第 15章 应用统计决策
Statistical Decision
本章概要
?The Payoff Table and Decision Trees
?Opportunity Loss
?Criteria for Decision Making
?Expected Monetary Value
?Expected Profit Under Certainty
?Return to Risk Ratio
?Decision Making with Sample Information
?Utility
Features of Decision Making
决策特征
?List Alternative Courses of Action
(Possible Events or Outcomes)
?Determine Payoffs?
(Associate a Payoff with Each Event or Outcome)
?Adopt Decision Criteria
(Evaluate Criteria for Selecting the Best Course of
Action)
List Possible Actions or Events
可能行为与事件清单
Payoff Table Decision Tree
Two Methods
of Listing
Event (Ei)
Cool Weather (E1) x11 =$50 x12 = $100
Warm Weather (E2) x21 = 200 x22 = 125
Payoff Table
盈利表
Consider a food vendor determining
whether to sell soft drinks or hot dogs.
Course of Action (Aj)
Sell Soft Drinks (A1) Sell Hot Dogs (A2)
xij = payoff (profit) for event i and action j
Decision Tree（决策树）,Example
Food Vendor Profit Tree Diagram
x11 = $50
x21 = 200
x22 =125
x12 = 100
Opportunity Loss（机会损失）,
Example
Highest possible profit for an event Ei
- Actual profit obtained for an action Aj
Opportunity Loss (lij )
Event,Cool Weather
Action,Soft Drinks Profit,$50
Alternative Action,Hot Dogs Profit,$100
Opportunity Loss = $100 - $50 = $50
Opportunity Loss,Table
Event Optimal Profit of Sell Soft Drinks Sell Hot Dogs
Action Optimal
Action
Cool Hot 100 100 - 50 = 50 100 - 100 = 0
Weather Dogs
Warm Soft 200 200 - 200 = 0 200 - 125 = 75
Weather Drinks
Alternative Course of Action
Decision Criteria
决策准则
Expected Monetary Value (EMV)
The expected profit for taking an action Aj
Expected Opportunity Loss (EOL)
The expected loss for not taking action Aj
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best
decision
Decision Criteria -- EMV
期望货币价值
Expected Monetary Value (EMV)
Sum (monetary payoffs of events) ?? (probabilities of the events)
Xij Pi??Vj? ?
N
EMVj = expected monetary value of action j
xi,j = payoff for action j and event i
Pi = probability of event i occurring
i = 1
Decision Criteria -- EMV Table
Example,Food Vendor
Pi Event Soft xijPi Hot xijPi
Drinks Dogs
.50 Cool $50 $50 ?.5 = $25 $100 $100?.50 = $50
.50 Warm $200 $200 ?.5 = 100 $125 $25?.50 = 62.50
EMV Soft Drink = $125 EMV Hot Dog = $112.50
Better alternative
Decision Criteria -- EOL
期望机会损失
Expected Opportunity Loss (EOL)
Sum (opportunity losses of events) ? (probabilities of events)
??Lj ? ?lij Pi
EOLj = expected monetary value of action j
li,j = payoff for action j and event i
Pi = probability of event i occurring
i =1
N
Decision Criteria -- EOL Table
Example,Food Vendor
Pi Event Op Loss lijPi OP Loss lijPi
Soft Drinks Hot Dogs
.50 Cool $50 $50?.50 = $25 $0 $0?.50 = $0
.50 Warm 0 $0 ?.50 = $0 $75 $75 ?.50 = $37.50
EOL Soft Drinks = $25 EOL Hot Dogs = $37.50
Better Choice
Decision Criteria -- EVPI
完美信息期望价值
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best decision
Represents the maximum amount you are willing
to pay to obtain perfect information
Expected Profit Under Certainty
- Expected Monetary Value of the Best Alternative
EVPI (should be a positive number)
EVPI Computation
Expected Profit Under Certainty
=,50($100) +,50($200)
= $150
Expected Monetary Value of the Best Alternative
= $125
EPVI = $25 The maximum you would be willing to
spend to obtain perfect information.
