Probabilistic reasoning

Question 1

Probability of h

Answer 1

pr(h),is a value between “0” (=certainly false) and “1” (=certainly true), .5 indicates that h has the same probability of being true or false

Question 2

Conditional probability of h in light of e

Answer 2

the probability that h will occur given the knowledge/assumption that e has already occurred
p(h | e) = p(ℎ&𝑒)/p(𝑒)

Question 3

Bayes theorem

Answer 3

Normative model. It expresses how a subjective degree of belief in h should rationally change to account for evidence e
𝑝(ℎ|𝑒)=𝑝(ℎ)x[p(𝑒|ℎ)/𝑝(𝑒)]

Question 4

Ellsberg paradox (Ellsberg, 1961)

Answer 4

Ellsberg paradox is generally taken to be evidence for ambiguity aversion (i.e, preference for known risks over unknown risks) (usually people change preference in the second choice to make even if the EV is the same)

Question 5

Monty hall

Answer 5

The contestant would double his/her odds of winning by switching doors. Updating of the odds from .33 to .5 (1/3 –> 1/2)

Question 6

Conservatism

Edwards (1968)

Answer 6

Most common answer: around 0.7 (= people updated beliefs conservatively, that is more slowly than Bayes’ theorem)

Question 7

Base rate fallacy

Answer 7

Opposite of conservatism. People update too much without taking into account the base rate probability and think that p (H|E) is > .5

Question 8

Conjunction fallacy

Answer 8

More primitive kind of fallacy. It  is a violation of the conjunction rule. 
Linda scenario (descrizione situa linda e varie frasi che descrivono possibilmente linda) -->  Artifact : participants are not irrational but people misunderstand something (need to rule it out for it to be a fallacy)

Question 9

Conjunction rule

Answer 9

p(h1 ^ h2) <= p(h1) because h1 ^ h2 logically imply h1

–>p(h1^h2)<=p(h1|e)

Question 10

possible errors that bring to the conjuction fallacy

Answer 10

1. Interpretation of the single conjunct as “h1 and not h2” (solution is to add the option of h1 and not h2 and to clarifly in the description)
2. The term probable could be interpreted in a non mathematical way (solution: use a betting paradigm)
3. the connective and could be misinterpreted as “or” (solution: Avoiding the word and i.e., using implicit conjunctions- Explicitly pointing out the conjunctive meaning of and- Controlling for the interpretation of andafter the CF task)

Question 11

Controversy on the nature of CF

Answer 11

From the very beginning the CF phenomenon has been described as a violationof ‘‘the simplest and the most basic qualitative law of probability’’(Tversky & Kahneman, 1983)
Gigerenzer (1994; et al. 1988; 1999):Lack of ecological validity in the CF paradigm (i.e. the task andresponse format used to explore the CF are not representative of those typically encountered in daily life) Is the CF a genuine reasoning error or is it an artifact?

Question 12

The “A–> B paradigm” (Tversky & Kahneman, 1983)

Answer 12

An implicit “causal model” M(background knowledge)
–> An added event A (hypothesis h2)(i.e., over 55)
–>A basic event B(hypothesis h1)which is highly “representative” of event A (i.e., one or more heart attacks) (A+B)
[Mr. F]

Question 13

The “M–>A paradigm” (Tversky & Kahneman, 1983)

Answer 13

A “causal model” M(evidence e)(i.e., Linda’s personality)

• ->An added event A (hypothesis h2)which is highly “representative” of model M (i.e., feminist) (M+A)
• ->A basic event B(hypothesis h1)which is “unrepresentative” of model M (i.e., bank teller)

Question 14

Why do people commit the conjunction fallacy?

Answer 14

Theories are needed to account for people’s actual reasoning behavior and its departures from rationality
Various explanations of CF, including:
•Costello (2008): systematic effect of random error
•Nilsson (2008): non-normative averaging rule as applied to the probability of conjuncts
•…all share the assumption that CF rates should rise as the probability of the added conjunct h2 rises

Question 15

BAYESIAN CONFIRMATION (or IMPACT)
(Evidential reasoning)

Answer 15

The support of evidence e on hypothesis h
•the technical meaning of confirmation departs from that of natural language, in which it usually implies to validate or ascertain
•confirmation only conveys the idea of positive support while,of course, impact can be negative as well (disconfirmation)
•in the psychological literature, the term confirmation has gained a negative connotation because of so-called confirmation bias(Nickerson,1998), a tendency to suboptimal reasoning depending on one’s target hypothesis

Question 16

Impact and posterior probability can be dissociated

Evidential reasoning

Answer 16

X is a male—————-X likes cigars
X is a male—————-X likes going to the cinema
Impact and posterior probability are both necessary to account for how inductive reasoning operates

Question 17

How are people’s impact judgements?

Evidential reasoning

Answer 17

Participants properly estimate the impact of certain evidence on a given hypothesis (even when their probability judgments are far from correct)
Participants properly estimate the impact of uncertain evidence
Moreover, classic fallacies in probabilistic reasoning can be explained in terms of confirmation.

In dealing with everyday uncertainty, people may rely more on detecting relations of inductive confirmation (i.e., the net impact of new evidence on the credibility of the hypotheses concerned) than on values of posterior probability(i.e., the overall credibility of hypotheses as updated on all given evidence)

Question 18

consistency (impact and prob.)

Evidential reasoning

Answer 18

both judgments are generally consistent, however confirmation judgments are more consistent than probability ones

Question 19

accuracy (impact and prob.)

Evidential reasoning

Answer 19

when impact is positive [negative], on average, participants overestimated [underestimate] the corresponding posterior probability of 6%

Question 20

What is the critical variable for classical conditioning to occur?The CS had to be a useful predictor of the US… but what makes the CS a useful predictor?
Rescorla (1968)

Answer 20

«Contiguity»
The number of times the CS is paired with the UCS
«Contingency» (information provided by the CS about the US)
-Positive contingency: the CS signals an increase in the probability that the US will occur (compared to before the CS)
–>excitatory conditioning: the subject learns to perform a certain response, like salivating when the bell is rung
-Zero contingency: CS predicts neither an increase nor a decrease in the probability of the US
-Negative contingency: the CS signals a decrease in the probability that the US will occur (compared to before the CS) inhibitory conditioning: the subject learns to with hold or suppress a certain response, like stop salivating when the bell rings(he salivates when the bell is not ringing)
According to Rescorla, learning only takes place with the positive and negative contingencies
Nozick’s confirmation measure:
n(ℎ,𝑒) =p(𝑒|ℎ)−p(𝑒|¬ℎ)

Question 21

Ubiquity of evidential reasoning

Answer 21

• Qualitative Adams’thesis: A simple indicative conditional“ifA,B” is acceptable to a person if her degree of belief in B given A, p(B|A), is high
• Douven&Verbrugge’s(2012) evidential support thesis: A simple indicative conditional “ifA,B” is acceptable to a person if her degree of belief in B given A, p(B|A), is not only high but also higher than p(B).
  –>
  -high value of p(B|A) not sufficient for the acceptability of “ifA,B”
• strong association between qualitative and quantitative confirmation (as measured with d(B,A)), and the acceptability of the conditional“ifA,B
  Krzyżanowska, Collins&Hahn(2017) clarified that the oddity of so-called missing-link conditionals does not depend on natural language pragmatics as much as on probabilistic relevance (i.e., a conditional is assertable only if its antecedent is relevant for the consequent)