Coursera概率图模型（Probabilistic Graphical Models）第一周编程作业分析

Computing probability queries in a Bayesian network

计算贝叶斯网络中的概率查询

1.基础因子操作

这一周的作业主要是熟悉一下基础操作。作业中因子的结构如下：

phi = struct('var', [3 1 2], 'card', [2 2 2], 'val', ones(1, 8));

其中：var表示因子中变量的标签及顺序，card代表基数，描述了各变量的状态数量，val表示各变量取不同值时对应的概率分布，其向量长度等于prod(card)。

FactorProduct.m 计算两个因子的积

输入:

FACTORS.INPUT(1) = struct('var', [1], 'card', [2], 'val', [0.11, 0.89]);

FACTORS.INPUT(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0.41, 0.22, 0.78]);

FACTORS.PRODUCT = FactorProduct(FACTORS.INPUT(1), FACTORS.INPUT(2));

期望输出：

FACTORS.PRODUCT = struct('var', [1, 2], 'card', [2, 2], 'val', [0.0649, 0.1958, 0.0451, 0.6942]);

我们知道，对贝叶斯网络而言，因子积其实就是表示贝叶斯链式法则。比如若FACTORS.INPUT(1) 表示学生的智力是否正常的分布，即

，FACTORS.INPUT(2)表示学生在其智力是否正常的条件下考试是否及格的分布，即

，则其联合概率分布可记为FACTORS.PRODUCT = FactorProduct(FACTORS.INPUT(1), FACTORS.INPUT(2))，即

。

计算步骤很简单，就是贝叶斯链式法则的步骤：



参考代码如下：

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

% YOUR CODE HERE:

% Correctly populate the factor values of C

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

for ii = 1 : length(C.val)

C.val(ii) = A.val(indxA(ii)) * B.val(indxB(ii));

end

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

FactorMarginalization.m 计算因子的边缘分布

输入:

FACTORS.INPUT(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0.41, 0.22, 0.78]);

FACTORS.MARGINALIZATION = FactorMarginalization(FACTORS.INPUT(2), [2]);

期望输出：

FACTORS.MARGINALIZATION = struct('var', [1], 'card', [2], 'val', [1 1]);

本质上，求边缘分布就是一个求和的过程。对相应变量的值求和就可以了。

参考代码如下：

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

% YOUR CODE HERE

% Correctly populate the factor values of B

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

for ii = 1 : length(unique(indxB))

B.val(ii) = sum(A.val(indxB == ii));

end

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

ObserveEvidence.m 变量观测

输入:

FACTORS.INPUT(1) = struct('var', [1], 'card', [2], 'val', [0.11, 0.89]);

FACTORS.INPUT(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0.41, 0.22, 0.78]);

FACTORS.INPUT(3) = struct('var', [3, 2], 'card', [2, 2], 'val', [0.39, 0.61, 0.06, 0.94]);

FACTORS.EVIDENCE = ObserveEvidence(FACTORS.INPUT, [2 1; 3 2]);

期望输出:

FACTORS.EVIDENCE(1) = struct('var', [1], 'card', [2], 'val', [0.11, 0.89]);

FACTORS.EVIDENCE(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0, 0.22, 0]);

FACTORS.EVIDENCE(3) = struct('var', [3, 2], 'card', [2, 2], 'val', [0, 0.61, 0, 0]);

在ObserveEvidence函数中，第二个参数为一个

的矩阵，第一列表示所观测的变量，第二列表示对应变量的取值。要求只保留因子中被观测变量所对应取值的概率，被观测变量的其他取值对应概率置0。未被观测变量不受影响。

参考代码如下：

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

% YOUR CODE HERE

% Adjust the factor F(j) to account for observed evidence

% Hint: You might find it helpful to use IndexToAssignment

% and SetValueOfAssignment

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

assignments = IndexToAssignment(1 : length(F(j).val), F(j).card);

F(j).val(assignments(:, indx) ~= x) = 0;

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

2.计算联合分布

ComputeJointDistribution.m 计算贝叶斯网络的联合概率分布

输入:

FACTORS.INPUT(1) = struct('var', [1], 'card', [2], 'val', [0.11, 0.89]);

FACTORS.INPUT(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0.41, 0.22, 0.78]);

FACTORS.INPUT(3) = struct('var', [3, 2], 'card', [2, 2], 'val', [0.39, 0.61, 0.06, 0.94]);

FACTORS.JOINT = ComputeJointDistribution(FACTORS.INPUT);

期望输出:

FACTORS.JOINT = struct('var', [1, 2, 3], 'card', [2, 2, 2], 'val', [0.025311, 0.076362, 0.002706, 0.041652, 0.039589, 0.119438, 0.042394, 0.652548]);

如前所述，在贝叶斯网络中，联合概率分布就是其因子积。下面是不同的表述：



参考代码如下：

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

% YOUR CODE HERE:

% Compute the joint distribution defined by F

% You may assume that you are given legal CPDs so no input checking is required.

%

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

Joint = F(1);

for ii = 2 : length(F)

Joint = FactorProduct(Joint, F(ii));

end

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

3.计算边缘分布

ComputeMarginal.m 计算贝叶斯网络的边缘概率分布

输入:

FACTORS.INPUT(1) = struct('var', [1], 'card', [2], 'val', [0.11, 0.89]);

FACTORS.INPUT(2) = struct('var', [2, 1], 'card', [2, 2], 'val', [0.59, 0.41, 0.22, 0.78]);

FACTORS.INPUT(3) = struct('var', [3, 2], 'card', [2, 2], 'val', [0.39, 0.61, 0.06, 0.94]);

FACTORS.MARGINAL = ComputeMarginal([2, 3], FACTORS.INPUT, [1, 2]);

期望输出:

FACTORS.MARGINAL = struct('var', [2, 3], 'card', [2, 2], 'val', [0.0858, 0.0468, 0.1342, 0.7332]);

相比之前计算因子的边缘分布，这里主要多了归一化的要求，同时还要注意合并相同变量的问题。

参考代码如下：

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

% YOUR CODE HERE:

% M should be a factor

% Remember to renormalize the entries of M!

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

Joint = ComputeJointDistribution(F);

M = FactorMarginalization(ObserveEvidence(Joint, E), setdiff(Joint.var, V));

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

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

var：表示变量的名称及它们之间的关系，这里（'var', [3,1]）表示因子描述的是phenotypeVar = 3的性状对genotypeVar = 1的基因型的条件概率分布。

card：是基数（cardinalities）的缩写，描述了元素间一一对应的集合（可能的）

这一周的作业主要是熟悉一下基础操作。作业中因子的结构如下：