Lecture 20 Bayesian Updating and Confirmation Bias

  • Bayesian Updating
    • Update beliefs in light of new evidence
    • In science, Update assessment about plausibility of a hypothesis or theory in light of evidence (from experiments, field studies, other sources)
    • \(H:\) Hypothesis. \(E:\) Evidence.
    • \(P(H)\) is prior. \(P(H|E)\) is posterior probability. (Given Evidence \(E\) is TRUE)
    • Narrowing down
    • Use Bayes' Rule to determine posterior.
    • If changing belief according to Bayes' rule, we say we engaged in Bayesian updating
    • "Two heads" or Fair coin Example
      • \(H:\) The coin has two heads.
        • Prior: \(P(H) = 0.01\) (say)
      • \(E:\) A head comes up.
        • \(P(E|H) = 1\) and \(P(E|\neg H) = 0.5\)
      • \(P(H|E) = 0.02\) (first Update)
      • \(P((H|E)|E) =0.04\) (second Update)1
      • This is called Bayesian updating
  • Washing out of Priors
    • After 15 flips: \(P(H) = 100\%\) (refers to "Washing out of Priors")
    • John and Wes roughly assign the same probability, independently of what each of their priors used to be.
    • Rational people exposed to the same evidence: come to agree regardless of their initial stance
    • IRL doesn't happen. Reasons:
      • Very different evidence (e.g. conservative newspapers/blogs, selected information)
      • Confirmation Bias
  • Confirmation Bias
    • Tendency to interpret evidence as supporting prior to a greater extent than warranted. \(P(H|E)\) will grow slower than rational if I see \(E\), and my prior belief was that the coin was unbiased.
    • Death Penalty
      • Two groups (for or against)
      • Same information provided (neutral)
      • Instead of coming to an agreement or the same conclusion... they strongly supported their prior beliefs
      • Individuals \(A\), \(B\) and \(C\)
        • A and B are rational, with different priors and agreed to the same point at the end
        • C doesn't budge, confirmation biased. He doesn't update his beliefs.
    • Explains why racist and sexist stereotypes exist
      • Sexist: Downplay evidence of girls being good at math and men being able to take care of children, but pick on quickly where they are not.
      • Racist: notices all people of other races going good, but focus on only those who do not.
    • Explains Gambling:
      • Belief: "I can predict"
      • Evidence: "You cannot"
      • Result: "I can still predict" (irrational)... He will notice all the cases where he did predict correctly.
    • Explains why people (think they) can beat the stock market
    • Explains How conspiracy theories survive in spite of overwhelming contradictory evidence (credit: conspiracy theorist who puts weight on morsels of evidence)
    • PhD Karl Popper: Scientists find evidence that support their theory everywhere.
    • It is easy to "confirm" just any theory.
    • Factors that contribute to confirmation bias:
      1. People sometimes fail to notice evidence that goes against their beliefs.
      2. Evidence vague \(\implies\) interpretation required. So, people tend to interpret in a way that supports their beliefs
      3. Tend to apply a much higher standard of proof to evidence contradicting their beliefs than otherwise.

  1. Note that after two flips and getting a head, the probability increases faster than the previous update ↩