貝葉斯網絡是一種概率網絡,它是基於概率推理的圖形化網絡,而貝葉斯公式則是這個概率網絡的基礎。貝葉斯網絡是基於概率推理的數學模型,所謂概率推理就是通過一些變量的信息來獲取其他的概率信息的過程,基於概率推理的貝葉斯網絡(Bayesian network)是為了解決不定性和不完整性問題而提出的,它對於解決複雜設備不確定性和關聯性引起的故障有很大的優勢,在多個領域中獲得廣泛應用。

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題目:Sum-product networks: A survey

摘要:和積網絡是一種基於有根無環有向圖的概率模型,其中終端節點表示單變量概率分布,非終端節點表示概率函數的凸組合(加權和)和乘積。它們與概率圖形模型密切相關,特別是與具有多種上下文特定獨立性的貝葉斯網絡。它們的主要優點是可以根據數據建立可處理的模型,即,該模型可以根據圖中鏈接的數量及時地執行多個推理任務。它們有點類似於神經網絡,可以解決類似的問題,如圖像處理和自然語言理解。本文綜述了SPN的定義、數據推理和學習的主要算法、主要應用、軟件庫的簡要介紹,並與相關模型進行了比較。

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We study the problem of learning the structure of an optimal Bayesian network $D$ when additional constraints are posed on the DAG $D$ or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class $\Pi$ by at most $k$ vertex deletions. We show that for $\Pi$ being the graphs with maximum degree $1$, an optimal network can be computed in polynomial time when $k$ is constant, extending previous work that gave an algorithm with such a running time for $\Pi$ being the class of edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when $\Pi$ is the set of graphs with maximum degree $2$ or when $\Pi$ is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if $k=0$. Finally, we show that learning an optimal network with at most $k$ edges in the moralized graph presumably has no $f(k)\cdot |I|^{\mathcal{O}(1)}$-time algorithm and that, in contrast, an optimal network with at most $k$ arcs in the DAG $D$ can be computed in $2^{\mathcal{O}(k)}\cdot |I|^{\mathcal{O}(1)}$ time where $|I|$ is the total input size.

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We study the problem of learning the structure of an optimal Bayesian network $D$ when additional constraints are posed on the DAG $D$ or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class $\Pi$ by at most $k$ vertex deletions. We show that for $\Pi$ being the graphs with maximum degree $1$, an optimal network can be computed in polynomial time when $k$ is constant, extending previous work that gave an algorithm with such a running time for $\Pi$ being the class of edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when $\Pi$ is the set of graphs with maximum degree $2$ or when $\Pi$ is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if $k=0$. Finally, we show that learning an optimal network with at most $k$ edges in the moralized graph presumably has no $f(k)\cdot |I|^{\mathcal{O}(1)}$-time algorithm and that, in contrast, an optimal network with at most $k$ arcs in the DAG $D$ can be computed in $2^{\mathcal{O}(k)}\cdot |I|^{\mathcal{O}(1)}$ time where $|I|$ is the total input size.

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