Paul,
I'm not aware of this being discussed anywhere but my observation is
that the information given makes TWC quite lousy -- the probability of
the forecast "70% chance of snow" is much too high when there is no
snow. It is a very specific piece of forecast and I would expect this
probability to be very small given that there is actually going to be no
snow. When you reduce this conditional probability, the forecast is
going to be more along the lines that you would expect.
I'm attaching a GeNIe model capturing your problem. To open it,
download GeNIe from http://genie.sis.pitt.edu/.
Cheers,
Marek
----------------------------------------------------------------------
Marek J. Druzdzel http://www.pitt.edu/~druzdzel
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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<?xml version="1.0" encoding="ISO-8859-1"?>
<smile version="1.0" id="SnowForecast" numsamples="1000" discsamples="10000">
<nodes>
<cpt id="It_Snows_on_Monday">
<state id="Snows" />
<state id="DoesNotSnow" />
<probabilities>0.05 0.95</probabilities>
</cpt>
<cpt id="Forecast70">
<state id="Snow70" />
<state id="Other" />
<parents>It_Snows_on_Monday</parents>
<probabilities>0.1 0.9 0.01 0.99</probabilities>
</cpt>
</nodes>
<extensions>
<genie version="1.0" app="GeNIe 2.0.3306.0" name="Paul Lehner's problem" faultnameformat="nodestate">
<node id="It_Snows_on_Monday">
<name>It Snows on Monday</name>
<interior color="e5f6f7" />
<outline color="000080" />
<font color="000000" name="Arial" size="10" bold="true" />
<position>142 14 250 81</position>
<barchart active="true" width="368" height="64" />
</node>
<node id="Forecast70">
<name>The Weather Channel Forecasts 70% Chance of Snow</name>
<interior color="e5f6f7" />
<outline color="000080" />
<font color="000000" name="Arial" size="10" bold="true" />
<position>147 213 249 276</position>
<barchart active="true" width="368" height="64" />
</node>
<textbox>
<caption>Paul Lehner's problem <pleh...@mitre.org>\n\nThe problem:\n\n1. Question: What is the chance that it will snow next Monday?\n2. My prior: 5% (because it typically snows about 5% of the days during the winter)\n3. Evidence: The Weather Channel (TWC) says there is a 70% chance of snow on Monday.\n4. TWC forecasts of snow are calibrated.\n\nMy initial answer is to claim that this problem is underspecified. So I add\n\n5. On winter days that it snows, TWC forecasts 70% chance of snow about 10% of the time\n6. On winter days that it does not snow, TWC forecasts 70% chance of snow about 1% of the time.\n\nSo 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.\n\nNow 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.</caption>
<font color="000000" name="Arial" size="10" bold="true" />
<position>406 16 1033 320</position>
</textbox>
</genie>
</extensions>
</smile>
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