Probabilistic reasoning Flashcards
Probability of h
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
Conditional probability of h in light of e
the probability that h will occur given the knowledge/assumption that e has already occurred
p(h | e) = p(ℎ&𝑒)/p(𝑒)
Bayes theorem
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)]
Ellsberg paradox (Ellsberg, 1961)
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)
Monty hall
The contestant would double his/her odds of winning by switching doors. Updating of the odds from .33 to .5 (1/3 –> 1/2)
Conservatism
Edwards (1968)
Most common answer: around 0.7 (= people updated beliefs conservatively, that is more slowly than Bayes’ theorem)
Base rate fallacy
Opposite of conservatism. People update too much without taking into account the base rate probability and think that p (H|E) is > .5
Conjunction fallacy
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)
Conjunction rule
p(h1 ^ h2) <= p(h1) because h1 ^ h2 logically imply h1
–>p(h1^h2)<=p(h1|e)
possible errors that bring to the conjuction fallacy
- 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)
- The term probable could be interpreted in a non mathematical way (solution: use a betting paradigm)
- 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)
Controversy on the nature of CF
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?
The “A–> B paradigm” (Tversky & Kahneman, 1983)
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]
The “M–>A paradigm” (Tversky & Kahneman, 1983)
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)
Why do people commit the conjunction fallacy?
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
BAYESIAN CONFIRMATION (or IMPACT) (Evidential reasoning)
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
Impact and posterior probability can be dissociated
Evidential reasoning
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
How are people’s impact judgements?
Evidential reasoning
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)
consistency (impact and prob.)
Evidential reasoning
both judgments are generally consistent, however confirmation judgments are more consistent than probability ones
accuracy (impact and prob.)
Evidential reasoning
when impact is positive [negative], on average, participants overestimated [underestimate] the corresponding posterior probability of 6%
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)
«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)
Ubiquity of evidential reasoning
- 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)
