Download Chapter 19

Chapter 19
Statistical Decision
Theory
©
Framework for a Decision
Problem
i.
Decision maker has available K possible courses of
action: a1, a2, . . ., aK. Actions are sometime called
alternatives.
ii. There are H possible uncertain states of nature: s1,
s2, . . ., sH. States of nature are the possible outcomes
over which the decision maker has no control.
Sometimes states of nature are called events.
iii. For each possible action-state of nature combination,
there is an associated outcome representing either
profit or loss, called the monetary payoff, Mij, that
corresponds to action ai and state of nature sj. The
table of all such outcomes for a decision problem is
called a payoff table.
Payoff Table for a Decision Problem
with K Possible Actions and H Possible
States of Nature
(Table 19.1)
ACTIONS
ai
STATES OF NATURE
s1
s2
...
sH
M11
M21
.
.
.
MK1
M12
M22
.
.
.
MK2
...
...
.
.
.
...
M1H
M2H
.
.
.
MKH
si
a1
a2
.
.
.
aK
Decision Rule Based on Maximin
Criterion
Suppose that a decision maker has to choose from K
admissible actions a1, a2, . . ., aK , given H possible states of
nature s1, s2, . . ., sH. Let Mij denote the payoff corresponding to
the ith action and jth state of nature. For each action, seek the
smallest possible payoff. For action a1, for example, this is the
smallest of M11, M12, . . .M1H . Let us denote the minimum M1*
where
M1*  Min (M11, M12 ,, M1H )
More generally, the smallest possible payoff for action ai is
given by
*
M i  ( M 11, M 12 ,, M 1H )
The maximin criterion then selects the action ai for which the
corresponding Mi* is largest (that is, the action for which the
minimum payoff is highest).
Regret or Opportunity Loss Table
Suppose that a payoff table is arranged as a
rectangular array, with rows corresponding to
actions and columns to states of nature. If each
payoff in the table is subtracted from the largest
payoff in its column, the resulting array is called
a regret table, or opportunity loss table.
Decision Rule Based on the
Minimax Criterion
Given the regret table, the action dictated by the
minimax regret criterion is found as follows:
(i) For each row (action), find the maximum regret.
(ii) Choose the action corresponding to the minimum
of these maximum regrets. The minimax criterion
selects the action for which the maximum regret is
smallest; that is, the minimax regret criterion
produces the smallest possible opportunity loss
that can be guaranteed.
Payoff s with State-of-Nature
Probabilities
(Table 19.6)
ACTIONS
ai
si
a1
a2
.
.
.
aK
STATES OF NATURE
s1
(1)
s2
(2)
...
sH
(H)
M11
M21
.
.
.
MK1
M12
M22
.
.
.
MK2
...
...
.
.
.
...
M1H
M2H
.
.
.
MKH
Expected Monetary Value (EMV)
Criterion
Suppose that a decision maker has K possible actions, a1, a2, . .
., aK and is faced with H states of nature. Let Mij denote the
payoff corresponding to the ith action and jth state and j the
H
probability of occurrence of the jth state of nature with  j  1.
j 1
The expected monetary value of action ai, EMV(ai) , is
H
EMV (a i )   1M i1   2 M i 2     H M iH    j M ij
j 1
The Expected Monetary Value Criterion adopts the action
with the largest expected monetary value; that is, given a
choice among alternation actions, the EMV criterion dictates
the choice of the action for which the EMV is highest.
Decision Trees
The tree diagram is a graphical device
that forces the decision-maker to examine
all possible outcomes, including
unfavorable ones.
Decision Trees
All decision trees contain:
Decision (or action) nodes
Event (or state-of-nature) nodes
Terminal nodes
Decision Trees
(Figure 19.3)
Actions
Process A
Process B
Process C
States of nature
Bayes’ Theorem
Let s1, s2, . . ., sH be H mutually exclusive and collectively
exhaustive events, corresponding to the H states of nature of a
decision problem. Let A be some other event. Denote the
conditional probability that si will occur, given that A occurs,
by P(si|A), and the probability of A, given si, by P(A|si).
Bayes’ Theorem states that the conditional probability of si,
given A, can be expressed as
P( A | si ) P( si )
P( si | A) 
P( A)
P( A | si ) P( si )
P( si | A) 
P( A | s1 ) P( s1 )  P( A | s2 ) P( s2 )    P( A | sH ) P( sH )
In the terminology of this section, P(si) is the prior probability
of si and is modified to the posterior probability, P(si|A),
given the sample information that event A has occurred.
