Coursera概率图模型（Probabilistic Graphical Models）第四周编程作业分析

Decision Making

作决策

 

这一周的内容在老版本的CS228课程中，是作为第六周的一个小节讲的（老版本的CS229只有9周的课程），而在概率图模型的教材里边对应的是第22章效用和决策。也就是说，这一周的课程更多的是对之前所学知识的一种应用。

 

1.记号和定义

 

使用影响图来表现本周所学的内容，如下图所示：

 

 

其中，X表示随机变量，D表示决策节点，U表示效用节点。

 

2.已知决策规则的期望效用

 

我们可以将随机变量与决策节点一起看做一个贝叶斯网络，并只保留效用节点的父节点，即对其它节点用VariableElimination函数进行变量消除，将所得因子的val值与效用节点的val值相乘即可。公式如下式所示：

 

 

SimpleCalcExpectedUtility.m 简单计算期望效用

 

参考代码如下：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% YOUR CODE HERE

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

V = unique([F(:).var]);

Z = setdiff(V, U.var);

Fnew = VariableElimination(F, Z);



Ffinal = Fnew(1);

for ii = 2 : length(Fnew)

    Ffinal = FactorProduct(Ffinal, Fnew(ii));

end



Ffinal.val = Ffinal.val / sum(Ffinal.val);



U_reorder_val = zeros(size(U.val));

for ii = 1 : prod(U.card)

    U_reorder_val(ii) = GetValueOfAssignment(U, IndexToAssignment(ii, U.card), Ffinal.var);

end



EU = Ffinal.val * U_reorder_val';

 

 

2.基于期望效用因子最大化期望效用

 

上一章的公式可以变形如下式：

 

 

我们需要计算的期望效用，就是式中的部分。这里我们可以将随机变量与效用节点一起看做一个贝叶斯网络，将网络中与决策节点无关的变量消除，其剩余因子的因子积就是我们要求的最大期望效用。

 

CalculateExpectedUtilityFactor.m 计算期望效用因子

 

参考代码如下：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% YOUR CODE HERE...

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

F = [I.RandomFactors, I.UtilityFactors];



V = unique([F(:).var]);

Z = setdiff(V, I.DecisionFactors(1).var);

Fnew = VariableElimination(F, Z);



EUF = Fnew(1);

for ii = 2 : length(Fnew)

    EUF = FactorProduct(EUF, Fnew(ii));

end

 

OptimizeMEU.m 优化最大期望效用

 

这里就是取效用最大的决策方案。注意一下多个决策节点时OptimalDecisionRule.var的含义。

 

参考代码如下：

OptimalDecisionRule = CalculateExpectedUtilityFactor(I);



if length(OptimalDecisionRule.var) == 1

    [MEU, index] = max(OptimalDecisionRule.val);

    OptimalDecisionRule.val = zeros(size(OptimalDecisionRule.val));

    OptimalDecisionRule.val(index) = 1;

else

    assignments = IndexToAssignment(1 : prod(OptimalDecisionRule.card(1 : end - 1)), OptimalDecisionRule.card(1 : end - 1));

    MEU = 0;

    for ii = 1 : OptimalDecisionRule.card(end)

        indices1 = AssignmentToIndex([assignments, ii * ones(size(assignments, 1), 1)], OptimalDecisionRule.card);

        [meu, indices2] = max(OptimalDecisionRule.val(indices1));

        MEU = MEU + meu;

        OptimalDecisionRule.val(indices1) = 0;

        OptimalDecisionRule.val(indices1(indices2)) = 1;

    end

end

 

3.多效用因子

 

我们还往往会遇到不止一个效用节点的情况，如下图所示：

 

 

这种情况下，我们当然是选择能够使多个效用节点期望值的和最大的决策啦。我们有两种处理方案：一种是直接求出每一个效用节点的期望值，之后直接加和即可；另一种是利用下式所示的公式对效用节点进行变形，之后直接计算最大期望效用即可。

 

上述的两种方法都会要用到因子求和函数，这个函数在这一周的作业中并没有提供……但是，可以在第七周的作业PA-Exact-Inference-Release中找到，函数名为FactorSum.m，直接把函数的全部内容复制在我们这一周的作业后边就好。

 

FactorSum.m 因子求和

 

第七周作业提供的代码如下：

function C = FactorSum(A, B)



if (isempty(A.var)), C = B; return; end;

if (isempty(B.var)), C = A; return; end;



[dummy iA iB] = intersect(A.var, B.var);

if ~isempty(dummy)

    assert(all(A.card(iA) == B.card(iB)), 'Dimensionality mismatch in factors');

end



C.var = union(A.var, B.var);





[dummy, mapA] = ismember(A.var, C.var);

[dummy, mapB] = ismember(B.var, C.var);



C.card = zeros(1, length(C.var));

C.card(mapA) = A.card;

C.card(mapB) = B.card;



C.val = zeros(1,prod(C.card));



assignments = IndexToAssignment(1:prod(C.card), C.card);

indxA = AssignmentToIndex(assignments(:, mapA), A.card);

indxB = AssignmentToIndex(assignments(:, mapB), B.card);



C.val = A.val(indxA) + B.val(indxB);



end

 

OptimizeWithJointUtility.m 联合效用优化

 

参考代码如下：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% YOUR CODE HERE

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

U = I.UtilityFactors(1);

for ii = 2 : length(I.UtilityFactors)

    U = FactorSum(U, I.UtilityFactors(ii));

end

I.UtilityFactors = U;

[MEU, OptimalDecisionRule] = OptimizeMEU(I);

 

OptimizeLinearExpectations.m 线性期望优化

 

参考代码如下：

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% YOUR CODE HERE

%

% A decision rule for D assigns, for each joint assignment to D's parents,

% probability 1 to the best option from the EUF for that joint assignment

% to D's parents, and 0 otherwise.  Note that when D has no parents, it is

% a degenerate case we can handle separately for convenience.

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

OptimalDecisionRule = struct('var', [], 'card', [], 'val', []);

for ii = 1 : length(I.UtilityFactors)

    I_ = I;

    I_.UtilityFactors = I.UtilityFactors(ii);

    EUF_ = CalculateExpectedUtilityFactor(I_);

    OptimalDecisionRule = FactorSum(OptimalDecisionRule, EUF_);

end



if length(OptimalDecisionRule.var) == 1

    [MEU, index] = max(OptimalDecisionRule.val);

    OptimalDecisionRule.val = zeros(size(OptimalDecisionRule.val));

    OptimalDecisionRule.val(index) = 1;

else

    assignments = IndexToAssignment(1 : prod(OptimalDecisionRule.card(1 : end - 1)), OptimalDecisionRule.card(1 : end - 1));

    MEU = 0;

    for ii = 1 : OptimalDecisionRule.card(end)

        indices1 = AssignmentToIndex([assignments, ii * ones(size(assignments, 1), 1)], OptimalDecisionRule.card);

        [meu, indices2] = max(OptimalDecisionRule.val(indices1));

        MEU = MEU + meu;

        OptimalDecisionRule.val(indices1) = 0;

        OptimalDecisionRule.val(indices1(indices2)) = 1;

    end

end

 

交作业的截图：

 

 

这两周诸事繁杂，所以概率图的学习进度放缓了些，下周开始要保持正常进度才行……