We propose the recursive autonomy identification (RAI) algorithm for constraint-based (CB) Bayesian network structure learning. The RAI algorithm learns the structure by sequential application of conditional independence (CI) tests, edge direction and structure decomposition into autonomous sub-structures. The sequence of operations is performed recursively for each autonomous substructure while simultaneously increasing the order of the CI test. While other CB algorithms d-separate structures and then direct the resulted undirected graph, the RAI algorithm combines the two processes from the outset and along the procedure. By this means and due to structure decomposition, learning a structure using RAI requires a smaller number of CI tests of high orders. This reduces the complexity and run-time of the algorithm and increases the accuracy by diminishing the curse-of-dimensionality. When the RAI algorithm learned structures from databases representing synthetic problems, known networks and natural problems, it demonstrated superiority with respect to computational complexity, run-time, structural correctness and classification accuracy over the

Bayesian networks are frequently used to model statistical dependencies in data. Without prior knowledge of dependencies in the data, the structure of a Bayesian network is learned from the data. Bayesian network structure learning is commonly posed as an optimization problem where search is used to find structures that maximize a scoring function. Since the structure search space is superexponential in the number of variables in a network, heuristics are applied to constrain the search space of high-dimensional networks. Greedy hill climbing is then applied in the reduced search space. The constrained search space of high-dimensional networks contains many local maxima that greedy hill climbing cannot overcome. This issue has only been addressed by augmenting greedy search with TABU lists or random moves. This is not a holistic solution to the problem. By using a search algorithm that is global in nature, we are not confined to results in a particular region of the search space, like previous approaches. We present Model-Based Search (MBS) [1] applied to Bayesian network structure learning. MBS uses information gained during search to explore promising search space regions. Maintaining this search space information keeps a global view of the search task and helps find structures at higher maxima than greedy hill climbing. We show that MBS performs better than hill climbing in the Max-Min Parents and Children (MMPC) [30] search space and can find better high-dimensional network structures than other leading structure learning algorithms. 1

In this paper, we propose a recursive method for structural learning of directed acyclic graphs (DAGs), in which a problem of structural learning for a large DAG is first decomposed into two problems of structural learning for two small vertex subsets, each of which is then decomposed recursively into two problems of smaller subsets until none subset can be decomposed further. In our approach, search for separators of a pair of variables in a large DAG is localized to small subsets, and thus the approach can improve the efficiency of searches and the power of statistical tests for structural learning. We show how the recent advances in the learning of undirected graphical models can be employed to facilitate the decomposition. Simulations are given to demonstrate the performance of the proposed method.

This paper introduces constraint relaxation, a new strategy for learning the structure of Bayesian networks. Constraint relaxation identifies and “relaxes ” possibly inaccurate independence constraints on the structure of the model. We describe a heuristic algorithm for constraint relaxation that combines greedy search in the space of undirected skeletons with edge orientation based on the constraints. This approach produces significant improvements in the structural accuracy of the learned models compared to four well-known structure learning algorithms in an empirical evaluation using data sampled from both real-world and randomly generated networks. 1

Abstract. Constraint-based Bayesian network (BN) structure learning algorithms typically controlthe False Positive Rate(FPR)of their skeleton identification phase. The False Discovery Rate (FDR), however, may be of greater interest and methods for its utilization by these algorithms have been recently devised. We present a unified approach to BN skeleton identification FDR estimation and control and experimentally evaluate the performance of FDR estimators in both tasks over several networks. We demonstrate that estimation is too conservative for most networks and strong control at common FDR thresholds is not achieved with some networks; finally, we identify the possible causes of this situation. 1

Hybrid algorithms for learning the structure of Bayesian networks combine techniques from both the constraintbased and search-and-score paradigms of structure learning. One class of hybrid approaches uses a constraintbased algorithm to learn an undirected skeleton identifying edges that should appear in the final network. This skeleton is used to constrain the model space considered by a search-and-score algorithm to orient the edges and produce a final model structure. At small sample sizes, the performance of models learned using this hybrid approach do not achieve likelihood as high as models learned by unconstrained search. Low performance is a result of errors made by the skeleton identification algorithm, particularly false negative errors, which lead to an over-constrained search space. These errors are often attributed to “noisy” hypothesis tests that are run during skeleton identification. However, at least three specific sources of error have been identified in the literature: unsuitable hypothesis tests, lowpower hypothesis tests, and unexplained d-separation. No previous work has considered these sources of error in combination. We determine the relative importance of each source individually and in combination. We identify that low-power tests are the primary source of false negative errors, and show that these errors can be corrected by a novel application of statistical power analysis. The result is a new hybrid algorithm for learning the structure of Bayesian networks which produces models with equivalent likelihood to models produced by unconstrained greedy search, using only a fraction of the time. 1

Abstract — A Bayes net has qualitative and quantitative aspects: The qualitative aspect is its graphical structure that corresponds to correlations among the variables in the Bayes net. The quantitative aspects are the net parameters. This paper develops a hybrid criterion for learning Bayes net structures that is based on both aspects. We combine model selection criteria measuring data fit with correlation information from statistical tests: Given a sample d, search for a structure G that maximizes score(G, d), over the set of structures G that satisfy the dependencies detected in d. We rely on the statistical test only to accept conditional dependencies, not conditional independencies. We show how to adapt local search algorithms to accommodate the observed dependencies. Simulation studies with GES search and the BDeu/BIC scores provide evidence that the additional dependency information leads to Bayes nets that better fit the target model in distribution and structure. I.

Partial orientation and local structural learning of causal networks for prediction

� Algorithms exist that scale up to problems with thousands of variables [7] � Decent quality of learning Experimental Evaluation

The ecological sciences have benefited greatly from recent advances in wireless sensor technologies. These technologies allow researchers to deploy networks of automated sensors, which can monitor a landscape at very fine temporal and spatial scales. However, these networks are subject to harsh conditions, which lead to malfunctions in individual sensors and failures in network communications. The resulting data streams often exhibit incorrect data measurements and missing values. Identifying and correcting these is time-consuming and error-prone. We present a method for real-time automated data quality control (QC) that exploits the spatial and temporal correlations in the data to distinguish sensor failures from valid observations. The model adapts to each deployment site by learning a Bayesian network structure that captures spatial relationships between sensors, and it extends the structure to a dynamic Bayesian network to incorporate temporal correlations. This model is able to flag faulty observations and predict the true values of the missing or corrupt readings. The performance of the model is evaluated on data collected by the SensorScope Project. The results show that the spatiotemporal model demonstrates clear advantages over models that include only temporal or only spatial correlations, and that the model is capable of accurately imputing corrupted values. Categories and Subject Descriptors: I.2.6 [Artificial Intelligence]: Learning—Parameter learning; I.5.1 [Pattern Recognition]: Models—Statistical; structural; G.3 [Mathematics of Computing]: Probability and Statistics—Distribution functions; Markov processes; multivariate statistics