Expected Value of Perfect
Information, EVPI
Suppose that a decision maker has to choose from among K
possible actions, in the face of H states of nature, s1, s2, . .
., sH. Perfect information corresponds to knowledge of
which state of nature will arise. The expected value of
perfect information is obtained as follows:
(i) Determine which action will be chosen if only the prior
probabilities P(s1), P(s2), . . ., P(sH) are used.
(ii) For each possible state of nature, si, find the difference,
Wi, between the payoff for the best choice of action, if it
were known that state would arise, and the payoff for the
action chosen if only prior probabilities are used. This is
the value of perfect information, when it is known that si
will occur.
(iii) The expected value of perfect information, EVPI, is then
EVPI  P(s1 )W1  P(s2 )W2    P(sH )WH
Expected Value of Sample
Information, EVSI
Suppose that a decision maker has to choose from among K
possible actions, in the face of H states of nature, s1, s2, . .
., sH. The decision-maker may obtain sample
information. Let there be M possible sample results, A1,
A2, . . . , AM. The expected value of sample information is
obtained as follows:
(i) Determine which action will be chosen if only the prior
probabilities were used.
(ii) Determine the probabilities of obtaining each sample
result:
P( Ai )  P( Ai | s1 ) P(s1 )  P( Ai | s2 ) P(s2 )    P( Ai | sH ) P(sH )
Expected Value of Sample
Information, EVSI
(continued)
(iii) For each possible sample result, Ai, find the difference, Vi,
between the expected monetary value for the optimal
action and that for the action chosen if only the prior
probabilities are used. This is the value of the sample
information, given that Ai was observed.
(iv) The expected value of sample information, EVSI, is
then:
EVPI  P(s1 )W1  P(s2 )W2    P(sH )WH
Obtaining a Utility Function
Suppose that a decision maker may receive several alternative payoffs.
The transformation from payoffs to utilities is obtained as
follows:
(i)
The units in which utility is measured are arbitrary.
Accordingly, a scale can be fixed in any convenient fashion. Let
L be the lowest and H the highest of all the payoffs. Assign
utility 0 to payoff L and utility 100 to payoff H.
(ii) Let I be any payoff between L and H. Determine the probability
 such that the decision-maker is indifferent between the
following alternatives:
(a) Receive payoff I with certainty
(b) Receive payoff H with probability  and payoff L with
probability (1 - )
(iii) The utility to the decision-maker of payoff I is then 100. The
curve relating utility to payoff is called a utility function.
Utility Functions; (a) Risk Aversion; (b)
Preference for Risk; (c) Indifference to Risk
Utility
Payoff
(a) Risk aversion
Utility
Utility
(Figure 19.11)
Payoff
(b) Preference for risk
Payoff
(c) Indifference to risk
The Expected Utility Criterion
Suppose that a decision maker has K possible actions, a1, a2, . . .,
aK and is faced with H states of nature. Let Uij denote the utility
corresponding to the ith action and jth state and j the
probability of occurrence of the jth state of nature. Then the
expected utility, EU(ai), of the action ai is
H
EU (a i )   1U i1   2U i 2     HU iH    jU ij
j 1
Given a choice between alternative actions, the expected utility
criterion dictates the choice of the action for which expected
utility is highest. Under generally reasonable assumptions, it
can be shown that the rational decision-maker should adopt
this criterion.
If the decision-maker is indifferent to risk, the expected
utility criterion and expected monetary value criterion are
equivalent.
Key Words
 Action
 Admissible action
 Aversion to Risk
 Bayes’ Theorem
 Decision nodes
 Decision Trees
 EMV
 Event nodes
 EVPI
 EVSI
 Expected Monetary
Value Criterion
 Expected net Value of
Sample Information
 Expected Utility
Criterion
 Inadmissible Action
 Indifference to Risk
 Maximin Criterion
 Minimax Regret
Criterion
 Opportunity Loss Table
 Payoff Table
 Perfect Information
Key Words
(continued)
 Preference for Risk
 Regret Table
 Sensitivity Analysis
 States of Nature
 Terminal Nodes
 Tree Plan
 Utility Function
 Value of Perfect Information
 Value of Sample Information