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A necessary step in the development of artificial intelligence is to enable a machine to represent how the world works, building an internal structure from data. This structure should hold a good trade-off between expressive power and querying efficiency. Bayesian networks have proven to be an effective and versatile tool for the task at hand. They have been applied to modeling knowledge in a variety of fields, ranging from bioinformatics to law, from image processing to economic risk analysis. A crucial aspect is learning the dependency graph of a Bayesian network from data. This task, called structure learning, is NP-hard and is the subject of intense, cutting-edge research. In short, it can be thought of as choosing one graph over the many candidates, grounding our reasoning over a collection of samples of the distribution generating the data. The number of possible graphs increases very quickly at the increase in the number of variables. Searching in this space, and selecting a graph over the others, becomes quickly burdensome. In this survey, we review the most relevant structure learning algorithms that have been proposed in the literature. We classify them according to the approach they follow for solving the problem and we also show alternatives for handling missing data and continuous variable. An extensive review of existing software tools is also given.

In this paper a new Monte-Carlo algorithm for the propagation of probabilities in Bayesian networks is proposed. This algorithm has two stages: in the ÿrst one an approximate propagation is carried out by means of a deletion sequence of the variables. In the second stage a sample is obtained using as sampling distribution the calculations of the ÿrst step. The di erent conÿgurations of the sample are weighted according to the importance sampling technique. We show how the use of probability trees to store and to approximate probability potentials, and a careful selection of the deletion sequence, make this algorithm able to propagate over large networks with extreme probabilities.

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