Dear Paul,
If the Weather Channel is Bayesian, then say they used that empricial
prior that you did (5%), and they observed evidence E to arrive at
their 70% for the snow S given E.
Their Bayes' ratio is 44.3. Yours, effectively, is 10 (assuming that
the event "They say 70%" coincides with "They observe evidence with a
Bayes ratio in the forties" - that is, they agree with you about the
empirical prior and are Bayesian).
So, having effectively disagreed with them about the import of what
they observed, you also disagreed with them about the conclusion.
Hope that helps,
Paul
2009/2/13 Lehner, Paul E. <pleh...@mitre.org>:
> 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
>
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>
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