Descriptive Theory of Probability
Subjective beliefs aren’t wrong, but they have to hang together coherently
– Still have normative models
For example, Bayes rule tells us how to update beliefs in light of new information:
p(D|H) p(H) p(H|D) = p(D|H) p(H) + p(D|H) p(H)
• I don’t know why anyone teaches it that way, because it looks like gobbledygook and most people won’t remember it. It’s easier to think off odds ratios and to remember that final beliefs (posterior odds) are just a product of all the ratios. Helpful to see an example:
It’s hard to know how to integrate a prior and a new piece of likelihood information. Birnbaum and Mellers (1983): How likely will a used car last for 3 more years? 90% of cars with the same year and model last for 3 years. A mechanic (independently!) says the car is in good shape. 60% of cars receiving this judgment last for 3 years.
Many people just average 90 and 60 because they’re not sure what to do with the numbers. But of course, each new piece of likelihood that’s better than 50% should increase your belief, not decrease it. Prior belief: 9:1 is the ratio of yes to no for lasting 3 years. New info: 6:4. 9:1*6:4 = 54:4. So, for every 54 times it lasts 3 years, 4 times it doesn’t. Can convert to a probability by computing 54/58 = .93
Of course, most of the time when we are judging a likelihood (perhaps implicitly), we aren’t consciously calculating. We may be relying on heuristics/intuitions that, while often useful, can sometimes take us far from the Bayesian standard.
Kahneman & Tversky: People use mental heuristics to judge probability. While these heuristics may often work well, they fail predictably in some circumstances. They identify three main forms of heuristic: representativeness, availability, and anchoring
“A person who follows this heuristic evaluates the probability of an uncertain event by the degree to which it is: (i) similar in essential properties to its parent population; and (ii) reflects the salient feature of the process by which it was generated” (p. 431).
Wait, what the…? Sounds like just a likelihood ratio to me. What happened to prior probabilities?
H = Hypothesis, "Teri has cancer" D = Datum, e.g., "Mammogram result is positive“ When Dr. Ayes examines Teri, she assigns a prior probability of 10% to the hypothesis H: p(H) = 0.1. The prior probability is the probability that the hypothesis is true, based on what we know prior to doing a test such as a mammogram. In thinking about whether to do the mammogram, Dr. Ayes knows: p(H) = 0.1 The prior probability is 10% p(D | H) = 0.9 Test has a "hit rate" of 90% p(D | not H) = 0.2 Test has a "false alarm rate" of 20% Dr. Ayes wants to know what the probability of Teri having cancer will be if the test is positive. Well?
Tom W. is of high intelligence, although lacking in true creativity. He has a need for order and clarity, and for neat and tidy systems in which every detail finds its appropriate place. His writing is rather dull and mechanical, occasionally enlivened by somewhat corny puns and by flashes of imagination of the sci-fi type. He has a strong drive for competence. He seems to have little feel and little sympathy for other people and does not enjoy interacting with others. Self-centered, he nonetheless has a deep moral sense.
Subjects’ judgments of Tom’s probability of having various majors almost exactly matched other subjects’ judgments of Tom’s representativeness of various majors. That could only be true if majors were equally likely, which a third group of subjects judged they weren’t.
• Base Rate Neglect
Subtle attempts to impugn the reliability of the description did nothing!
Cab problem (Tversky and Kahneman) A cab was involved in a hit and run accident at night. Two cab 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 the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time. What is the probability that the cab involved in the accident was Blue rather than Green?
Bayes' Theorem calculation
p(D|H) p(H) p(H|D) = p(D|H) p(H) + p(D|H) p(H)
.80 .15 .12 p(H|D) = .41 .80 .15 + .2 .85 .12 .17
Representativeness can lead to base rate neglect
Base rate neglect In probability judgments, relying on likelihood information (the diagnostic ratio) while largely ignoring prior probabilities
p(H|D) p(H) p(D|H) = p(D|H) p(H) p(D|H)
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. Judged probability of Jack being an engineer is insensitive to being told he was randomly drawn from a population that was 7:3 lawyers vs. 7:3 engineers
The law of small numbers
Which sequence of coin flips is more random? HHHTTT HTHTHT
Sequences should “look” random in that they conform to binomial probabilities even in small slices.
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.
Linda is a teacher in an elementary school. Linda works in a bookstore and takes Yoga classes. Linda is active in the feminist movement. [F] Linda is a psychiatric social worker. Linda is a member of the League of Women Voters. Linda is a bank teller. [B] Linda is an insurance salesperson. Linda is a bank teller and is active in the feminist movement. [B and F].
Dilution effects? A coin is biased such that when spun, it has a 60% chance of landing with a particular side up. Which of these makes you more confident that it is heads? 4 spins: 4 heads
18 spins: 11 heads, 7 tails
Ratio form of Bayes’ Rule
p(H|D) p(H) p(D|H) = p(D|H) p(H) p(D|H)
Both yield the same likelihood of coin being biased heads, but the larger sample is more trust worthy
4 – 0 “looks” more like biased heads
Judge the likelihood (or frequency) of an event by the ease with which it is imagined or remembered
Not a bad shortcut in principle, but there are many situations in which we remember more easily (recency, primacy, salience) even if the event is not more frequent
Judge these frequencies:
English words that end _ n _ English words that end i n g Words ending ing are easier to think of, though obviously a subset of _n_
Judged frequency of causes of death
The judged likelihood of an event is enhanced when a good scenario is provided.
This heuristic is closely linked with the conjunction fallacy, as adding details makes the conjunction of events less likely, but makes people consider feel it is more likely.
Tversky to a group of forecasters: which is more likely? In the next decade, a deadly flood will occur killing 1,000 people in the U.S.
In the next decade, an earthquake in California will cause a deadly flood to occur, killing 1,000 people in the U.S.
Scenario availability is purported to be a heuristic, but are there probability judgments for which it is the best/only approach? How do we judge the probabilities of various terror attacks?
Anchoring and (insufficient) adjustment Judgment is “anchored” on a particular numerical value which is known to be wrong, yet still exerts influence on the judgment, which isn’t adjusted far enough away from the anchor. This phenomenon works with even arbitrary anchors
Anchoring examples K&T spun a wheel of fortune and then asked people: is the % of African nations in the U.N. higher or lower than this number? What is it? wheel lands on 10: median 25
wheel lands on 65: median 45
Estimate these quantities in less than 5 seconds:
1 2 3 4 5 6 7 8
m = 512
8 7 6 5 4 3 2 1
m = 2,250
Is using these heuristics rational? Von Winterfeldt and Edwards: Real problems, but not irrational anymore than not knowing how to do a physics problem
Brunswickian view: adaptive
Cohen: Rules of probability have no normative status, so violating them is not irrational per se
-On Sunday night, you are given a drug that makes you sleep for 3 days. A coin is then flipped. -On Monday, you are awoken briefly but you don’t know what day it is. You are put back to sleep and will not remember waking up. -If the flip was heads, you will be woken up on Tuesday in the same manner. -On Wednesday, you wake up knowing it is Wednesday and the experiment is over. -Given that you wake up and don’t know what day it is, what should you believe is the probability the coin flip was tails?
More updating troubles
You remember that a couple has 2 children and that at least one is a girl. Given that, what is the probability that they have 2 girls?
You remember that a couple has 2 children and that at least one is a girl with an unusual name (“Florida”). Given that, what is the probability that they have 2 girls?