INC 551 Artificial Intelligence

Lecture 8

Models of Uncertainty

Inference by Enumeration

Bayesian Belief Network Model

Causes -> Effect

Graph Structure shows dependency

Burglar Alarm ExampleMy house has a burglar alarm but sometimes it rings becauseof an earthquake. My neighbors, John and Mary promise meto call if they hear the alarm. However, their ears are not perfect.

One Way to Create BBN

Computing Probability

= 0.90x0.70x0.001x0.999x0.998 = 0.00062

BBN Construction

There are many ways to construct BBN ofa problem because the events depend oneach other (related).

Therefore, it depends on the order of eventsthat you consider.

The most simple is the most compact.

Not compact

Inference Problem

)|( eEXP i Find

),|( trueMaryCallstrueJohnCallsBurglaryP

For example, find

Inference by Enumeration

y

yeXPeXPeP

eXPeXP ),,(),(

)(

),()|(

Note: let

α is called “Normalized Constant”

)(

1

eP

y are other events

),|( trueMaryCallstrueJohnCallsBurglaryP

(summation of all events)

Calculation Tree

Inefficient, Compute P(j|a)P(m|a) for every value of e

001.0

998.))06.05.01(.)94.9.7((.

002.))05.05.01(.)95.9.7((.

),,(),|(

mjbPmjbP

Next, we have to find P(~b|j,m) and find α = 1/P(j,m)

Approximate Inference

Idea: Count from real examples

We call this procedure “Sampling”

Sampling = get real examples from the world model

Sampling Example

Cloudy = มี�เมีฆSprinkler = ละอองน้ำ��

?)?,?,,(

),,,(

T

WetGrassRainSprinklerCloudy

?)?,?,,(

),,,(

T

WetGrassRainSprinklerCloudy

?)?,,,(

),,,(

FT

WetGrassRainSprinklerCloudy

?),,,(

),,,(

TFT

WetGrassRainSprinklerCloudy

?),,,(

),,,(

TFT

WetGrassRainSprinklerCloudy

),,,(

),,,(

TTFT

WetGrassRainSprinklerCloudy

1 Sample

Rejection Sampling

Idea: Count only the sample that agree with e

To find )|( eXP i

Rejection Sampling

Drawback: There are not many samples that agree with e

From the example above, from 100 samples, only 27 areUsable.

Likelihood Weighting

Idea: Generate only samples that are relevant with e

However, we must use “weighted sampling”

e.g. Find ),|( trueWetGrasstrueSprinklerRainP

Fix sprinkler = TRUE Wet Grass = TRUE

Weighted Sampling

Sample from P(Cloudy)=0.5 , suppose we get “true”

Sprinkler already has value = true,therefore we multiply with weight = 0.1

Sample from P(Rain)=0.8 , suppose we get “true”

WetGrass already has value = true,therefore we multiply with weight = 0.99

Finally, we got a sample (t,t,t,t)With weight = 0.099

Temporal Model (Time)

When the events are tagged with timestamp

1dayRain 2dayRain 3dayRain

Each node is considered “state”

Markov Process

Let Xt = stateFor Markov processes, Xt depends only on finite numberof previous xt

Hidden Markov Model (HMM)

Each state has observation, Et.

We cannot see “state”, but we see “observation”.