Probabilistic networks are now fairly well established as practical representations of knowledge for reasoning under uncertainty, as demonstrated by an increasing number of successful applications in such domains as (medical) diagnosis and prognosis, planning, vision,

We investigate methods for parameter learning from incomplete data that is not missing at random. Likelihood-based methods then require the optimization of a profile likelihood that takes all possible missingness mechanisms into account. Optimizing this profile likelihood poses two main difficulties: multiple (local) maxima, and its very high-dimensional parameter space. In this paper a new method is presented for optimizing the profile likelihood that addresses the second difficulty: in the proposed AI&M (adjusting imputation and maximization) procedure the optimization is performed by operations in the space of data completions, rather than directly in the parameter space of the profile likelihood. We apply the AI&M method to learning parameters for Bayesian networks. The method is compared against conservative inference, which takes into account each possible data completion, and against EM. The results indicate that likelihoodbased inference is still feasible in the case of unknown missingness mechanisms, and that conservative inference is unnecessarily weak. On the other hand, our results also provide evidence that the EM algorithm is still quite effective when the data is not missing at random.

The Bayesian network (BN) formalism is a powerful representation for encoding domains characterised by uncertainty. However, before it can be used it must first be constructed, which is a major challenge for any real-life problem. There are two broad approaches, namely the hand-crafted approach, which relies on a human expert, and the data-driven approach, which relies on data. The former approach is useful, however issues such as human bias can introduce errors into the model. We have conducted a literature review of the expert-driven approach, and we have cherry-picked a number of common methods, and engineered a framework to assist non-BN experts with expert-driven construction of BNs. The latter construction approach uses algorithms to construct the model from a data set. However, construction from data is provably NP-hard [45]. To solve this problem, approximate, heuristic algorithms have been proposed; in particular, algorithms that assume an order between the nodes, therefore reducing the search space. However, traditionally, this approach relies on an expert providing the order