Taking Account of
Variability,Food Vendor
?2 for Soft Drink
= (50 -125)2 ?.5 + (200 -125)2 ?.5 = 5625
? for Soft Drink = 75
CVfor Soft Drinks = (75/125) ? 100% = 60%
?2 for Hot Dogs = 156.25 ? for Hot dogs = 12.5
CVfor Hot dogs = 11.11%
Return to Risk Ratio
收益风险比
Expresses the relationship between the return
(payoff) and the risk (standard deviation).
RRR = Return to Risk Ratio =
j
jEM V
?
RRRSoft Drinks = 125/75 = 1.67 RRRHot Dogs = 9
You might wish to choose Hot Dogs,Although Soft
Drinks have the higher Expected Monetary Value,Hot
Dogs have a much larger return to risk ratio and a
much smaller CV.
Decision Making with
Sample Information
?Permits Revising Old
Probabilities Based on New
Information
New
Information
Revised
Probability
Prior
Probability
校正
Additional Information,Weather forecast is COOL.
When the weather is cool,the forecaster was correct 80% of the time.
When it has been warm,the forecaster was correct 70% of the time.
Revised Probabilities （校正概率）
Example,Food Vendor
Prior Probability
先验概率
F1 = Cool forecast
F2 = Warm forecast
E1 = Cool Weather = 0.50
E2 = Warm Weather = 0.50
P(F1 | E1) = 0.80 P(F1 | E2) = 0.30
Revising Probabilities
Example:Food Vendor
P(E1 | F1) = P(cool) ?P(cool forecast | cool)P(cool forecast)
= =,73(.50) (.80) (.80)(.50) + (.30)(.50)
Revised Probability
P(F1 | E1) = 0.80 P(F1 | E2) = 0.30 E1 = 0.50 E2 = 0.50
P(E2 | F1) = P(warm) ?P(cool forecast | warm)
P(cool forecast)
=,27
Revised EMV Table
Example,Food Vendor
Pi Event Soft xijPi Hot xijPi
Drinks Dogs
.73 Cool $50 $36.50 $100 $73
.27 Warm $200 54 125 33.73
EMV Soft Drink = $90.50 EMV Hot Dog = $106.75
Better alternative
Revised EOL Table
Example,Food Vendor
Pi Event Op Loss lijPi OP Loss lijPi
Soft Drink Hot Dogs
.73 Cool $50 $36.50 $0 0
.27 Warm 0 $0 75 20.25
EOL Soft Drinks = 36.50 EOL Hot Dogs = $20.25
Better Choice
Revised EVPI Computation
Expected Profit Under Certainty
=,73($100) +,27($200)
= $127
Expected Monetary Value of the Best Alternative
= $106.75
EPVI = $20.25 The maximum you would be willing to spend to obtain perfect information.
Taking Account of Variability,
Revised Computation
?2 for Soft Drinks
= (50 -90.5)2 ?.73 + (200 -90.5)2 ?.27 = 4434.75
? for Soft Drinks = 66.59
CVfor Soft Drinks = (66.59/90.5) ? 100% = 73.6%
?2 for Hot Dogs = 123.1875
? for Hot dogs = 11.10
CVfor Hot dogs = (11.10/106.75) ? 100% = 10.4%
Revised Return to Risk Ratio
Expresses the relationship between the return
(payoff) and the risk (standard deviation).
RRR = Return to Risk Ratio =
j
jEMV
?
RRRSoft Drinks = 90.50/66.59 = 1.36 RRRHot Dogs = 9.62
You might wish to choose Hot Dogs,Hot Dogs have a
much larger return to risk ratio.
Utility（效用）
Utility is the idea that each incremental $1 of
profit does not have the same value to every
individual:
? A risk averse person,once reaching a goal,
assigns less value to each incremental $1.
? A risk seeker assigns more value to each
incremental $1.
? A risk neutral person assigns the same value
to each $1.
