I was working on a set of instructions to teach simple
two-hypothesis/one-evidence Bayesian updating. I came across a problem that
perplexed me. This can't be a new problem so I'm hoping someone will clear
things up for me.
The problem
1. Question: What is the chance that it will snow next Monday?
2. My prior: 5% (because it typically snows about 5% of the days during
the winter)
3. Evidence: The Weather Channel (TWC) says there is a "70% chance of
snow" on Monday.
4. TWC forecasts of snow are calibrated.
My initial answer is to claim that this problem is underspecified. So I add
5. On winter days that it snows, TWC forecasts "70% chance of snow" about
10% of the time
6. On winter days that it does not snow, TWC forecasts "70% chance of
snow" about 1% of the time.
So now from P(S)=.05; P("70%"|S)=.10; and P("70%"|S)=.01 I apply Bayes rule and
deduce my posterior probability to be P(S|"70%") = .3448.
Now it seems particularly odd that I would conclude there is only a 34% chance
of snow when TWC says there is a 70% chance. TWC knows so much more about
weather forecasting than I do.
What am I doing wrong?
Paul E. Lehner, Ph.D.
Consulting Scientist
The MITRE Corporation
(703) 983-7968
pleh...@mitre.org<mailto:pleh...@mitre.org>
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