Data
There are three types. In all types, we need the following:
| A=thevectorofactions | (1×m) | m | =thenumberofactions |
| PH:PH(i)=P(H=ui) | (1×s) | s | =thenumberofvaluesofH |
| PXH:PXH(i,j)=P(X=xj|H=ui) | (s×q) | q | =thenumberofvaluesofX |
- Type 1: The usual type. In addition to the above, we need
L=[L(a,yk)] (m×n) m =thenumberofactions PYH:PYH(i,k)]=P(Y=yk|H=ui) (s×n) n =thenumberofvaluesofY - Type 2: The matrix RH=[r(a,i)] is given. L and PYH are not needed.
- Type 3: Sometimes Y=H. In this case RH=L, which we need, in addition to the above.
Calculated quantities
- RH=[r(a,i)](m×s) [Risk function = expected loss, given H] r(a,i)=E[L(a,Y)|H=ui]=∑L(a,k)P(Y=yk|H=ui) MATLAB: RH = L*PYH'
- PX(1×q) PX(j)=P(X=xj)=∑P(H=ui)P(X=j|H=ui) MATLAB:
PX = PH*PXH - PHX(q×s) PHX(i,)=P(H=uj|X=xi)=P(X=xi|H=ui)P(H=uj)/P(X=Xi) MATLAB: [a,b] = meshgrid(PH,PX) PHX = PXH'.*a./b
- RX=[R(a,j)](m×q) [Expected risk, given X] R(a,j)=E[r(a,H)|X=xj]=∑r(a,i)P(H=ui|X=xj) MATLAB: RX = RH*PHX'
- Select d* from RX: d*(j) is the action a (row number) for minimum expected loss, given X=j. Set D=[d*(1),d*(2),⋯d*(q)].
- Calculate the Bayesian risk BD for d*. BD=E[R(d*(X),X)]=∑RX(D(j),j)PX(j) MATLAB: RD*PX'
NOTE:
Actions are represented in calculations by action number (position in the matrix). In some cases, each action has a value other than its position number. The actual values can be presented in the final display.file dec.m
% file dec.m
% Version of 12/12/95
disp('Decision process with experimentation')
disp('There are three types, according to the data provided.')
disp('In all types, we need the row vector A of actions,')
disp('the row vector PH with PH(i) = P(H = u_i),')
disp('the row vector X of test random variable values, and')
disp('the matrix PXH with PXH(i,j) = P(X = x_j|H = u_i).')
disp('Type 1. Loss matrix L of L(a,k)')
disp(' Matrix PYH with PYH(i,k) = P(Y = y_k|H = u_i)')
disp('Type 2. Matrix RH of r(a,i) = E[L(a,Y)|H = u_i].')
disp(' L and PYH are not needed for this type.')
disp('Type 3. Y = H, so that only RH = L is needed.')
c = input('Enter type number ');
A = input('Enter vector A of actions ');
PH = input('Enter vector PH of parameter probabilities ');
PXH = input('Enter matrix PXH of conditional probabilities ');
X = input('Enter vector X of test random variable values ');
s = length(PH);
q = length(X);
if c == 1
L = input('Enter loss matrix L ');
PYH = input('Enter matrix PYH of conditional probabilities ');
RH = L*PYH';
elseif c == 2
RH = input('Enter matrix RH of expected loss, given H ');
else
L = input('Enter loss matrix L ');
RH = L;
end
PX = PH*PXH; % (1 x s)(s x q) = (1 x q)
[a,b] = meshgrid(PH,PX);
PHX = PXH'.*a./b; % (q x s)
RX = RH*PHX'; % (m x s)(s x q) = (m x q)
[RD D] = min(RX); % determines min of each col
% and row on which min occurs
S = [X; A(D); RD]';
BD = RD*PX'; % Bayesian risk
h = [' Optimum losses and actions'];
sh = [' Test value Action Loss'];
disp(' ')
disp(h)
disp(sh)
disp(S)
disp(' ')
disp(['Bayesian risk B(d*) = ',num2str(BD),])
EXAMPLE 1: General case
% file dec1.m
% Data for Problem 22-11
type = 1;
A = [10 15]; % Artificial actions list
PH = [0.3 0.2 0.5]; % PH(i) = P(H = i)
PXH = [0.7 0.2 0.1; % PXH(i,j) = P(X = j|H= i)
0.2 0.6 0.2;
0.1 0.1 0.8];
X = [-1 0 1];
L = [1 0 -2; % L(a,k) = loss when action number is a, outcome is k
3 -1 -4];
PYH = [0.5 0.3 0.2; % PYH(i,k) = P(Y = k|H = i)
0.2 0.5 0.3;
0.1 0.3 0.6];
dec1
dec
Decision process with experimentation
There are three types, according to the data provided.
