(1"!)l

SLIDE 23

Log likelihood

Itʼs actually easier to work with   the log-likelihood:

CS440/ECE448: Intro AI

L(d |!) = log P(d|!) = logP(di !)

j=1 N

!

= c log! +l log(1"!)

SLIDE 24

Maximizing Log-likelihood

CS440/ECE448: Intro AI

dL(D |!) d! = c ! ! l 1!! = 0 "! = c c +l = c N

SLIDE 25

Maximum likelihood estimation

We can simply count how many cherry candies we see.

This is also called the relative frequency

estimate.

It is appropriate when we have complete

data (i.e. we know the flavor of each candy).

CS440/ECE448: Intro AI

SLIDE 26

Todayʼs reading

Chapter 13.5, Chapter 20.1 and 20.2.1

CS440/ECE448: Intro AI

Whatʼs the probability that the next candy is lime?

We donʼt know which bag of candy we got, so we have to assume it could be any one of them:

P(X | D) = ∑i P(X | D, hi )P(hi |D) = ∑i P(X | hi )P(hi |D)

Lecture 19 Learning graphical models

juliahmr@illinois.edu

The Burglary example

Learning Bayes Nets

How do we know the parameters

We want to estimate the parameters based

variables in the Bayes Net.

Surprise Candy

There are two flavors of Surprise Candy: cherry and lime. Both have the same wrapper.

look the same) that Surprise Candy is sold in:

Surprise Candy

You just bought a bag of Surprise Candy. Which kind of bag did you get?

D1 = cherry, D2 = lime, …., DN= ….

(evidence)?

Conditional probability refresher

P(X |Y )P(Y ) = P(Y | X)P(X) P(X |Y ) = P(X,Y ) P(Y ) P(X |Y )P(Y ) = P(X,Y )

Bayes Rule

P(cause | effect): posterior probability of cause. P(effect | cause): likelihood of effect

P(cause | effect)! P(effect | cause)P(cause) P(cause | effect) = P(effect | cause)P(cause) P(effect)

Bayes Rule

P(h | D): posterior probability of hypothesis. P(D | h): likelihood of data, given hypothesis

P(h | D)! P(D | h)P(h) P(h | D) = P(D | h)P(h) P(D)

Bayes Rule

P(h | D): posterior probability of hypothesis. P(D | h): likelihood of data, given hypothesis

Bayesian learning

Use Bayes rule to calculate the probability of each hypothesis given the data.

likelihood?

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

The prior P(h)

Sometimes we know P(h) in advance.

– Surprise Candy: (0.1, 0.2, 0.4, 0.2, 0.1)

(e.g. a uniform prior, when we donʼt know anything)

The likelihood P(D|h)

We typically assume that each observation di is drawn “i.i.d.” - independently from the same (identical) distribution.

P(D | h) = P(di | h)

i

!

The posterior P(h|D)

Assume weʼve seen 10 lime candies:

Whatʼs the probability that the next candy is lime?

This probability will eventually (if we had an infinite amount of data) agree with the true hypothesis.

Bayes optimal prediction

We donʼt know which hypothesis is true, so we marginalize them out:

This is guaranteed to converge to the true hypothesis.

Maximum a-posteriori (MAP)

We assume the hypothesis with the maximum posterior probability hMAP = argmaxh P(h|D) is true:

Maximum likelihood (ML)

We assume a uniform prior P(h). We then choose the hypothesis that assigns the highest likelihood to the data hML = argmax h P(D|h) P(X | D) = P(X | hML ) This is commonly used in machine learning.

Surprise candy again

Now the manufacturer has been bought up by another company.

proportions θ (= P(cherry)) anymore.

flavor

Maximum likelihood learning

Given data D, we want to find the parameters that maximize P(D | θ).

c candies are cherry. l = (N-c) candies are lime.

Maximum likelihood learning

Out of N candies, c are cherry, (N-c) lime.

P(d |!) = P(d j |

!

!) = ! c(1"!)l

Log likelihood

Itʼs actually easier to work with the log-likelihood:

L(d |!) = log P(d|!) = logP(di !)

!

= c log! +l log(1"!)

Maximizing Log-likelihood

dL(D |!) d! = c ! ! l 1!! = 0 "! = c c +l = c N

Maximum likelihood estimation

We can simply count how many cherry candies we see.

estimate.

data (i.e. we know the flavor of each candy).

Todayʼs reading

Chapter 13.5, Chapter 20.1 and 20.2.1

Lecture 19  Learning graphical models

Conditional probability  refresher

We donʼt know which hypothesis is true,   so we marginalize them out:

We assume the hypothesis with the maximum posterior probability     hMAP = argmaxh P(h|D) is true:

We assume a uniform prior P(h). We then choose the hypothesis that assigns the highest likelihood to the data    hML = argmax h P(D|h) P(X | D) = P(X | hML ) This is commonly used in machine learning.

Itʼs actually easier to work with   the log-likelihood: