Koehler: The Bayesian Calculation

From: HARNAD Stevan (harnad@coglit.soton.ac.uk)
Date: Thu May 28 1998 - 20:28:55 BST


Hi All,

Here is the fully worked out version of the cab problem using Bayes'
rule. By Christine McCarthy at:

http://www.ed.uiuc.edu/EPS/PES-Yearbook/93_docs/MCCARTHY.HTM

The same applies to the rare/common disease problem. (She has a
good discussion of some of the issues at the above URL, based on
a much earlier BBS article by Jonathan Cohen.)

THE CAB PROBLEM

A cab was involved in a hit and run accident. Two cab companies,
the Green and the Blue, operate in the city. You know that:

  (a) 85% of the cabs in the city are Green; 15% are Blue.
  (b) a witness says the cab involved was Blue.
  (c) when tested, the witness correctly identified the two colors
      80% of the time.

The question is, How probable is it that the cab involved in the
accident was Blue, as the witness reported, rather than Green?
According to Kahneman and Tversky, this is a problem that "permits the
calculation of the correct posterior probability under some reasonable
assumptions."11

The Bayesian calculation that is taken to be required is:

      P(Blue cab|"blue report") =

      [P(B) x P("b"|B)] /
      {[P(B) x P("b"|B)] + [P(-B) x P("b"|-B)]}
                          
= [(.15 x .80)] /
      {[.15 x .80] + [.85 x .20]}

= .12/(.12 + .17)

= .41

and hence the probability that the errant cab was Green, not Blue, is
.59. So, according to Kahneman and Tversky, "...in spite of the
witness's report...the hit-and-run cab is more likely to be Green than
Blue, because the base-rate is more extreme than the witness is
credible."

Most subjects, however, fail to make use of the "base rate" data,
i.e., the 85% Green and 15% Blue figures, which are taken by Kahneman
and Tversky to represent the prior probability of involvement in the
accident. "The...answer [subjects give] is typically .80, a value
which coincides with the credibility of the witness, and is apparently
unaffected by the relative frequency of Blue and Green cabs."12 This
phenomenon, dubbed "ignoring the base-rate," is often reported.
Bar-Hillel writes, "The genuineness, the robustness, and the
generality of the base-rate fallacy are matters of established
fact."13



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