In all types, we need the row vector A of actions,
the row vector PH with PH(i) = P(H = i),
the row vector X of test random variable values, and
the matrix PXH with PXH(i,j) = P(X = j|H = i).
Type 1. Loss matrix L of L(a,k)
Matrix PYH with PYH(i,k) = P(Y = k|H = i)
Type 2. Matrix RH of r(a,i) = E[L(a,Y)|H = i].
L and PYH are not needed in this case.
Type 3. Y = H, so that only RH = L is needed.
Enter type number type
Enter vector A of actions A
Enter vector PH of parameter probabilities PH
Enter matrix PXH of conditional probabilities PXH
Enter vector X of test random variable values X
Enter loss matrix L L
Enter matrix PYH of conditional probabilities PYH
Optimum losses and actions
Test value Action Loss
-1.0000 15.0000 -0.2667
0 15.0000 -0.9913
1.0000 15.0000 -2.1106
Bayesian risk B(d*) = -1.3
Intermediate steps in solution of Example 1, to show results of various operations
RH
RH = 0.1000 -0.4000 -1.1000
0.4000 -1.1000 -2.4000
PX
PX = 0.3000 0.2300 0.4700
a
a = 0.3000 0.2000 0.5000
0.3000 0.2000 0.5000
0.3000 0.2000 0.5000
b
b = 0.3000 0.3000 0.3000
0.2300 0.2300 0.2300
0.4700 0.4700 0.4700
PHX
PHX = 0.7000 0.1333 0.1667
0.2609 0.5217 0.2174
0.0638 0.0851 0.8511
RX
RX = -0.1667 -0.4217 -0.9638
-0.2667 -0.9913 -2.1106
EXAMPLE 2: RH given
% file dec2.m
% Data for type in which RH is given
type = 2;
A = [1 2];
X = [-1 1 3];
PH = [0.2 0.5 0.3];
PXH = [0.5 0.4 0.1; % PXH(i,j) = P(X = j|H = i)
0.4 0.5 0.1;
0.2 0.4 0.4];
RH = [-10 5 -12;
5 -10 -5]; % r(a,i) = expected loss when
% action is a, given H = i
dec2
dec
Decision process with experimentation
------------------- Instruction lines edited out
Enter type number type
Enter vector A of actions A
Enter vector PH of parameter probabilities PH
Enter matrix PXH of conditional probabilities PXH
Enter vector X of test random variable values X
Enter matrix RH of expected loss, given H RH
Optimum losses and actions
Test value Action Loss
-1.0000 2.0000 -5.0000
1.0000 2.0000 -6.0000
3.0000 1.0000 -7.3158
Bayesian risk B(d*) = -5.89
EXAMPLE 3: Example 3
Carnival example (type in which Y=H)
A carnival is scheduled to appear on a given date. Profits to be earned depend heavily on the weather. If rainy, the carnival loses $15 (thousands); if cloudy, the loss is $5 (thousands); if sunny, a profit of $10 (thousands) is expected. If the carnival sets up its equipment, it must give the show; if it decides not to set up, it forfeits $1,000. For an additional cost of $1,000, it can delay setup until the day before the show and get the latest weather report.
Actual weather H=Y is 1 rainy, 2 cloudy, or 3 sunny.
The weather report X has values 1, 2, or 3, corresponding to predictions rainy, cloudy, or sunny respectively.
Reliability of the forecast is expressed in terms of P(X=j|H=i)– see matrix PXH
Two actions: 1 set up; 2 no set up.
Possible losses for each action and weather condition are in matrix L.
% file dec3,m
% Carnival problem
type = 3; % Y = H (actual weather)
A = [1 2]; % 1: setup 2: no setup
X = [1 2 3]; % 1; rain, 2: cloudy, 3: sunny
L = [16 6 -9; % L(a,k) = loss when action number is a, outcome is k
2 2 2]; % --with premium for postponing setup
PH = 0.1*[1 3 6]; % P(H = i)
PXH = 0.1*[7 2 1; % PXH(i,j) = P(X = j|H = i)
2 6 2;
1 2 7];
dec3
dec
Decision process with experimentation
------------------- Instruction lines edited out
Enter case number case
Enter vector A of actions A
Enter vector PH of parameter probabilities PH
Enter matrix PXH of conditional probabilities PXH
Enter vector X of test random variable values X
Enter loss matrix L L
Optimum losses and actions
Test value Action Loss
1.0000 2.0000 2.0000
2.0000 1.0000 1.0000
3.0000 1.0000 -6.6531
Bayesian risk B(d*) = -2.56
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