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Given the Bayesian network on the left hand side in the following figure, it shows that the random variable $$B$$ is dependent on $$A$$ and $$C$$, and the Bayesian network $$G$$ can be factorized as: $$P(G) = p(A) \times p(C) \times p(B|A,C)$$. The Bayesian network $$G$$ can be moralized to a Markov network $$M$$ for junction tree algorithm and it is shown on the right hand side of the figure. In $$M$$, all the existing edges in $$G$$ are changed to undirected ones and it adds a new undirected edge between the parents of $$B$$. Is it correct to use the same probability $$P(G) = p(A) \times p(C) \times p(B|A,C)$$ to factorize $$M$$ (assuming the factors over $$A$$, $$B$$, and $$C$$ are probabilities)? I understand that a new edge is added between $$A$$ and $$C$$, but I think the edge is added to capture the dependencies in the Markov network, is it compulsory to introduce an edge factor/potential in the factorization of $$M$$?

Given the Bayesian network on the left hand side in the following figure, it shows that the random variable $$B$$ is dependent on $$A$$ and $$C$$, and the Bayesian network $$G$$ can be factorized as: $$P(G) = p(A) \times p(C) \times p(B|A,C)$$. The Bayesian network $$G$$ can be moralized to a Markov network $$M$$ and it is shown on the right hand side of the figure. In $$M$$, all the existing edges in $$G$$ are changed to undirected ones and it adds a new undirected edge between the parents of $$B$$. Is it correct to use the same probability $$P(G) = p(A) \times p(C) \times p(B|A,C)$$ to factorize $$M$$ (assuming the factors over $$A$$, $$B$$, and $$C$$ are probabilities)? I understand that a new edge is added between $$A$$ and $$C$$, but I think the edge is added to capture the dependencies in the Markov network, is it compulsory to introduce an edge factor/potential in the factorization of $$M$$?

Given the Bayesian network on the left hand side in the following figure, it shows that the random variable $$B$$ is dependent on $$A$$ and $$C$$, and the Bayesian network $$G$$ can be factorized as: $$P(G) = p(A) \times p(C) \times p(B|A,C)$$. The Bayesian network $$G$$ can be moralized to a Markov network $$M$$ for junction tree algorithm and it is shown on the right hand side of the figure. In $$M$$, all the existing edges in $$G$$ are changed to undirected ones and it adds a new undirected edge between the parents of $$B$$. Is it correct to use the same probability $$P(G) = p(A) \times p(C) \times p(B|A,C)$$ to factorize $$M$$ (assuming the factors over $$A$$, $$B$$, and $$C$$ are probabilities)? I understand that a new edge is added between $$A$$ and $$C$$, but I think the edge is added to capture the dependencies in the Markov network, is it compulsory to introduce an edge factor/potential in the factorization of $$M$$?

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Moralized graph factorization from Bayesian network

Given the Bayesian network on the left hand side in the following figure, it shows that the random variable $$B$$ is dependent on $$A$$ and $$C$$, and the Bayesian network $$G$$ can be factorized as: $$P(G) = p(A) \times p(C) \times p(B|A,C)$$. The Bayesian network $$G$$ can be moralized to a Markov network $$M$$ and it is shown on the right hand side of the figure. In $$M$$, all the existing edges in $$G$$ are changed to undirected ones and it adds a new undirected edge between the parents of $$B$$. Is it correct to use the same probability $$P(G) = p(A) \times p(C) \times p(B|A,C)$$ to factorize $$M$$ (assuming the factors over $$A$$, $$B$$, and $$C$$ are probabilities)? I understand that a new edge is added between $$A$$ and $$C$$, but I think the edge is added to capture the dependencies in the Markov network, is it compulsory to introduce an edge factor/potential in the factorization of $$M$$?