Reasoning about probabilities Flashcards
What is debated amongst mathematicians regarding the nature of probability?
Whether it is Bayesian or Frequentist.
How do Bayesians define probability?
Probability refers to a subjective degree of confidence, and as one can express confidence that a single event will occur, one can express the probability of a single event.
How do Frequentists define probability?
Probability is always defined over a reference class such as infinite number of coin tosses. Single events don’t belong to a reference class so they cannot have a relative frequency or probability.
What do psychologists mean when they refer to normative probabilities?
If the probability is normative, the output of the system is the same that would be returned by a Bayesian machine. This is not related to whether the internal process is Bayesian.
What has much psychological research been conducted with reference to?
Single event probabilities (posterior probability).
What is posterior probability?
The conditional probability that is assigned after the relevant evidence is taken into account; in psychology the probability of a hypothesis (H) given data (D) = p(H|D).
What did Kahneman and Tversky (1972) write about humans and Bayesian evaluation of evidence (probability calculating)?
Humans are “not Bayesian at all”.
What did Gould (1992) state about the human mind and probability?
“…our minds are not built…to work by the rules of probability”
What can the heuristics and biases literature be used to demonstrate about probabilistic reasoning?
Essentially, humans are irrational when it comes to probabilistic reasoning.
Can we reason according to Bayes Theorem according to evolutionary psychologists?
Yes, as long as the information is presented in a format which we have evolved to process (i.e. not probabilities, a modern form of mathematical notation).
What is Kahneman and Tversky’s view on why humans are irrational when it comes to probabilistic reasoning?
Bayesian reasoning is too complex so we use heuristics, which are necessary because of the poverty of input. The mechanism and/or task are too complex and we’re limited by our inability to reason.
What is Cosmides and Tooby’s view on why humans are irrational when it comes to probabilistic reasoning?
We cannot do probabilistic reasoning because the information is in the wrong format (not frequency). The mechanism needed is simple (a few lines of computer code), and the task is therefore no more complex.
What is the problem with Cosmides and Tooby’s argument regarding probabilistic reasoning?
It is a tautology and gives no evolutionary reason why Bayesian reasoning shouldn’t have developed - even simple sea slugs exhibit habituation and all vertebrates can be classically conditioned. These processes can be described as Bayesian inferences - animal learning approximates Bayes Theorem.
What processes can be described as Bayesian inferences?
Habituation and classical conditioning - learning approximates Bayes Theorem.
What are the advantages of the frequentist format?
- The number of events (sample size), which indicates the reliability of the decision, is retained
- Permits easy updating as new information is collected
- Reference classes can be constructed post-hoc according to new information as the reference class changes
What did Tversky and Kahneman (1982) study?
Base rate neglect in the city cabs experiment.
Describe Tversky and Kahneman (1982)’s experiment.
Participants are told that a cab was involved in a hit and run accident in a place where 85% of the cabs are green and 15% are blue. A witness identified the cab involved as blue, and the court found their reliability to be 80% correct. Participants were asked: “what is the probability that the cab involved was blue not green?”
What did Tversky and Kahneman (1982) find?
The actual probability that the taxi was blue = 0.12/(0.12+0.17) = 0.41, however most participants think the taxi was more likely to be blue e.g. >0.50 and most stated p=0.80.
What did Tversky and Kahneman (1982) attribute base rate neglect to?
The representativeness heuristic - participants focus on the witness’ accuracy and neglect the base rate of the city’s cabs.
What did Casscells et al. (1978) study?
Base rate neglect using the medical diagnosis problem.
What did Casscells et al. (1978) do?
Asked medical students: If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease? Assuming that you know nothing about the person’s symptoms or signs.
What did Casscells et al. (1978) find?
18% responded 2% (correct Bayesian inference)
45% responded 95% (neglected base rate)
What did Casscells et al. (1978) conclude?
That even medical students ignore base rates for diagnostic problems.
What did Cosmides & Tooby (1996) do?
Presented a similar medical diagnosis problem to Casscells et al. (1978) in both frequency and probability formats. Participants had to estimate how many of those out of 1000 randomly selected who tested positive actually have the disease.
