cognitive systems 300: probability and causality...

Cognitive Systems 300:

Probability and Causality (cont.)

David Poole and Peter Danielson

University of British Columbia

Fall 2013

1 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)

The story so far....

Agents act in environments.

A controller considers:

what should the agent do nowwhat should the agent remember or believe

as a function of percepts and previous memory.

Memories are symbol structure; reasoning is search.

Hierarchical systems reduce complexity.

Acting is gambling.probabilities: possible worlds + conditioning


Learning Objectives

At the end of the class you should be able to:

know how to compute marginals and apply Bayes’theorem

build a belief network for a domain

predict the inferences for a belief network

explain the predictions of a causal model


Bayes’ theorem

The definition of conditioning and the commutativity ofconjunction (h ∧ e is equivalent to e ∧ h) gives us:

P(h ∧ e) =

P(h | e)× P(e)

= P(e | h)× P(h).

If P(e) 6= 0, divide the right hand sides by P(e):

P(h | e) =P(e | h)× P(h)

P(e).

This is Bayes’ theorem.


Bayes’ theorem


P(h ∧ e) = P(h | e)× P(e)

=

P(e | h)× P(h).


P(h | e) =P(e | h)× P(h)

P(e).



Bayes’ theorem


P(h ∧ e) = P(h | e)× P(e)

= P(e | h)× P(h).


P(h | e) =

P(e | h)× P(h)

P(e).



Bayes’ theorem


P(h ∧ e) = P(h | e)× P(e)

= P(e | h)× P(h).


P(h | e) =P(e | h)× P(h)

P(e).



Exercise

A cab was involved in a hit-and-run accident at night. Twocab companies, the Green and the Blue, operate in the city.You are given the following data:

85% of the cabs in the city are Green and 15% are Blue.

A witness identified the cab as Blue. The court tested thereliability of the witness in the circumstances that existedon the night of the accident and concluded that thewitness correctly identifies each one of the two colours80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accidentwas Blue?

From D. Kahneman, Thinking Fast and Slow, 2011, p. 166.


Exercise


85% of the cabs in the city are Green and 15% are Blue.


What is the probability that the cab involved in the accidentwas Blue?From D. Kahneman, Thinking Fast and Slow, 2011, p. 166.


Exercise


The two companies operate the same number of cabs,but Green cabs are involved in 85% of the accidents.


What is the probability that the cab involved in the accidentwas Blue?

From D. Kahneman, Thinking Fast and Slow, 2011, p. 167.Chapter 16 “Causes trump statistics”


Exercise


The two companies operate the same number of cabs,but Green cabs are involved in 85% of the accidents.


What is the probability that the cab involved in the accidentwas Blue?From D. Kahneman, Thinking Fast and Slow, 2011, p. 167.Chapter 16 “Causes trump statistics”


Tversky and Kahneman’s Linda [1983]

Linda is thirty-one years old, single, outspoken,and very bright. She majored in philosophy. As astudent, she was deeply concerned with issues ofdiscrimination and social justice, and alsoparticipated in antinuclear demonstrations.

Which is more probable:

(a) Linda is a bank teller.

(b) Linda is a bank teller and is active in the feministmovement.

85% to 95% of undergraduates at several major universitieschose the second option.From Tversky and Kahneman, Psychological Review, 1983.See D. Kahneman, Thinking Fast and Slow, 2011, p. 156.


Psychology of probability assessments

Which is more probable:

(a) A massive flood somewhere in North America next year,in which more than 1000 people drown.

(b) An earthquake in California sometime next year, causing aflood in which more than 1000 people drown.

From D. Kahneman, Thinking Fast and Slow, 2011, p. 159.


Psychology of probability assessments

There is a regular six-sided die with four green faces and twored faces. Here are three sequences of greens (G) and reds (R)and you have to choose one. Which do you choose:

(a) RGRRR

(b) GRGRRR

(c) GRRRRR

From D. Kahneman, Thinking Fast and Slow, 2011, p. 162.almost two-thirds preferred (b)....until it was explained to them


Conditional independence

Random variable X is independent of random variable Y givenrandom variable Z if

P(X | Y ,Z )

= P(X | Z ).

That is, knowledge of Y ’s value doesn’t affect the belief in X ,given a value of Z .


Example domain — lights and switches in a house

light

two-wayswitch

switch

off

on

poweroutlet

circuit�breaker

outside power

�

l1

l2

w1

w0

w2

w4

w3

w6

w5

p2

p1

cb2

cb1s1

s2s3


Bayesian Belief networks

Directed acyclic graph where the nodes are randomvariables.

We generate the variables one at a time.When generating X , the parents of X are those alreadygenerated variables upon which X directly depends.

Represents the conditional (in)dependence assumption:a variable is independent of its non-descendants given itsparents.


Example: fire alarm belief network

Variables:

Fire: there is a fire in the building

Tampering: someone has been tampering with the firealarm

Smoke: what appears to be smoke is coming from anupstairs window

Alarm: the fire alarm goes off

Leaving: people are leaving the building en masse.

Report: a colleague says that people are leaving thebuilding en masse. (A noisy sensor for leaving.)


Components of a belief network

A belief network consists of:

a directed acyclic graph with nodes labeled with randomvariables

a domain for each random variable

a set of conditional probability tables for each variablegiven its parents (including prior probabilities for nodeswith no parents).


Example: wet grass on a summers day

Variables:

Shoes wet after walking on grass

Sprinkler was on last night

Grass wet

Rained last night

Grass shiny and appears to be wet

http://artint.info/tutorials/sprinkler.xml


http://artint.info/tutorials/sprinkler.xml

cognitive systems 300: probability and causality...

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