Embed Size (px)
Citation preview
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 hypothesise = the evidence for hPr(h) = the statistical probability of hPr(e|h) = the true positive rate of e as evidence for
hPr(e|~h) = the false positive rate of e as evidence
for h
The Table Method:
h ~h Total
e True Positives
False Positives
Pr(e)*Pop.
~e False Negatives
True Negatives
Pr(~e)*Pop.
Total Pr(h)*Pop. Pr(~h)*Pop. Pop. = 10^n
n = sum of decimal places in two most specific probabilities.
The Table Method:
h ~h Total
e = Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~e = below - above
= below - above
Total of this row
Total Pr(h)*Pop. Pr(~h)*Pop. Pop.
The Table Method for Wendy:
h ~h Total
e = Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~e = below - above
= below - above
Total of this row
Total Pr(h)*Pop. Pr(~h)*Pop. Pop.
The Table Method for Wendy:
has CC ~ have CC Total
e = Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~e = below - above
= below - above
Total of this row
Total Pr(h)*Pop. Pr(~h)*Pop. Pop.
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
= Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
= below - above
= below - above
Total of this row
Total Pr(h)*Pop. Pr(~h)*Pop. Pop.
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
= Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
= below - above
= below - above
Total of this row
Total .003*Pop. .997*Pop. 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
= Pr(e|h) * [Pr(h)*Pop.]
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
= below - above
= below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
= True Positive Rate (.9) * 300
= Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
= below - above
= below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 = Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
= below - above
= below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 = Pr(e|~h) * [Pr(~h)*Pop.]
Total of this row
~ test positive
30 = below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 = False positive rate (.03) * 99,700
Total of this row
~ test positive
30 = below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 2,991 Total of this row
~ test positive
30 = below - above
Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 2,991 Total of this row
~ test positive
30 96,709 Total of this row
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 2,991 3,261
~ test positive
30 96,709 96,739
Total 300 99,700 100,000
The Table Method for Wendy:
has CC ~ have CC Total
tests positive
270 (true positive)
2,991 (false positive)
3,261
~ test positive
30 (false negative)
96,709 (true negative)
96,739
Total 300 99,700 100,000
What are Wendy’s chances?
has CC ~ have CC Totaltests positive
270 (true positive)
2,991 (false positive)
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_fallacy
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] = $8This 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?
