Structual Learning Of Multiple Bayesian Networks

Probability graph model is one of the effective research tools in the field of statistics.In the graph model,nodes represent random variables,and the edge be-tween nodes reflect the relationship between random variables.The graph model can be divided into the directed graph or the undirected graph.Undirected graph is called a markov random field,the edge of the presence represents conditional independence between the random variables.The directed acyclic graph that has a certain probability is called a bayesian network.For the bayesian network,its all edges have directions and cannot constitute any loop,the edges expresses the causal relationship between the random variables.The research object of this pa-per is the directed acyclic graph(DAG)with gaussian probability-the gaussian bayesian network.It has been widely used in physics research and biological engi-neering.By the exhaustive method,estimating the network that has n nodes,the complexity of the network is O(2~n),Heuristic algorithm can behave better when the number of nodes is small.But with the number of nodes increasing,the complexity of the algorithm exemplified by the heuristics method grows up geometrically.So estimating directed graph becomes a NP-hard problem.Series of algorithms based on the model of the low dimension structure of high dimension data.Those kind of algorithms utilizes the maximum likelihood estimation of Gaussian distribution to construct the optimization function besides using the low rank of high-dimensional data to estimate the weights between nodes by regression algorithm,so the connectivity between nodes can be estimated.The resulting high-dimensional data tends to be structurally related when sampling in different scenarios.Thus these correlations can reduce the degree of illness caused by the amount of data in the regression problem.A series of algorithms for joint estimation of Gaussian directed graphs are also born.Although there are more mature algorithms to estimate Gaussian graphs,it is a ill-conditional problem to recover directed graphs only by observing data due to the observational equiva-lence of directed graphs.A regression model based on the maximum likelihood estimation with similar structure penalty term is proposed in this paper to es-timate the adjacency matrices of the networks.Computationally,the model is solved by coordinate descent method with complexity O(nk~2p).Compared with PC algorithm,numerical experiments demonstrates the algorithm proposed in this paper performs well.natural ordering between nodes and use the similarity struc-ture between different graphs.A regression model based on the maximum likeli-hood estimation with similar structure penalty term is proposed in this paper to estimate the adjacency matrices of the networks.Computationally,the model is solved by coordinate descent method with complexity O(nk~2p).Compared with PC algorithm,numerical experiments demonstrates the algorithm proposed in this paper performs well.