Bayes Lecture

Bayes’ Theorem

Psyc 339 02/07/2006

Outline „ „ „ „ „

Sample problems Bayes’ Theorem History of Bayes’ Theorem Applications Practice questions

One Example „ „ „

This is a hypothetical problem Suppose 1% of the population has HIV A test for HIV is said to be 90% accurate …

If a person has HIV, the test can identify 90% of the times correctly … If a person does not have HIV, the test can reject 90% of the times correctly. „ „

If someone gets back a positive test, what’s the probability that he/she has HIV? Choices: A. Below 10%; B. 10%~30%; C. 30%~50%; D.50%~70%; E. 70%~90%; F. 90%; G.90%~100%

One approach to calculate… „

Consider 1000 people

0.01

0.1

Negative: 1

0.9

Positive: 9

HIV: 10 108

0.99

No 0.1 HIV: 990 0.9

P(H|+)=9/(99+9)=0.083

Positive:99

Negative: 891

Or, look at the problem this way… Test +

People with HIV

Hit 90%

People without HIV

False Alarm 10%

Test -

Miss 10% Correct Rejection 90%

Bayes' Theorem Hit Rate

Base Rate

p(+ | H) p(H) p(H | +) = p(+ | H) p(H) + p(+ | H) p(H) False Alarm Rate

So, another approach to calculate…

(0.9)(0.1) p(H | +) = = 0.083 (0.9)(0.1) + (0.1)(0.99) Bayesian Calculators: • http://psych.rice.edu/online_stat/ • http://psych.fullerton.edu/mbirnbaum/calculators/Bayes Calc.htm

Base Rate Neglect „

In the HIV example, if your guesstimate is way over 8.3%, then you probably neglect the base rate.

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Base rate neglect is a persistent phenomenon in which people do not place sufficient weight on the probabilities of occurrence of relevant events.

Bayes’ Theorem: A more general form

p(H & D) p(D | H) p(H) p(H | D) = = p(D) p(D) p(H|D): the probability of H given D p(H&D): the probability of H and D together p(D): the probability of D (including all the possibilities that D can happen)

Monty Hall Problem „

Suppose you are on a game show, and you’re given a choice of three doors. …

Behind one door is a car; behind the others, goats. … You pick a door, and the host, who knows what’s behind the doors, offers to open a second door, which has a goat. … After that, you can switch to the third door, or you can stay with your original choice. … Do you have a better chance to win the car if you switch?

Imagine the situation… „

Let us suppose that A is the door you pick first, and B and C are the two other doors.

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If you don’t ask the host to open the door, your chance of winning is 1/3.

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Suppose you ask the host to open a door, and he opens B, revealing a goat. Then, what is the probability of this datum Db given the three hypotheses, Ha, Hb, and Hc?

Finding P(Db) „

If the car were in A, he would pick one of the other two doors at random, so p(Db|Ha)=1/2

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If the car were in B, he would not pick B, so p(Db|Hb)=0

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If the car were in C, he would only pick B, so p(Db|Hc)=1

Apply Bayes’ Theorem „

Now we can apply the Bayes’ Theorem to calculate p(Hc|Db). p(Db | Hc )p(Hc ) p(Hc | Db ) = p(Db ) p(Db | Hc )p(Hc ) = p(Db | Ha )p(Ha ) + p(Db | Hb )p(Hb ) + p(Db | Hc )p(Hc ) (1)(1/3) = = 2/3 (1/2)(1/3) + (0)(1/3) + (1)(1/3)

For those who don’t believe… „

Here is the simulation for Monty Hall problem.

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http://psych.rice.edu/online_stat/

Bayes' Theorem History The theorem was named after Thomas Bayes' (1702-1761), who first recognized the importance of personal probability.

Application: Spam filter „

Bayesian spam filters calculate the probability of a message being spam based on its contents.

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Unlike simple content-based filters, Bayesian spam filter does not classify an email as spam rigidly.

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Bayesian spam filters can also learn from spam and from good mails and returns hardly any false positives.

Other Applications: „ „ „ „ „

Microsoft Office Assistant Google Search Engine Autonomy Systems Modeling how neurons behave in very complicated systems Any other applications?

Practice Question 1 „

A manufacturer claims that its drug test will detect steroid use (that is, show positive for an athlete who uses steroids) 95% of the time. Your friend on the basketball team has just tested positive. The probability that he uses steroids is: … A. 0.95 … B. At most 0.95 … C. At least 0.95 … D. Not possible to say, based on the information

Practice Question 2 „

In a population, 70% of the people have a certain condition. A test is developed that has a 40% chance of detecting the condition in a person who has it and a 10% chance of falsely indicating it in a person who does not have it. If a person gets a positive test result, roughly what is the probability they have the condition?

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Choices: 0.10; 0.25; 0.33; 0.50; 0.67; 0.75; 0.90

Practice Question 3 Toss a fair coin. If it lands head up, draw a ball from box 1; otherwise, draw a ball from box 2. If the ball is blue, what is the probability that it is drawn from box 2? Box1 p(box1) = .5 p(red ball | box1) = .4 p(blue ball | box1) = .6

Box2 p(box2) = .5 p(red ball | box2) = .5 p(blue ball | box2) = .5

Key to Q3 p(box1) = .5 P(red ball | box1) = .4 P(blue ball | box1) = .6

p(box2) = .5 P(red ball | box2) = .5 P(blue ball | box2) = .5

p(box2)p(blue ball | box2) p(box2 | blue ball) = p(blue ball) p(box2)p(blue ball | box2) = p(box1)p(blue ball | box1) + p(box2)p(blue ball | box2) .5 * .5 = = .25 = 0.4545454545... .5 * .6 + .5 * .5 .55

Any Questions?