CRP 834: Decision Analysis
Week One Notes
Jean-Michel Guldmann
Sumei Zhang
Statistical Decision Theory
• Problem setup:
– Two alternatives about the state of nature: A null
hypothesis ( H 0 ) and an alternative one (H1 );
• Decision rule:
– Make decision based on a critical value;
• Action:
– Reject or accept the null hypothesis based on a sample;
• Type I vs. Type II Error
Statistical Decision Theory
• Example:
College students’ IQ score follows a normal distribution
with mean 125, standard deviation 5. 100 students from
OSU make the sample. Their average IQ score is 135.
Statistical Decision Theory
• Comments:
– No account of the seriousness of the consequences of
committing type I and type II errors;
– No information about the states of nature;
– Choice between two alternatives.
Decision Rules Under Uncertainty
• Elements of Decision Making
–
–
–
–
–
–
–
–
Problem
Objective
Alternative
Consequences
Tradeoffs
Uncertainty
Risk Tolerance
Interaction Decision Making
Decision Rules Under Uncertainty
• Concepts
– Loss
– Regret
– Risk
• Example
– Loss Function
Loss = l (State, Action)
Action
A1: take umbrella
A2: do not take umbrella
States of nature
W1: Rain
W2: no Rain
$2
$5
$10
$0
Decision Rules Under Uncertainty
– Regret Function
r (Wi , A j )  l (Wi , A j )  Min l (Wi , A j )
A
– Additional Information: probabilities
W 1( rain)
W 2(no rain)
F air
0.1
0.6
N o F orecast
0.2
0.3
R ain
0.7
0.1
Decision Rules Under Uncertainty
– 8 Possible Decision Rules
D1
D2
D3
D4
D5
D6
D7
D8
F air
A1
A1
A1
A2
A1
A2
A2
A2
N o F orecast
A1
A1
A2
A1
A2
A1
A2
A2
R ain
A1
A2
A1
A1
A2
A2
A1
A2
P (A1/W 1)= 1 - α
1
0.3
0.8
0.9
0.1
0.2
0.7
0
P (A2/W 1)= α
0
0.7
0.2
0.1
0.9
0.8
0.3
1
P (A1/W 2)= β
1
0.9
0.7
0.4
0.6
0.3
0.1
0
P (A2/W 2)= 1 - β
0
0.1
0.3
0.6
0.4
0.7
0.9
1
Decision Rules Under Uncertainty
– Risk Function
Risk = g [ Loss ] = g [ f (State, Action) ]
In case of the example:
When the state of nature is W1=Rain:
R(W1 , d j )  ($2)(1   )  ($10)( )
When the state of nature is W2=No Rain:
R(W2 , d j )  ($5)(  )  ($0)(1   )
Decision Rules Under Uncertainty
• Decision Rules
– Look at the average of the risks
– Look at the Expected risk (Bayes Risk)
Decision Rules Under Uncertainty
– Comments about Bayes Risk
• Incorporates the losses due to committing Type I and Type II
errors;
• Provide room for a policy maker’s subjective evaluation;
• Evaluation of the Bayes risk can be improved by use of the
Bayes theorem;
P( Ai B j ) 
P( Ai ) * P( B j Ai )
 P( A ) * P( B
k 1 m
k
j
Ak )
Decision Rules Under Uncertainty
• Example (using Bayes Theorem): Of all applicants for a job, it
is felt that 75% are able to do the job, and 25% are not. To aid
in the selection process, an aptitude test is designed such that a
capable applicant has a probability 0.8 of passing test while an
incapable one a probability of 0.4 of passing it.
An applicant passes the test—what is the probability that he
will be able to do the job?
Decision Rules Under Uncertainty
– More Decision Rules
• The Maximin criterion
• The Maximax criterion
• The Hurwicz criterion
• The Bayes (Laplace) Criterion
• The Minimax regret criterion
• Mixed strategy