Natures strategy, your strategy, Bayes strategy, and randomization strategy

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Natures strategy, your strategy, Bayes
strategy, and randomization strategy
10/15/09 Yoon G Kim Department of
Mathematics Humboldt State University Arcata, CA
95521
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unknown
known
I want to learn about
(i.i.d. r.s.)
Neyman-Pearson Lemma
Use
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Generalize to include
(1) I think H0 is more likely than H1 (prior
information). (2) It is much more expensive to
wrongly conclude H1 than H0 (loss information).
We will let (1) (2) influence our decision
optimally.
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Outline

• Gambles, Utility theory, Game theory
• Decision theory
• Statistical applications
• Minimax theory, Bayes procedures
• IV. Empirical Bayes procedures

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System xi of alternatives
(xi is not preferred to xj) properties
Enrich preferences with gambles
will stand for probability p of getting xi, (1?p)
of getting xj.
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Rules for gambles
Deeper rules
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Subtle axioms
II. ltltlt Archimedean axiom gtgtgt
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Pascals Wager (or Pascals Gambit)
Live a bad life, there is hell
Live a bad life, there is no hell
Behave
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Historically, Pascal's Wager was groundbreaking
as it had charted new territory in probability
theory, was one of the first attempts to make use
of the concept of infinity, marked the first
formal use of decision theory. Alan Hájek,
Stanford Encyclopedia of Philosophy
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Two Lemmas
Lemma.
Lemma.
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Proofs
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Tools for a choice under uncertainty

• Expected value Blaise Pascal, Pensées 1670
• Utility function Daniel Bernoulli, Exposition
  of a New Theory on the Measurement of Risk 1738
• Loss function, risk function, admissible decision
  rules, a priori distribution, Bayes decision
  rules, Minimax decision rules Abraham Wald, A
  new formula for the index of cost of living1939

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Utility numerical representation of preference
is a number so that
(i.e., utility of a gamble expected utility)
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Define a loss function
be natures strategy and
A be your strategy,
is your (negative) utility
To play a game, consider the risk function
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Loss Table
Nature

• If you knew natures mind, obvious.
• If not,
• conservative strategy
• If I choose I, I might lose up to 4.
• If I choose II, I might lose up to 2.
• So, I choose II.

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Nature

• Assume you know how likely nature is to pick each
  strategy,
• say, p0.5 for 1 and 2.
• If I choose I, I might lose up to
  (0.5)(-1)(0.5)(4)1.5
• If I choose II, I might lose up to
  (0.5)(2)(0.5)(-5)-1.5.
• So, I choose II.

Bayes strategy
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Bayes strategy
Nature
Nature has a priority
You choose I or II so that either one of the
following is smaller
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(No Transcript)
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Randomization strategy
Nature has a fixed unknown choices 1 2. I am
going to randomize I choose I with prob q and II
with prob (1?q).
What happens?
If 1 is natures choice, my risk is
If 2 is natures choice, my risk is
Nature
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Lower value 0.25 Upper value
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Homework 1 (Due Thursday 10/22/09)
Assume R(1, I) lt R(2, I), R(1, II) gt R(2,
II). Prove that
Upper value Lower value
Answer
ltltlt Handout gtgtgt
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(Q1) Which one would you choose?
A. A gift of 240, guaranteed B. A 25 chance
to win 1000 and a 75 chance of getting nothing
(Q2) How about this?
C. A sure loss of 740 D. A 75 chance to lose
1000 and a 25 chance to lose nothing
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Appealing choice isnt the same as a mathematical
one.

• A versus B majority chose sure gain A. Expected
  value under choice B is 250, higher than sure
  gain of 240 in A, yet people prefer A.
• C versus D majority chose gamble rather than
  sure loss. Expected value under D is 750, a
  larger expected loss than 740 in C.
• People value sure gain, but willing to take risk
  to prevent loss.

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Bayes in the spam filtering business
Make it simple, The Economist, Oct. 30, 2004
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Formal Decision Theory
In a statistical experiment, there is an unknown
law of nature , You perform a random
experiment X, whose distribution depends on ?.
You make a decision d that depends on X.
You have a loss function
Your risk is
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(Ex) Arcata, San Francisco
Decision
Truth

• Experiment Take temperature at 2PM on 10/16/09
• Arcata temperature N (60, 6)
• SF temperature N (70, 6)

Decision rule
Prob(decisionSF Arcata)
Prob(decisionArcata SF)
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Homework 2 (Due Thursday 10/29/09) Due to Mr.
Ryan Battleson (former student in 2002) in Oregon
Someone claims to have sighted several rare, shy
Alligator Thrushes on some property on which you
hope to build. On the other hand, it may just be
the ordinary Common Thrush. You construct an
experiment to decide. You cant catch such a shy
bird, but you find ten droppings, which will be
measured for their nitrogen contents x1, , x10.
The Alligator Thrushes dropping has average
nitrogen content 0.057. The average Common Thrush
dropping has nitrogen content 0.065. In each
case, the distribution is roughly normal with
standard deviation 0.01.
Since the Alligator Thrush is rare, you believe
it is three times as likely that the birds
sighted were Common Thrushes.
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Homework 2
If you decide you have the Common Thrush, and go
ahead and build, and you are wrong, you will lose
about 30,000 in later penalties.
If you decide you have the Alligator Thrush, and
fail to build, and you are wrong, you will lose
about 50,000 in lost opportunity costs.
What would Bayes do here? Whats Bayes risk?
(Extra Credit)

• You can ignore a prior opinion as to the
  probability of one or the other.
• Find the risk set for this problem and graph it.
• Develop a complete class of strategies.
• Find a minimax strategy for this problem.

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ltltlt Handout Printout gtgtgt