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Heuristics & Biases. Bayes Rule. Prior Beliefs. Posterior Probability. Evidence. Medical Test. In the 1980’s in the US, a HIV test was used that had the following properties: There were 4% false positives There were 100% true positives About 0.4% of the male population was HIV positive

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Bayes rule
Bayes Rule

Prior Beliefs

Posterior Probability

Evidence


Medical test
Medical Test

  • In the 1980’s in the US, a HIV test was used that had the following properties:There were 4% false positivesThere were 100% true positives

  • About 0.4% of the male population was HIV positive

  • If a man tested HIV positive, what is the probability he is actually HIV positive?


Representation
Representation

  • P( positive | no HIV ) = .04 (4% false positives)

  • P( positive | HIV ) = 1 (100% true positives)

  • P( HIV ) = .004 (0.4% HIV positive rate)

  • want: P( HIV | positive ) = ?

HIV no HIV

Positive

Negative

P( positive | HIV)P( HIV ) P(positive | noHIV )P( noHIV )

P( negative | HIV)P( HIV ) P(negative | noHIV )P( noHIV)


Representation1
Representation

  • P( positive | no HIV ) = .04 (4% false positives)

  • P( positive | HIV ) = 1 (100% true positives)

  • P( HIV ) = .004 (0.4% HIV positive rate)

  • want: P( HIV | positive ) = ?

HIV no HIV

P( positive | HIV)P( HIV )= P(positive | noHIV )P( noHIV )=

(1)(.004) = .004 (.04)(.996) = .03984

P( negative | HIV)P( HIV )= P(negative | noHIV )P( noHIV)=

(0)(.004) = 0 (.96)(.996) = .95616

Positive

Negative


Solution
Solution

  • P( HIV | positive ) = .004 / ( .004 + .03984 ) = .091

HIV no HIV

P( positive | HIV)P( HIV )= P(positive | noHIV )P( noHIV )=

(1)(.004) = .004 (.04)(.996) = .03984

P( negative | HIV)P( HIV )= P(negative | noHIV )P( noHIV)=

(0)(.004) = 0 (.96)(.996) = .95616

Positive

Negative


The taxi problem version 1
The Taxi Problem: version 1

  • A witness sees a crime involving a taxi in Carborough. The witness says that the taxi is blue. It is known from previous research that witnesses are correct 80% of the time when making such statements.

  • What is the probability that a blue taxi was involved in the crime?


The taxi problem version 2
The Taxi Problem: version 2

  • A witness sees a crime involving a taxi in Carborough. The witness says that the taxi is blue. It is known from previous research that witnesses are correct 80% of the time when making such statements.

  • The police also know that 15% of the taxis in Carborough are blue, the other 85% being green.

  • What is the probability that a blue taxi was involved in the crime?


Normative model
Normative Model

  • Bayes rule tells you how you should reason with probabilities – it is a prescriptive (i.e., normative) model

  • But do people reason like Bayes?

    (Tversky & Kahneman)

    • Bayes rate neglect

    • Conservatism


Base rate neglect 2
Base Rate Neglect (2)

  • Kahneman & Tversky (1973).

    group A: 70 engineers and 30 lawyers

    group B: 30 engineers and 70 lawyers

  • What is probability of picking an engineer in group A and B? Subjects can do this …


Provide some evidence

“Jack is a 45 year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies, which include home carpentry, sailing, and mathematical puzzles”

What now is probability Jack is an engineer?

Estimates for both group A and group B was P = .9

Provide some evidence …


Tversky kahneman
Tversky & Kahneman children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies, which include home carpentry, sailing, and mathematical puzzles”

Much of decision making may be based on:

Biases and heuristics (mental short-cuts)

Lowers cognitive load, but more errors

 Representativeness heuristic

 Availability heuristic


Heuristics biases

All the families having exactly six children in a particular city were surveyed. In 72 of the families, the exact order of the births of boys and girls was:

G B G B B G

What is your estimate of the number of families surveyed in which the exact order of births was:

B G B B B B

Answer: a) < 72 b) 72 c) >72


Representativeness heuristic
Representativeness Heuristic city were surveyed. In 72 of the families, the exact order of the births of boys and girls was:

The sequence “G B G B B G” is seen as

A) more representative of all possible birth sequences.

B) better reflecting the random process of B/G


Heuristics biases

A coin is flipped. What is a more likely sequence? city were surveyed. In 72 of the families, the exact order of the births of boys and girls was:

A) H T H T T H

B) H H H H H H

A) #H = 3 and #T = 3 (in some order)

B) #H = 6

Gambler’s fallacy: wins are perceived to be more likely after a string of losses


Does the hot hand phenomenon exist
Does the “hot hand” phenomenon exist? city were surveyed. In 72 of the families, the exact order of the births of boys and girls was:

Most basketball coaches/players/fans refer to players having a “Hot hand” or being in a “Hot zone” and show “Streaky shooting”

However, there is little statistical evidence that basketball players switch between a state of “hot hand” and “cold hand”

People often see structure in sequences that are statistically purely random (and nonchanging)

(Gilovich, Vallone, & Tversky, 1985)


Availability heuristic
Availability Heuristic city were surveyed. In 72 of the families, the exact order of the births of boys and girls was:

  • Are there more words in the English language that begin with the letter V or that have V as their third letter?

  • What about the letter R, K, L, and N?

(Tversky & Kahneman, 1973)


Heuristics biases

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Rate the likelihood that the following statements about Linda are true:

a) Linda is active in the feminist movement

b) Linda is a bank teller

c) Linda is a bank teller and is active in the feminist movement

CONJUNCTION FALLACY


Are heuristics wrong
Are heuristics wrong? She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

No, we use mental shortcuts because they are often right.

Availability and representativeness are often ecologically valid cues.