Hi Paul,
Your calculation is correct, but the numbers in the example are odd. If
TWC really only manage to predict snow 10% of the time (90% false negative
rate), you would be right not to assign much value to their predictions
(you do assign _some_, hence the seven-fold increase from your prior to
your posterior, but with prediction performance like that TWC cannot
possibly think there is really a 70% chance of snow).
Change the 10% true positives to 90%, and your posterior goes up to 82.6%
- much more believable.
Also, it's important not to think the figure of 70% has any bearing on the
problem. I appreciate that you put it in as a red herring to challenge the
students, but be aware that it may also lead to confusion.
Konrad
On Fri, 13 Feb 2009, Lehner, Paul E. wrote:
> 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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