Hi Paul,
Your calculations are correct (although I note you really mean
P("70%"|not S) = 0.01 in the calc below).
^^^
Sometimes it helps to think about what the numbers actually
mean. First 0.05 prob of snow is quite a low prior.
You need to have quite "certain" evidence to move that up higher.
A posterior of 0.35 means that snow is now *7 times* more likely
given the evidence than it was before you knew anything, which
is still quite a large shift up.
It *sounds* like you have strong evidence with TWC 70% chance of snow.
However, you also have a conditional probability that even when there
is snow, TWC only says 70% chance of snow one in 10 times. That means that
9 in ten times it doesn't say that. So when you entered such evidence
it gets discounted (because it is so often wrong!).
Another side point about the way you have modelled this problem is
your second variable is TWC70%ChanceOfSnow, a true/false variable.
So TWC's confidence isn't really being modelled in the Bayesian
updating, only in the way you've structured your variables.
It might be better instead to have the second variable be
TWCPredictsSnow (True/False) and then incorporate their 70% confidence
as virtual (uncertain) evidence on that variable. But then you'd
need to know P(TWCPredictsSnow|Snow) and P(TWCPredictsSnow|notSnow)...
Hope this helps.
regards,
Ann
On Fri, Feb 13, 2009 at 04:28:41PM -0500, 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>
> _______________________________________________
> uai mailing list
> uai@ENGR.ORST.EDU
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--
A/Prof. Ann Nicholson
Clayton School _--_|\ www.csse.monash.edu.au/~annn/
of Information Technology, / \ phone: +61 3 9905 5211
Monash University, VIC 3800 \_.--.*/ fax: +61 3 9905 5146
Australia v ann.nichol...@infotech.monash.edu.au
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