Phil 148

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Phil 148
Bayes’s Theorem/Choice Theory
You may have noticed:
• The previously discussed rules of probability
involved each of the logical operators:
negation, disjunction, and conjunction, except
for conditional.
• Bayes’s Theorem is a theorem of conditional
probability.
• You’ll notice that we are now progressing
beyond a priori probability, and into statistical
probability.
An Example:
• Wendy has tested positive for colon cancer.
• Colon cancer occurs in .3% of the population
(.003 probability)
• If a person has colon cancer, there is a 90%
chance that they will test positive (.9 probability
of a true positive)
• If a person does not have colon cancer, then
there is a 3% chance that they will test positive
(3% chance of a false positive)
• Given that Wendy has tested positive, what are
her chances of having colon cancer?
Answer:
• The correct answer is 8.3%
• Most people assume that the chances are
much better than they really are that Wendy
has colon cancer. The reason for this is that
they forget that a test must be absurdly
specific to give a high probability of having a
rare condition.
Formal statement of Bayes’s Theorem:
BT: Pr(h | e) =
___Pr(h) * Pr(e|h)___
[Pr(h) * Pr(e|h)] + [Pr(~h) * Pr(e|~h)]
h = the hypothesis
e = the evidence for h
Pr(h) = the statistical probability of h
Pr(e|h) = the true positive rate of e as evidence for
h
Pr(e|~h) = the false positive rate of e as evidence
for h
The Table Method:
e
~e
Total
h
~h
Total
True
Positives
False
Negatives
Pr(h)*Pop.
False
Positives
True
Negatives
Pr(~h)*Pop.
Pr(e)*Pop.
Pr(~e)*Pop.
Pop. = 10^n
n = sum of decimal places in two most
specific probabilities.
The Table Method:
h
e
~e
Total
~h
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
Pr(h)*Pop.
Pr(~h)*Pop.
Total
Total of this
row
Total of this
row
Pop.
The Table Method for Wendy:
h
e
~e
Total
~h
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
Pr(h)*Pop.
Pr(~h)*Pop.
Total
Total of this
row
Total of this
row
Pop.
The Table Method for Wendy:
has CC
e
~e
Total
~ have CC
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
Pr(h)*Pop.
Pr(~h)*Pop.
Total
Total of this
row
Total of this
row
Pop.
The Table Method for Wendy:
has CC
tests
positive
~ test
positive
Total
~ have CC
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
Pr(h)*Pop.
Pr(~h)*Pop.
Total
Total of this
row
Total of this
row
Pop.
The Table Method for Wendy:
has CC
tests
positive
~ test
positive
Total
~ have CC
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
.003*Pop.
.997*Pop.
Total
Total of this
row
Total of this
row
100,000
The Table Method for Wendy:
has CC
tests
positive
~ test
positive
Total
~ have CC
= Pr(e|h) * = Pr(e|~h) *
[Pr(h)*Pop.] [Pr(~h)*Pop.
]
= below = below above
above
300
99,700
Total
Total of this
row
Total of this
row
100,000
The Table Method for Wendy:
tests
positive
~ test
positive
Total
has CC
~ have CC
Total
= True
Positive Rate
(.9) * 300
= below above
300
= Pr(e|~h) *
[Pr(~h)*Pop.
]
= below above
99,700
Total of this
row
Total of this
row
100,000
The Table Method for Wendy:
has CC
~ have CC
Total
tests
positive
270
Total of this
row
~ test
positive
Total
= below above
300
= Pr(e|~h) *
[Pr(~h)*Pop.
]
= below above
99,700
Total of this
row
100,000
The Table Method for Wendy:
has CC
~ have CC
Total
tests
positive
270
Total of this
row
~ test
positive
Total
30
= Pr(e|~h) *
[Pr(~h)*Pop.