本章小结
?Described The Payoff Table and Decision Trees
?Opportunity Loss
?Provided Criteria for Decision Making
?Expected Monetary Value
?Expected Profit Under Certainty
?Return to Risk Ratio
?Discussed Decision Making with Sample
Information
?Addressed the Concept of Utility
Statistical Decision
本章概要
?The Payoff Table and Decision Trees
?Opportunity Loss
?Criteria for Decision Making
?Expected Monetary Value
?Expected Profit Under Certainty
?Return to Risk Ratio
?Decision Making with Sample Information
?Utility
Features of Decision Making
决策特征
?List Alternative Courses of Action
(Possible Events or Outcomes)
?Determine Payoffs?
(Associate a Payoff with Each Event or Outcome)
?Adopt Decision Criteria
(Evaluate Criteria for Selecting the Best Course of
Action)
List Possible Actions or Events
可能行为与事件清单
Payoff Table Decision Tree
Two Methods
of Listing
Event (Ei)
Cool Weather (E1) x11 =$50 x12 = $100
Warm Weather (E2) x21 = 200 x22 = 125
Payoff Table
盈利表
Consider a food vendor determining
whether to sell soft drinks or hot dogs.
Course of Action (Aj)
Sell Soft Drinks (A1) Sell Hot Dogs (A2)
xij = payoff (profit) for event i and action j
Decision Tree（决策树）,Example
Food Vendor Profit Tree Diagram
x11 = $50
x21 = 200
x22 =125
x12 = 100
Opportunity Loss（机会损失）,
Example
Highest possible profit for an event Ei
- Actual profit obtained for an action Aj
Opportunity Loss (lij )
Event,Cool Weather
Action,Soft Drinks Profit,$50
Alternative Action,Hot Dogs Profit,$100
Opportunity Loss = $100 - $50 = $50
Opportunity Loss,Table
Event Optimal Profit of Sell Soft Drinks Sell Hot Dogs
Action Optimal
Action
Cool Hot 100 100 - 50 = 50 100 - 100 = 0
Weather Dogs
Warm Soft 200 200 - 200 = 0 200 - 125 = 75
Weather Drinks
Alternative Course of Action
Decision Criteria
决策准则
Expected Monetary Value (EMV)
The expected profit for taking an action Aj
Expected Opportunity Loss (EOL)
The expected loss for not taking action Aj
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best
decision
Decision Criteria -- EMV
期望货币价值
Expected Monetary Value (EMV)
Sum (monetary payoffs of events) ?? (probabilities of the events)
Xij Pi??Vj? ?
N
EMVj = expected monetary value of action j
xi,j = payoff for action j and event i
Pi = probability of event i occurring
i = 1
Decision Criteria -- EMV Table
Example,Food Vendor
Pi Event Soft xijPi Hot xijPi
Drinks Dogs
.50 Cool $50 $50 ?.5 = $25 $100 $100?.50 = $50
.50 Warm $200 $200 ?.5 = 100 $125 $25?.50 = 62.50
EMV Soft Drink = $125 EMV Hot Dog = $112.50
Better alternative
Decision Criteria -- EOL
期望机会损失
Expected Opportunity Loss (EOL)
Sum (opportunity losses of events) ? (probabilities of events)
??Lj ? ?lij Pi
EOLj = expected monetary value of action j
li,j = payoff for action j and event i
Pi = probability of event i occurring
i =1
N
Decision Criteria -- EOL Table
Example,Food Vendor
Pi Event Op Loss lijPi OP Loss lijPi
Soft Drinks Hot Dogs
.50 Cool $50 $50?.50 = $25 $0 $0?.50 = $0
.50 Warm 0 $0 ?.50 = $0 $75 $75 ?.50 = $37.50
EOL Soft Drinks = $25 EOL Hot Dogs = $37.50
Better Choice
Decision Criteria -- EVPI
完美信息期望价值
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best decision
Represents the maximum amount you are willing
to pay to obtain perfect information
Expected Profit Under Certainty
- Expected Monetary Value of the Best Alternative
EVPI (should be a positive number)
EVPI Computation
Expected Profit Under Certainty
=,50($100) +,50($200)
= $150
Expected Monetary Value of the Best Alternative
= $125
EPVI = $25 The maximum you would be willing to
spend to obtain perfect information.
