Probabilistic reasoning

Quote

”Probability theory is nothing but common sense reduced to calculation.” – Laplace

Probabilistic reasoning uses subjective probabilities, which can be computed by Bayes’ theorem. From a cognitive science perspective, Bayes’ theorem encodes two facts about how our beliefs change in response to new evidence: if we believe an event has a low probability, then the probably is still low in spite of reliable evidence; and if new evidence is unreliable, then our beliefs will change very little.

Related: Bayesian models of cognition, Metacognition, Conditional probability and Bayes’ theorem

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Key terms §

• Subjective probability = probabilities of events that are assigned in reasoning.
• Bayesian optimality = for discrepant cues, the probability of a cue is determined by taking a weighted average, where each weight depends on the reliability of that cue.
• Heuristic model of Bayesian suboptimality = using heuristics to make incorrect probability judgements.
  • Representativeness = when asked the probability that A belongs to category B, people often rely on the degree to which A is a paradigmatic example of B.
  • Availability = when asked the probability of A, people often rely on the ease with which instances of A can be recalled.

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Bayes’ theorem from a cognitive perspective §

Bayes’ theorem

$\text{Pr}(H\mid E)=\frac{P(E\mid H)\times P(H)}{P(E)}$

• Posterior $\text{Pr}(H\mid E)$ = probability assigned to the possibility of an event, which is hypothesis $H$, when new evidence $E$ is gathered.
• Prior = probability based on previous knowledge of likelihood.
  • Likelihood $P(E\mid H)\times P(H)$ = also known as reliability; the possibility of new evidence $E$ if hypothesis $H$ is taken to be true.
  • Normalizing constant $P(E)$ = equal to $P(E\mid H)+P(E\mid \neg H)$.

Bayes’ theorem encodes two facts about how beliefs change:

• If you believe an event is almost impossible (i.e., prior probability of $H$ is low), new evidence will not make much of a difference (i.e., posterior probability will also be low).
• If you believe your evidence is worthless (i.e., $P(E\mid H)$ and $P(E\mid \neg H)$ are similar), you will stick with what you already believed (i.e., the posterior will be close to the prior).

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Bayesian suboptimality §

Subjective probabilities are Bayesian optimal when they come from taking a weighted average of past events. However, empirical evidence shows that people deviate from Bayesian optimality, or have errors in reasoning, in two systematic ways: base rate neglect, or underreacting to the prior, and conservatism, or underreacting to the likelihood.

One theory of Bayesian suboptimality is that people use heuristics to overcome limits in cognitive processing (e.g., representativeness and availability). On the other hand, Rational process models claim that judgement errors come from approximations of rational reasoning (e.g., hypothesis sampling). These theories are overlapping, as heuristics can be considered approximation methods.

In “A Theory of Learning to Infer,” the authors propose that Bayesian optimality is not an appropriate standard at all. Rather, the brain should be evaluated on how well it handles the trade-off between accurate probabilistic reasoning and computational costs.