Consider the following revised version.
The TWC problem
1. Question: What is the chance that it will snow next Monday?
2. My subjective prior: 5%
3. Evidence: The Weather Channel (TWC) says there is a “70%
chance of snow” on Monday.
4. TWC forecasts of snow are calibrated.
Notice that I did not justify by subjective prior with a base rate.
From P(S)=.05 and P(S|”70%”) = .7 I can deduce that P(“70%”|S)/
P(“70%”|~S) = 44.33. So now I can “deduce” from my prior and
evidence odds that P(S|”70%”) = .7. But this seems silly. Suppose
my subjective prior was 20%. Then P(“70%”|S)/P(“70%”|~S) = 9.33333
and again I can “deduce” P(S|”70%”)=.7.
My latest quandary is that it seems odd that my subjective
conditional probability of the evidence should depend on my
subjective prior. This may be coherent, but is too counter
intuitive for me to easily accept. It would also suggest that when
receiving a single evidence item in the form of a judgment from a
calibrated source, my posterior belief does not depend on my prior
belief. In effect, when forecasting snow, one should ignore priors
and listen to The Weather Channel.
Is this correct? If so, does this bother anyone else?
paull
I think the source of your discomfort is the fact that in practice,
TWC likely has a different prior than us. This model does not seem to
allow for that possibility (since TWC's forecasts are calibrated), and
I'm not sure of a way to create a model taking that possibility into
account which is not under-specified (we'd need to include
relationships between TWC's priors and ours, for instance). Consider:
1. Our initial probability of snow: P(S | us) = 0.05 ("us" here
represents whatever information we have)
2. The weather channel's probability of snow: P(S | TWC) = 0.7 ("TWC"
here represents whatever information TWC has).
3. Question: what do we think the probability of snow is given that
P(S | TWC) is 0.7. That is, what is P(S | us and TWC') where P(S |
TWC') = 0.7 (we're not assuming we have knowledge of the information
the weather channel used to make their prediction).
I think the traditional assumption is that whatever information the
weather channel has, it includes our information as well, meaning that
for any information TWC' held by the weather channel, P(S | us and
TWC') = P(S | TWC').
Hope that helps,
Austin Parker
From: uai-boun...@engr.orst.edu [mailto:uai-boun...@engr.orst.edu]
On Behalf Of Lehner, Paul E.
Sent: Friday, February 13, 2009 4:29 PM
To: uai@ENGR.ORST.EDU
Subject: [UAI] A perplexing problem
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
5. Question: What is the chance that it will snow next Monday?
6. My prior: 5% (because it typically snows about 5% of the
days during the winter)
7. Evidence: The Weather Channel (TWC) says there is a “70%
chance of snow” on Monday.
8. TWC forecasts of snow are calibrated.
My initial answer is to claim that this problem is underspecified.
So I add
9. On winter days that it snows, TWC forecasts “70% chance of
snow” about 10% of the time
10. 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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