Bayesian network learning with cutting planes

The problem of learning the structure of Bayesian networks from complete discrete data with a limit on parent set size is considered. Learning is cast explicitly as an optimisation problem where the goal is to find a BN structure which maximises log marginal likelihood (BDe score). Integer programming, specifically the SCIP framework, is used to solve this optimisation problem. Acyclicity constraints are added to the integer program (IP) during solving in the form of cutting planes. Finding good cutting planes is the key to the success of the approach -the search for such cutting planes is effected using a sub-IP. Results show that this is a particularly fast method for exact BN learning

First-order integer programming for MAP problems

Finding the most probable (MAP) model in SRL frameworks such as Markov logic and Problog can, in principle, be solved by encoding the problem as a `grounded-out' mixed integer program (MIP). However, useful first-order structure disappears in this process motivating the development of first-order MIP approaches. Here we present mfoilp, one such approach. Since the syntax and semantics of mfoilp is essentially the same as existing approaches we focus here mainly on implementation and algorithmic issues. We start with the (conceptually) simple problem of using a logic program to generate a MIP instance before considering more ambitious exploitation of first-order representations.Comment: corrected typo

Finding Minimal Cost Herbrand Models with Branch-Cut-and-Price

Given (1) a set of clauses $T$ in some first-order language $\cal L$ and (2) a cost function $c : B_{{\cal L}} \rightarrow \mathbb{R}_{+}$, mapping each ground atom in the Herbrand base $B_{{\cal L}}$ to a non-negative real, then the problem of finding a minimal cost Herbrand model is to either find a Herbrand model $\cal I$ of $T$ which is guaranteed to minimise the sum of the costs of true ground atoms, or establish that there is no Herbrand model for $T$. A branch-cut-and-price integer programming (IP) approach to solving this problem is presented. Since the number of ground instantiations of clauses and the size of the Herbrand base are both infinite in general, we add the corresponding IP constraints and IP variables `on the fly' via `cutting' and `pricing' respectively. In the special case of a finite Herbrand base we show that adding all IP variables and constraints from the outset can be advantageous, showing that a challenging Markov logic network MAP problem can be solved in this way if encoded appropriately

Online Causal Structure Learning in the Presence of Latent Variables

We present two online causal structure learning algorithms which can track changes in a causal structure and process data in a dynamic real-time manner. Standard causal structure learning algorithms assume that causal structure does not change during the data collection process, but in real-world scenarios, it does often change. Therefore, it is inappropriate to handle such changes with existing batch-learning approaches, and instead, a structure should be learned in an online manner. The online causal structure learning algorithms we present here can revise correlation values without reprocessing the entire dataset and use an existing model to avoid relearning the causal links in the prior model, which still fit data. Proposed algorithms are tested on synthetic and real-world datasets, the latter being a seasonally adjusted commodity price index dataset for the U.S. The online causal structure learning algorithms outperformed standard FCI by a large margin in learning the changed causal structure correctly and efficiently when latent variables were present.Comment: 16 pages, 9 figures, 2 table