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1
Two Envelopes Revisited
•
The “two envelopes” problem setup
Two envelopes: one contains $X, other contains $2X
You select an envelope and open it
o
Let Y = $ in envelope you selected
o
Let Z = $ in other envelope
Before opening envelope, think either equally
good
o
So, what happened by opening envelope?
E[Z  Y] above assumes all values X (where 0 <
X
<
)
are equally likely
o
Note: there are infinitely many values of X
o
So, not true probability distribution over X (doesn’t integrate to 1)
Y
Y
Y
Z
E
Y
4
5
2
1
2
2
1
2
]

[
Subjectivity of Probability
•
Belief about contents of envelopes
Since implied distribution over X is not a true probability
distribution, what is our distribution over X?
o
Frequentist
: play game infinitely many times and see how often
different values come up.
o
Problem
: I only allow you to play the game
once
Bayesian probability
o
Have prior
belief of distribution for X (or anything for that matter)
o
Prior belief is a
subjective
probability
•
By extension,
all
probabilities are subjective
o
Allows us to answer question when we have no/limited data
•
E.g., probability a coin you’ve never flipped lands on heads
o
As we get more data, prior belief is “swamped” by data
The Envelope, Please
•
Bayesian
: have prior distribution over X, P(X)
Let Y = $ in envelope you selected
Let Z = $ in other envelope
Open your envelope to determine Y
If Y > E[Z  Y], keep your envelope, otherwise switch
o
No inconsistency!
Opening envelop provides data to compute P(X  Y)
and thereby compute E[Z  Y]
Of course, there’s the issue of how you determined
your prior distribution over X…
o
Bayesian: Doesn’t matter how you determined prior, but you
must
have one (whatever it is)
o
Imagine if envelope you opened contained $17.51
Revisting Bayes Theorem
•
Bayes Theorem (
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This document was uploaded on 12/24/2011.
 Spring '09

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