Coursera概率圖模型（Probabilistic Graphical Models）第一周編程作業分析
Computing probability queries in a Bayesian network
計算貝葉斯網絡中的概率查詢
1.基礎因子操作
作業中因子的結構
phi = struct(‘var‘, [3 1 2], ‘card‘, [2 2 2], ‘val‘, ones(1, 8));
其中：var表示因子中變量的標簽及順序，card代表基數，描述了各變量的狀態數量，val表示各變量取不同值時對應的概率分布，其向量長度等於prod(card)。
FactorProduct.m 計算兩個因子的積
輸入:
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));
期望輸出：
FACTORS.PRODUCT = struct(‘var‘, [1, 2], ‘card‘, [2, 2], ‘val‘, [0.0649, 0.1958, 0.0451, 0.6942]);
我們知道，對貝葉斯網絡而言，因子積其實就是表示貝葉斯鏈式法則。比如若FACTORS.INPUT(1) 表示學生的智力是否正常的分布，即
，則其聯合概率分布可記為FACTORS.PRODUCT = FactorProduct(FACTORS.INPUT(1), FACTORS.INPUT(2))，即
。
計算步驟很簡單，就是貝葉斯鏈式法則的步驟：
參考代碼如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 計算因子的邊緣分布
輸入:
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]);
期望輸出：
FACTORS.MARGINALIZATION = struct(‘var‘, [1], ‘card‘, [2], ‘val‘, [1 1]);
本質上，求邊緣分布就是一個求和的過程。對相應變量的值求和就可以了。
參考代碼如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 變量觀測
輸入:
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]);
期望輸出:
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函數中，第二個參數為一個
的矩陣，第一列表示所觀測的變量，第二列表示對應變量的取值。要求只保留因子中被觀測變量所對應取值的概率，被觀測變量的其他取值對應概率置0。未被觀測變量不受影響。
參考代碼如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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.計算聯合分布
ComputeJointDistribution.m 計算貝葉斯網絡的聯合概率分布
輸入:
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);
期望輸出:
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，在貝葉斯網絡中，聯合概率分布就是其因子積。下面是不同的表述：
參考代碼如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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.計算邊緣分布
ComputeMarginal.m 計算貝葉斯網絡的邊緣概率分布
輸入:
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]);
期望輸出:
FACTORS.MARGINAL = struct(‘var‘, [2, 3], ‘card‘, [2, 2], ‘val‘, [0.0858, 0.0468, 0.1342, 0.7332]);
相比之前計算因子的邊緣分布，這裏主要多了歸一化的要求，同時還要註意合並相同變量的問題。
參考代碼如下：
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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：表示變量的名稱及它們之間的關系，這裏（‘var‘, [3,1]）表示因子描述的是phenotypeVar = 3的性狀對genotypeVar = 1的基因型的條件概率分布。
card：是基數（cardinalities）的縮寫，描述了元素間一一對應的集合（可能的）
Coursera概率圖模型（Probabilistic Graphical Models）第一周編程作業分析