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. Optimzing 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 mazimization) 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 likelihood-based 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.

We study the problem of learning the best Bayesian network structure with respect to a decomposable score such as BDe, BIC or AIC. This problem is known to be NP-hard, which means that solving it becomes quickly infeasible as the number of variables increases. Nevertheless, in this paper we show that it is possible to learn the best Bayesian network structure with over 30 variables, which covers many practically interesting cases. Our algorithm is less complicated and more efficient than the techniques presented earlier. It can be easily parallelized, and offers a possibility for efficient exploration of the best networks consistent with different variable orderings. In the experimental part of the paper we compare the performance of the algorithm to the previous state-of-the-art algorithm. Free source-code and an online-demo can be found at http://b-course.hiit.fi/bene.

A central task in many applications is reasoning about processes that change in a continuous time. The mathematical framework of Continuous Time Markov Processes provides the basic foundations for modeling such systems. Recently, Nodelman et al introduced continuous time Bayesian networks (CTBNs), which allow a compact representation of continuous-time processes over a factored state space. In this paper, we introduce continuous time Markov networks (CTMNs), an alternative representation language that represents a different type of continuous-time dynamics. In many real life processes, such as biological and chemical systems, the dynamics of the process can be naturally described as an interplay between two forces - the tendency of each entity to change its state, and the overall fitness or energy function of the entire system. In our model, the first force is described by a continuous-time proposal process that suggests possible local changes to the state of the system at different rates. The second force is represented by a Markov network that encodes the fitness, or desirability, of different states; a proposed local change is then accepted with a probability that is a function of the change in the fitness distribution. We show that the fitness distribution is also the stationary distribution of the Markov process, so that this representation provides a characterization of a temporal process whose stationary distribution has a compact graphical representation. This allows us to naturally capture a different type of structure in complex dynamical processes, such as evolving biological sequences. We describe the semantics of the representation, its basic properties, and how it compares to CTBNs. We also provide algorithms for learning such models from data, and discuss its applicability to biological sequence evolution.

We consider in this paper the formulation of approximate inference in Bayesian networks as a problem of exact inference on an approximate network that results from deleting edges (to reduce treewidth). We have shown in earlier work that deleting edges calls for introducing auxiliary network parameters to compensate for lost dependencies, and proposed intuitive conditions for determining these parameters. We have also shown that our method corresponds to IBP when enough edges are deleted to yield a polytree, and corresponds to some generalizations of IBP when fewer edges are deleted. In this paper, we propose a different criteria for determining auxiliary parameters based on optimizing the KL-divergence between the original and approximate networks. We discuss the relationship between the two methods for selecting parameters, shedding new light on IBP and its generalizations. We also discuss the application of our new method to approximating inference problems which are exponential in constrained treewidth, including MAP and nonmyopic value of information.

We observe that certain large-clique graph triangulations can be useful to reduce both computational and space requirements when making queries on mixed stochastic/deterministic graphical models. We demonstrate that many of these large-clique triangulations are non-minimal and are thus unattainable via the variable elimination algorithm. We introduce ancestral pairs as the basis for novel triangulation heuristics and prove that no more than the addition of edges between ancestral pairs need be considered when searching for state space optimal triangulations in such graphs. Empirical results on random and real world graphs show that the resulting triangulations that yield significant speedups are almost always non-minimal. We also give an algorithm and correctness proof for determining if a triangulation can be obtained via elimination, and we show that the decision problem associated with finding optimal state space triangulations in this mixed stochastic/deterministic setting is NP-complete.

Tasks such as record linkage and multi-target tracking, which involve reconstructing the set of objects that underlie some observed data, are particularly challenging for probabilistic inference. Recent work has achieved efficient and accurate inference on such problems using Markov chain Monte Carlo (MCMC) techniques with customized proposal distributions. Currently, implementing such a system requires coding MCMC state representations and acceptance probability calculations that are specific to a particular application. An alternative approach, which we pursue in this paper, is to use a general-purpose probabilistic modeling language (such as BLOG) and a generic Metropolis-Hastings MCMC algorithm that supports user-supplied proposal distributions. Our algorithm gains flexibility by using MCMC states that are only partial descriptions of possible worlds; we provide conditions under which MCMC over partial worlds yields correct answers to queries. We also show how to use a context-specific Bayes net to identify the factors in the acceptance probability that need to be computed for a given proposed move. Experimental results on a citation matching task show that our general-purpose MCMC engine compares favorably with an application-specific system.

