MCMC methods for sampling from the space of DAGs can mix poorly due to the local nature of the proposals that are commonly used. It has been shown that sampling from the space of node orders yields better results [FK03, EW06]. Recently, Koivisto and Sood showed how one can analytically marginalize over orders using dynamic programming (DP) [KS04, Koi06]. Their method computes the exact marginal posterior edge probabilities, thus avoiding the need for MCMC. Unfortunately, there are four drawbacks to the DP technique: it can only use modular priors, it can only compute posteriors over modular features, it is difficult to compute a predictive density, and it takes exponential time and space. We show how to overcome the first three of these problems by using the DP algorithm as a proposal distribution for MCMC in DAG space. We show that this hybrid technique converges to the posterior faster than other methods, resulting in more accurate structure learning and higher predictive likelihoods on test data.

In this paper we propose a novel gradient algorithm to learn a policy from an expert's observed behavior assuming that the expert behaves optimally with respect to some unknown reward function of a Markovian Decision Problem. The algorithm's aim is to find a reward function such that the resulting optimal policy matches well the expert's observed behavior. The main difficulty is that the mapping from the parameters to policies is both nonsmooth and highly redundant. Resorting to subdifferentials solves the first difficulty, while the second one is over- come by computing natural gradients. We tested the proposed method in two artificial domains and found it to be more reliable and efficient than some previous methods.

Scaled sparse linear regression jointly estimates the regression coefficients and noise level in a linear model. It chooses an equilibrium with a sparse regression method by iteratively estimating the noise level via the mean residual square and scaling the penalty in proportion to the estimated noise level. The iterative algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty levels above a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a convex minimization of a penalized joint loss function for the regression coefficients and noise level. Under mild regularity conditions, we prove that the scaled lasso simultaneously yields an estimator for the noise level and an estimated coefficient vector satisfying certain oracle inequalities for prediction, the estimation of the noise level and the regression coefficients. These inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise level estimator, including certain cases where the number of variables is of greater order than the sample size. Parallel results are provided for the least squares estimation after model selection by the scaled lasso. Numerical results demonstrate the superior performance of the proposed methods over an earlier proposal of joint convex minimization.

BDeu marginal likelihood score is a popular model selection criterion for selecting a Bayesian network structure based on sample data. This non-informative scoring criterion assigns same score for network structures that encode same independence statements. However, before applying the BDeu score, one must determine a single parameter, the equivalent sample size alpha. Unfortunately no generally accepted rule for determining the alpha parameter has been suggested. This is disturbing, since in this paper we show through a series of concrete experiments that the solution of the network structure optimization problem is highly sensitive to the chosen alpha parameter value. Based on these results, we are able to give explanations for how and why this phenomenon happens, and discuss ideas for solving this problem.

Sparse coding is an unsupervised learning algorithm that learns a succinct high-level representation of the inputs given only unlabeled data; it represents each input as a sparse linear combination of a set of basis functions. Originally applied to modeling the human visual cortex, sparse coding has also been shown to be useful for self-taught learning, in which the goal is to solve a supervised classification task given access to additional unlabeled data drawn from different classes than that in the supervised learning problem. Shift-invariant sparse coding (SISC) is an extension of sparse coding which reconstructs a (usually time-series) input using all of the basis functions in all possible shifts. In this paper, we present an efficient algorithm for learning SISC bases. Our method is based on iteratively solving two large convex optimization problems: The first, which computes the linear coefficients, is an L1-regularized linear least squares problem with potentially hundreds of thousands of variables. Existing methods typically use a heuristic to select a small subset of the variables to optimize, but we present a way to efficiently compute the exact solution. The second, which solves for bases, is a constrained linear least squares problem. By optimizing over complex-valued variables in the Fourier domain, we reduce the coupling between the different variables, allowing the problem to be solved efficiently. We show that SISC's learned high-level representations of speech and music provide useful features for classification tasks within those domains. When applied to classification, under certain conditions the learned features outperform state of the art spectral and cepstral features.

Rating prediction is an important application, and a popular research topic in collaborative filtering. However, both the validity of learning algorithms, and the validity of standard testing procedures rest on the assumption that missing ratings are missing at random (MAR). In this paper we present the results of a user study in which we collect a random sample of ratings from current users of an online radio service. An analysis of the rating data collected in the study shows that the sample of random ratings has markedly different properties than ratings of user-selected songs. When asked to report on their own rating behaviour, a large number of users indicate they believe their opinion of a song does affect whether they choose to rate that song, a violation of the MAR condition. Finally, we present experimental results showing that incorporating an explicit model of the missing data mechanism can lead to significant improvements in prediction performance on the random sample of ratings.

Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly identify all arrow orientations shared by all DAGs in a Markov equivalence class, given a member of that class. For DAG models with latent variables, maximal ancestral graphs (MAGs) provide a neat representation that facilitates model search. Earlier work (Ali et al. 2005) has identified a set of orientation rules sufficient to construct all arrowheads common to a Markov equivalence class of MAGs. In this paper, we provide extra rules sufficient to construct all common tails as well. We end up with a set of orientation rules sound and complete for identifying commonalities across a Markov equivalence class of MAGs, which is particularly useful for causal inference.

Distance metric learning is an important component for many tasks, such as statistical classification and content-based image retrieval. Existing approaches for learning distance metrics from pairwise constraints typically suffer from two major problems. First, most algorithms only offer point estimation of the distance metric and can therefore be unreliable when the number of training examples is small. Second, since these algorithms generally select their training examples at random, they can be inefficient if labeling effort is limited. This paper presents a Bayesian framework for distance metric learning that estimates a posterior distribution for the distance metric from labeled pairwise constraints. We describe an efficient algorithm based on the variational method for the proposed Bayesian approach. Furthermore, we apply the proposed Bayesian framework to active distance metric learning by selecting those unlabeled example pairs with the greatest uncertainty in relative distance. Experiments in classification demonstrate that the proposed framework achieves higher classification accuracy and identifies more informative training examples than the non-Bayesian approach and state-of-the-art distance metric learning algorithms.

Relational Markov Decision Processes are a useful abstraction for complex reinforcement learning problems and stochastic planning problems. Recent work developed representation schemes and algorithms for planning in such problems using the value iteration algorithm. However, exact versions of more complex algorithms, including policy iteration, have not been developed or analyzed. The paper investigates this potential and makes several contributions. First we observe two anomalies for relational representations showing that the value of some policies is not well defined or cannot be calculated for restricted representation schemes used in the literature. On the other hand, we develop a variant of policy iteration that can get around these anomalies. The algorithm includes an aspect of policy improvement in the process of policy evaluation and thus differs from the original algorithm. We show that despite this difference the algorithm converges to the optimal policy.

Counterfactual statements, e.g., "my headache would be gone had I taken an aspirin" are central to scientific discourse, and are formally interpreted as statements derived from "alternative worlds". However, since they invoke hypothetical states of affairs, often incompatible with what is actually known or observed, testing counterfactuals is fraught with conceptual and practical difficulties. In this paper, we provide a complete characterization of "testable counterfactuals," namely, counterfactual statements whose probabilities can be inferred from physical experiments. We provide complete procedures for discerning whether a given counterfactual is testable and, if so, expressing its probability in terms of experimental data.