02 Probability, Bayes Theorem and the Monty Hall Problem

7/31/2019 02 Probability, Bayes Theorem and the Monty Hall Problem

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Probability, Bayes Theorem

and the Monty Hall Problem


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PSYC 6130, PROF. J. ELDER 2

Probability Distributions

A random variable is a variable whose value is uncertain.

For example, the height of a randomly selected person in this classis a random variableI wont know its value until the person isselected.

Note that we are not completely uncertain about most randomvariables.

For example, we know that height will probably be in the 5-6 range.

In addition, 56 is more likely than 50 or 60 (for women).

The function that describes the probability of each possible value ofthe random variable is called a probability distribution.


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Probability distributions are closely related to frequencydistributions.


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Dividing each frequency by the total number of scoresand multiplying by 100 yields a percentage distribution.


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Dividing each frequency by the total number of scoresyields a probability distribution.


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For a discrete distribution, the probabilities over allpossible values of the random variable must sum to 1.


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For a discrete distribution, we can talk about the probability of a particular

score occurring, e.g., p(Province = Ontario) = 0.36. We can also talk about the probability of any one of a subset of scores

occurring, e.g., p(Province = Ontario or Quebec) = 0.50.

In general, we refer to these occurrences as events.


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For a continuous distribution, the probabilities over all possiblevalues of the random variable must integrate to 1 (i.e., the areaunder the curve must be 1).

Note that the height of a continuous distribution can exceed 1!

Shadedarea=0.683 Shadedarea=0.954 Shadedarea=0.997


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Continuous Distributions

For continuous distributions, it does not make sense to talk about theprobability of an exact score.

e.g., what is the probability that your height is exactly 65.485948467 inches?

55 60 65 70 750

0.02

0.04

0.060.08

0.1

0.12

0.14

0.16

Height (in)

Probabilityp

Normal Approximation to probability distribution for height of Canadian females(parameters from General Social Survey, 1991)

5'3.8"

2.6"s

?


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Continuous Distributions

It does make sense to talk about the probability of observing a score that falls within a certainrange

e.g., what is the probability that you are between 53 and 57?

e.g., what is the probability that you are less than 510?

55 60 65 70 750

0.02

0.04

0.060.08

0.1

0.12

0.14

0.16

Height (in)

Probabilityp

Normal Approximation to probability distribution for height of Canadian females(parameters from General Social Survey, 1991)

5'3.8"

2.6"s

Valid events


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Probability of Combined Events

More generally, if and are mutually exclusive,( or ) ( ) ( ) ( an

notd )

A Bp A B p A p B p A B

Canadian Community Health Survey, SleepingExam Haple: bits

Let event that respondent sleeps less than 6 hours per night.A

Let event that respondent reports trouble sleeping most or all of the timeB

( ) 0.139p A

( ) 0.152p B

( and ) 0.061p A B

Thus

( or ) 0.139 0.152 0.061 0.230p A B


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Exhaustive Events

Two or more events are said to be exhaustive if at leastone of them must occur.

For example, if A is the event that the respondent sleeps

less than 6 hours per night and B is the event that therespondent sleeps at least 6 hours per night, then A andBare exhaustive.

(Although A is probably the more exhausted!!)


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Independence

Two events are if the occurence of one

in no way affects the probability of

ind

the

ependent

other.

If events and are independent, then

( and ) ( ) ( )

A B

p A B p A p B

If events and are not independent, then

( and ) ( ) ( | )

A B

p A B p A p B A

: pick a card, anyExam cple ard.


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An Example: The Monty Hall Problem
http://www.nytimes.com/2008/04/08/science/08monty.html


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Problem History

When problem first appeared in Parade, approximately10,000 readers, including 1,000 PhDs, wrote claimingthe solution was wrong.

In a study of 228 subjects, only 13% chose to switch.


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Intuition

Before Monty opens any doors, there is a 1/3 probabilitythat the car lies behind the door you selected (Door 1),and a 2/3 probability it lies behind one of the other twodoors.

Thus with 2/3 probability, Monty will be forced to open aspecific door (e.g., the car lies behind Door 2, so Montymust open Door 3).

This concentrates all of the 2/3 probability in theremaining door (e.g., Door 2).


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http://www.youtube.com/watch?v=hcFkic2I8zU


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Analysis

Switchingloses withprobability 1/6

Switching wins with probability 2/3Switching loses with

probability 1/3

Switching wins with probability1/3

Switching wins with probability1/3

Switchingloses withprobability 1/6

Host must open Door 2Host must open Door 3Host opens either Door 2 or 3

Player initially picks Door 1

Car hidden behind Door 3Car hidden behind Door 2Car hidden behind Door 1


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Notes

It is important that

Monty must open a door that reveals a goat

Monty cannot open the door you selected

These rules mean that your choice may constrain whatMonty does.