]
= below above
99,700
300
Total of this
row
100,000
The Table Method for Wendy:
has CC
~ have CC
Total
tests
positive
270
Total of this
row
~ test
positive
Total
30
= False
positive rate
(.03) *
99,700
= below above
99,700
300
Total of this
row
100,000
The Table Method for Wendy:
tests
positive
~ test
positive
Total
has CC
~ have CC
Total
270
2,991
30
= below above
99,700
Total of this
row
Total of this
row
100,000
300
The Table Method for Wendy:
tests
positive
~ test
positive
Total
has CC
~ have CC
Total
270
2,991
30
96,709
300
99,700
Total of this
row
Total of this
row
100,000
The Table Method for Wendy:
tests
positive
~ test
positive
Total
has CC
~ have CC
Total
270
2,991
3,261
30
96,709
96,739
300
99,700
100,000
The Table Method for Wendy:
tests
positive
~ test
positive
Total
has CC
~ have CC
Total
270 (true
positive)
30 (false
negative)
300
2,991 (false 3,261
positive)
96,709 (true 96,739
negative)
99,700
100,000
What are Wendy’s chances?
tests
positive
has CC
270 (true
positive)
~ have CC
2,991 (false
positive)
Total
3,261
•Wendy’s Chances are the true positives divided by
the number of total tests. That is, 270/3261, which is
.083 (8.3%).
•Those who misestimate that probability forget that
colon cancer is rarer than a false positive on a test.
How about a second test?
• Note that testing positive (given the test accuracy
specified) raises one’s chances of having the
condition from .003(the base rate) to .083.
• If we use .083 as the new base rate, those who
again test positive then have a 73.1% chance of
having the condition.
• A third positive test (with .731 as the new base
rate) raises the chance of having the condition to
98.8%
Another example:
• I highly recommend reading the discussion
question that runs from p.299-302.
• See also this excellent Wikipedia write-up that
contains an update to the Sally Clark case:
http://en.wikipedia.org/wiki/Prosecutor's_fallac
y
Choice Theory:
• The relationship between probability and
action is often complex, however we can use
simple mathematical operations (so far all
we’ve used have been the four arithmetic
operations) to assist in making good choices.
• The first principles we will look at are:
Expected Monetary Value and Expected
Overall Value.
Expected Monetary Value:
• EMV = [Pr(winning) * net gain ($)] – [Pr(losing)
* net loss ($)]
• Example, Lottery:
• EMV = [(1/20,000,000) * $9,999,999] –
[(19,999,999/20,000,000) * $1]
• That comes out to -$0.50
• That means that you lose 50 cents on the dollar you
invest; this is a bad bet.
Expected Monetary Value:
• Consider an example with twice the odds of
winning and twice the jackpot:
• Example 2, Lottery:
• EMV = [(1/10,000,000) * $19,999,999] –
[(9,999,999/10,000,000) * $1]
• That comes out to $1
• That means that you gain a dollar for every dollar you
invest; this is a good bet.
Expected Overall Value
• Monetary value is not the only kind of value. This is
because money is not an intrinsic value, but only
extrinsic. It is only valuable for what it can be
exchanged for.
• If the fun of fantasizing about winning is worth losing
50 cents on the dollar, then the overall value of the
ticket justifies its purchase.
• In general, gamblers always lose money. If viewed as a
form of entertainment that is worth the expenditure, it
has a good value. If people lose more than they can
afford, or if the loss hurts them, it has negative value.
Diminishing marginal value:
• This is a concept that affects expected overall
value.
• Diminishing marginal value occurs when an
increase in something becomes less valuable
per increment of increase.
• Examples: sleep, hamburgers, shoes, even
money (for discussion, how does diminishing
marginal value affect tax policy?)
Decisions under risk:
• When a person has an idea of what different
potential outcomes are, but does not know
what the chances of such outcomes are, there
are a number of strategies that can guide a
decision.
• Consider the following table:
Outcomes (1-4) given choice (A-C)
1
2
3
4
A
11
3
3
3
B
5
5
5
5
C
6
6
6
3
•Dominance is when one choice is as good or better in every outcome as any
competing choice.
•There is no dominant choice in the above.
•If we do not know the probabilities of ourcomes 1-4, we may assume they are equally
probable to generate an expected utility. The EU of A and B are equal, at 5. C comes
out slightly better at 5.25.
•Other strategies that make sense are:
•Maximax: Choose the strategy with the best maximum (in this case, A)
•Maximin: Choose the strategy with the best minimum (in this case, B)
•Which strategy choice makes most sense depends on how risk-averse the situation is.
Ch. 12, Exercise III:
1. EMV = [Pr(winning) * net gain ($)] – [Pr(losing) *
net loss ($)]
That is: EMV = [.9 * $10 ] – [.1 * $10] = $8
This is a good bet, but would you be willing to risk
your friend’s life on it? I should say not. So the
EMV is positive, but the EOV is not. In other
words, the stakes are SO high for failure that it
makes sense to use a maximin strategy, which is
not to bet.
2. Your own example?

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