Taking Account of
Variability,Food Vendor
?2 for Soft Drink
= (50 -125)2 ?.5 + (200 -125)2 ?.5 = 5625
? for Soft Drink = 75
CVfor Soft Drinks = (75/125) ? 100% = 60%
?2 for Hot Dogs = 156.25 ? for Hot dogs = 12.5
CVfor Hot dogs = 11.11%
Return to Risk Ratio
收益风险比
Expresses the relationship between the return
(payoff) and the risk (standard deviation).
RRR = Return to Risk Ratio =
j
jEM V
?
RRRSoft Drinks = 125/75 = 1.67 RRRHot Dogs = 9
You might wish to choose Hot Dogs,Although Soft
Drinks have the higher Expected Monetary Value,Hot
Dogs have a much larger return to risk ratio and a
much smaller CV.
Decision Making with
Sample Information
?Permits Revising Old
Probabilities Based on New
Information
New
Information
Revised
Probability
Prior
Probability
校正
Additional Information,Weather forecast is COOL.
When the weather is cool,the forecaster was correct 80% of the time.
When it has been warm,the forecaster was correct 70% of the time.
Revised Probabilities （校正概率）
Example,Food Vendor
Prior Probability
先验概率
F1 = Cool forecast
F2 = Warm forecast
E1 = Cool Weather = 0.50
E2 = Warm Weather = 0.50
P(F1 | E1) = 0.80 P(F1 | E2) = 0.30
Revising Probabilities
Example:Food Vendor
P(E1 | F1) = P(cool) ?P(cool forecast | cool)P(cool forecast)
= =,73(.50) (.80) (.80)(.50) + (.30)(.50)
Revised Probability
P(F1 | E1) = 0.80 P(F1 | E2) = 0.30 E1 = 0.50 E2 = 0.50
P(E2 | F1) = P(warm) ?P(cool forecast | warm)
P(cool forecast)
=,27
Revised EMV Table
Example,Food Vendor
Pi Event Soft xijPi Hot xijPi
Drinks Dogs
.73 Cool $50 $36.50 $100 $73
.27 Warm $200 54 125 33.73
EMV Soft Drink = $90.50 EMV Hot Dog = $106.75
Better alternative
Revised EOL Table
Example,Food Vendor
Pi Event Op Loss lijPi OP Loss lijPi
Soft Drink Hot Dogs
.73 Cool $50 $36.50 $0 0
.27 Warm 0 $0 75 20.25
EOL Soft Drinks = 36.50 EOL Hot Dogs = $20.25
Better Choice
Revised EVPI Computation
Expected Profit Under Certainty
=,73($100) +,27($200)
= $127
Expected Monetary Value of the Best Alternative
= $106.75
EPVI = $20.25 The maximum you would be willing to spend to obtain perfect information.
Taking Account of Variability,
Revised Computation
?2 for Soft Drinks
= (50 -90.5)2 ?.73 + (200 -90.5)2 ?.27 = 4434.75
? for Soft Drinks = 66.59
CVfor Soft Drinks = (66.59/90.5) ? 100% = 73.6%
?2 for Hot Dogs = 123.1875
? for Hot dogs = 11.10
CVfor Hot dogs = (11.10/106.75) ? 100% = 10.4%
Revised Return to Risk Ratio
Expresses the relationship between the return
(payoff) and the risk (standard deviation).
RRR = Return to Risk Ratio =
j
jEMV
?
RRRSoft Drinks = 90.50/66.59 = 1.36 RRRHot Dogs = 9.62
You might wish to choose Hot Dogs,Hot Dogs have a
much larger return to risk ratio.
Utility（效用）
Utility is the idea that each incremental $1 of
profit does not have the same value to every
individual:
? A risk averse person,once reaching a goal,
assigns less value to each incremental $1.
? A risk seeker assigns more value to each
incremental $1.
? A risk neutral person assigns the same value
to each $1.
本章小结
?Described The Payoff Table and Decision Trees
?Opportunity Loss
?Provided Criteria for Decision Making
?Expected Monetary Value
?Expected Profit Under Certainty
?Return to Risk Ratio
?Discussed Decision Making with Sample
Information
?Addressed the Concept of Utility