We consider a Bayesian method for learning the Bayesian network structure from complete data. Recently, Koivisto and Sood (2004) presented an algorithm that for any single edge computes its marginal posterior probability in O(n 2^n) time, where n is the number of attributes; the number of parents per attribute is bounded by a constant. In this paper we show that the posterior probabilities for all the n (n - 1) potential edges can be computed in O(n 2^n) total time. This result is achieved by a forward-backward technique and fast Moebius transform algorithms, which are of independent interest. The resulting speedup by a factor of about n^2 allows us to experimentally study the statistical power of learning moderate-size networks. We report results from a simulation study that covers data sets with 20 to 10,000 records over 5 to 25 discrete attributes

Most causal inference algorithms in the literature (e.g., Pearl (2000), Spirtes et al. (2000), Heckerman et al. (1999)) exploit an assumption usually referred to as the causal Faithfulness or Stability condition. In this paper, we highlight two components of the condition used in constraint-based algorithms, which we call "Adjacency-Faithfulness" and "Orientation-Faithfulness". We point out that assuming Adjacency-Faithfulness is true, it is in principle possible to test the validity of Orientation-Faithfulness. Based on this observation, we explore the consequence of making only the Adjacency-Faithfulness assumption. We show that the familiar PC algorithm has to be modified to be (asymptotically) correct under the weaker, Adjacency-Faithfulness assumption. Roughly the modified algorithm, called Conservative PC (CPC), checks whether Orientation-Faithfulness holds in the orientation phase, and if not, avoids drawing certain causal conclusions the PC algorithm would draw. However, if the stronger, standard causal Faithfulness condition actually obtains, the CPC algorithm is shown to output the same pattern as the PC algorithm does in the large sample limit. We also present a simulation study showing that the CPC algorithm runs almost as fast as the PC algorithm, and outputs significantly fewer false causal arrowheads than the PC algorithm does on realistic sample sizes. We end our paper by discussing how score-based algorithms such as GES perform when the Adjacency-Faithfulness but not the standard causal Faithfulness condition holds, and how to extend our work to the FCI algorithm, which allows for the possibility of latent variables.

We introduce priors and algorithms to perform Bayesian inference in Gaussian models defined by acyclic directed mixed graphs. Such a class of graphs, composed of directed and bi-directed edges, is a representation of conditional independencies that is closed under marginalization and arises naturally from causal models which allow for unmeasured confounding. Monte Carlo methods and a variational approximation for such models are presented. Our algorithms for Bayesian inference allow the evaluation of posterior distributions for several quantities of interest, including causal effects that are not identifiable from data alone but could otherwise be inferred where informative prior knowledge about confounding is available.

In many cases it makes sense to model a relationship symmetrically, not implying any particular directionality. Consider the classical example of a recommendation system where the rating of an item by a user should symmetrically be dependent on the attributes of both the user and the item. The attributes of the (known) relationships are also relevant for predicting attributes of entities and for predicting attributes of new relations. In recommendation systems, the exploitation of relational attributes is often referred to as collaborative filtering. Again, in many applications one might prefer to model the collaborative effect in a symmetrical way. In this paper we present a relational model, which is completely symmetrical. The key innovation is that we introduce for each entity (or object) an infinite-dimensional latent variable as part of a Dirichlet process (DP) model. We discuss inference in the model, which is based on a DP Gibbs sampler, i.e., the Chinese restaurant process. We extend the Chinese restaurant process to be applicable to relational modeling. Our approach is evaluated in three applications. One is a recommendation system based on the MovieLens data set. The second application concerns the prediction of the function of yeast genes/proteins on the data set of KDD Cup 2001 using a multi-relational model. The third application involves a relational medical domain. The experimental results show that our model gives significantly improved estimates of attributes describing relationships or entities in complex relational models.