If you initially selected a door concealing a goat, then there isonly one door Monty can open.

One can rigorously account for the Monty Hall problemusing a Bayesian analysis


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End of Lecture 2

Sept 17, 2008


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Conditional Probability

To understand Bayesian inference, we first need to understand the

concept of conditional probability.

What is the probability I will roll a 12 with a pair of (fair) dice?

What if I first roll one die and get a 6? What now is the probabilitythat when I roll the second die they will sum to 12?

Let be the state of die 1

Let B be the state of die 2

Let be the sum of die 1 and 2

A

C

( 6) __?p A

( 6) __?p B

( 12) __?p C

( 12 | 6) __?p C A

Probability of C givenA


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Conditional Probability

The conditional probabilityof A given B is thejointprobabilityof A and B, divided by the marginalprobabilityof B.

Thus if A and B are statistically independent,

However, if A and B are statistically dependent, then

( , )( | )

( )

p A Bp A B

p B

( , ) ( ) ( )( | ) ( ).

( ) ( )

p A B p A p B p A B p A

p B p B

( | ) ( ).p A B p A


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Bayes Theorem

Bayes Theorem is simply a consequence of the

definition of conditional probabilities:

( , )

( | ) ( , ) ( | ) ( )( )

p A Bp A B p A B p A B p B

p B

( , )

( | ) ( , ) ( | ) ( )( )

p A Bp B A p A B p B A p A

p A

Thus ( | ) ( ) ( | ) ( )p A B p B p B A p A

( | ) ( )

( | )( )

p B A p Ap A B

p B

Bayes Equation


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Bayes Theorem

Bayes theorem is most commonly used to estimate the

state of a hidden, causal variable Hbased on themeasured state of an observable variable D:

( | ) ( )

( | )( )

p D H p H p H D

p D

Evidence

PriorLikelihood

Posterior


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Bayesian Inference

Whereas the posterior p(H|D) is often difficult to estimatedirectly, reasonable models of the likelihood p(D|H) canoften be formed. This is typically because H is causal onD.

Thus Bayes theorem provides a means for estimating

the posterior probability of the causal variable Hbasedon observations D.


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Marginalizing

To calculate the evidence p(D)in Bayes equation, wetypically have to marginalizeover all possible states ofthe causal variable H.

( | ) ( )( | )( )

p D H p H p H Dp D

1 2

1 1 2 2

( ) ( , ) ( , ) ( , )

( | ) ( ) ( | ) ( ) ( | ) ( )n

n n

p D p D H p D H p D H

p D H p H p D H p H p D H p H


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The Full Monty

Lets get back to The Monty Hall Problem.

Lets assume you initially select Door 1.

Suppose that Monty then opens Door 2 to reveal a goat.

We want to calculate the posteriorprobability that a carlies behind Door 1 after Monty has provided these newdata.


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The Full Monty

Let represent the state that the car lies behind Door , [1,2,3].iC i i

Let represent the event that Monty opens door , [1,2,3],

revealing a goat.

iM i i

2 1 11 2

2

( | ) ( )We seek ( | )( )

p M C p C p C Mp M


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But were not on Lets Make a Deal!

Why is the Monty Hall Problem Interesting?

It reveals limitations in human cognitive processing ofuncertainty

It provides a good illustration of many concepts of probability

It get us to think more carefully about how we deal with andexpress uncertainty as scientists.

What else is Bayes theorem good for?


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Clinical Example

Christiansen et al (2000) studied the mammogram results of 2,227women at health centers of Harvard Pilgrim Health Care, a largeHMO in the Boston metropolitan area.

The women received a total of 9,747 mammograms over 10 years.Their ages ranged from 40 to 80. Ninety-three different radiologists

read the mammograms, and overall they diagnosed 634mammograms as suspicious that turned out to be false positives.

This is a false positive rate of 6.5%.

The false negative rate has been estimated at 10%.


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Clinical Example

There are about 58,500,000 women between the ages of40 and 80 in the US

The incidence of breast cancer in the US is about

184,200 per year, i.e., roughly 1 in 318.


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PSYC 6130 PROF J ELDER 34

Clinical Example

0Let represent the absence of cancer.C

0Let represent a negative mammogram result.M

1Let represent a positive mammogram result.M

1Let represent the presence of cancer.C

1 1 1 1Remember: ( | ) ( | )!p C M p M C

Suppose your friend receives a positive mammogram result.

What quantity do you want to compute?

02 Probability, Bayes Theorem and the Monty Hall Problem

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