Dear Paul,
Your numerical application of Bayes rule is correct. Thus given your
model, your estimate is accurate assuming the numbers you assigned to
your prior and conditional probabilities are accurate for your
location.
However, you model the information provided by TWC as a binary
variable (Either TWC says "70% chance of snow", or they say something
else). This model has two weaknesses: First, it discards very relevant
evidence actually provided to you by TWC --so you should not expect as
accurate predictions as them-- , and second, more subtly, it does so
in a particularly "weak" way, by aggregating together in the
"something else" pieces of evidence that influence the probability of
snow in opposite directions from "70% chance of snow". A model of
identical complexity to yours that would not suffer from this flaw
would still be a binary TWC variable with the values: TWC says "Chance
of snow greater or equal to 70%" vs TWC says "Chance of snow less than
70%".
A solution that maintains the structural simplicity of your model, and
that at the same time captures a much larger fraction of the
information provided by TWC that is relevant to your problem would be
to model the TWC variable as a multinomial variable that can take the
values TWC says: "100% chance of snow", "90% chance of snow", ... ,
"0% chance of snow".
Best regards,
Jorge Moraleda, Ph.D.
Senior Research Scientist
Ricoh Innovations, Inc.
2882 Sand Hill Road, Suite 115
Menlo Park, CA 94025-7054
Tel 650-496-5716
Fax 650-854-8740
>
> 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>
> -------------- next part --------------
> An HTML attachment was scrubbed...
> URL:
> <https://secure.engr.oregonstate.edu/mailman/private/uai/attachments/20090213/f1ccafc0/attachment-0001.html>
>
> ------------------------------
>
> _______________________________________________
> uai mailing list
> uai@ENGR.ORST.EDU
> https://secure.engr.oregonstate.edu/mailman/listinfo/uai
>
>
> End of uai Digest, Vol 50, Issue 12
> ***********************************
>
>
>
_______________________________________________
uai mailing list
uai@ENGR.ORST.EDU
https://secure.engr.oregonstate.edu/mailman/listinfo/uai
