REVIEWER 5: Yes, clarifying that we assume chordality is useful, and will revise the title, abstract and elsewhere to emphasize this assumption. REVIEWER 6: The reviewer's summary of the proof of Lemma 4 about the balancing condition is accurate. We may have been a bit pedantic in spelling out the details of the proof, but on the other hand, simply saying that the balancing condition "obviously" holds because of the running intersection property would not be very informative either, and we would rather err on the side of giving too much details rather than too little. The standard Bayesian approach we use for model learning is statistically consistent for choosing the correct dimensionality, since prior distribution assigned to model parameters acts as a regularizer. This property is so widely established in the literature that we did not consider it to be necessary to emphasize the aspect in the paper.

We investigate the problem of learning the structure of a Markov network from data. It is shown that the structure of such networks can be described in terms of constraints which enables the use of existing solver technology with optimization capabilities to compute optimal networks starting from initial scores computed from the data. To achieve efficient encodings, we develop a novel characterization of Markov network structure using a balancing condition on the separators between cliques forming the network. The resulting translations into propositional satisfiability and its extensions such as maximum satisfiability, satisfiability modulo theories, and answer set programming, enable us to prove optimal certain networks which have been previously found by stochastic search.

Sampling inference methods are computationally difficult to scale for many models in part because global dependencies can reduce opportunities for parallel computation. Without strict conditional independence structure among variables, standard Gibbs sampling theory requires sample updates to be performed sequentially, even if dependence between most variables is not strong. Empirical work has shown that some models can be sampled effectively by going "Hogwild" and simply running Gibbs updates in parallel with only periodic global communication, but the successes and limitations of such a strategy are not well understood. As a step towards such an understanding, we study the Hogwild Gibbs sampling strategy in the context of Gaussian distributions. We develop a framework which provides convergence conditions and error bounds along with simple proofs and connections to methods in numerical linear algebra. In particular, we show that if the Gaussian precision matrix is generalized diagonally dominant, then any Hogwild Gibbs sampler, with any update schedule or allocation of variables to processors, yields a stable sampling process with the correct sample mean.

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parameterization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increases quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.

We describe a set of fast, tractable methods for characterizing neural responses to high-dimensional sensory stimuli using a model we refer to as the generalized quadratic model (GQM). The GQM consists of a low-rank quadratic function followed by a point nonlinearity and exponential-family noise. The quadratic function characterizes the neuron's stimulus selectivity in terms of a set linear receptive fields followed by a quadratic combination rule, and the invertible nonlinearity maps this output to the desired response range.

What if there is a teacher who knows the learning goal and wants to design good training data for a machine learner? We propose an optimal teaching framework aimed at learners who employ Bayesian models. Our framework is expressed as an optimization problem over teaching examples that balance the future loss of the learner and the effort of the teacher. This optimization problem is in general hard. In the case where the learner employs conjugate exponential family models, we present an approximate algorithm for finding the optimal teaching set.

With simultaneous measurements from ever increasing populations of neurons, there is a growing need for sophisticated tools to recover signals from individual neurons. In electrophysiology experiments, this classically proceeds in a two-step process: (i) threshold the waveforms to detect putative spikes and (ii) cluster the waveforms into single units (neurons). We extend previous Bayesian nonparametric models of neural spiking to jointly detect and cluster neurons using a Gamma process model. Importantly, we develop an online approximate inference scheme enabling real-time analysis, with performance exceeding the previous state-of-theart. Via exploratory data analysis--using data with partial ground truth as well as two novel data sets--we find several features of our model collectively contribute to our improved performance including: (i) accounting for colored noise, (ii) detecting overlapping spikes, (iii) tracking waveform dynamics, and (iv) using multiple channels. We hope to enable novel experiments simultaneously measuring many thousands of neurons and possibly adapting stimuli dynamically to probe ever deeper into the mysteries of the brain.

Designing a principled and effective algorithm for learning deep architectures is a challenging problem. The current approach involves two training phases: a fully unsupervised learning followed by a strongly discriminative optimization. We suggest a deep learning strategy that bridges the gap between the two phases, resulting in a three-phase learning procedure. We propose to implement the scheme using a method to regularize deep belief networks with top-down information. The network is constructed from building blocks of restricted Boltzmann machines learned by combining bottom-up and top-down sampled signals. A global optimization procedure that merges samples from a forward bottom-up pass and a top-down pass is used.

This paper introduces a method for finding Bayesian networks for continuous variables in high-dimensional spaces. The paper assumes a Gaussian distribution of any particular random variable when conditioned on its parent nodes. A LASSO objective function is used to construct a sparse set of parent nodes for each random variable, subject to an additional constraint that the resulting structure be an acyclic graph. The network structure constraint is framed as an ordering problem, and an A* search algorithm is proposed which finds a directed acyclic graph which maximizes the LASSO objective function. The LASSO objective function, minus the DAG constraint, is used as an admissible heuristic in the A* search.

We address the problem of learning a sparse Bayesian network structure for continuous variables in a high-dimensional space. The constraint that the estimated Bayesian network structure must be a directed acyclic graph (DAG) makes the problem challenging because of the huge search space of network structures. Most previous methods were based on a two-stage approach that prunes the search space in the first stage and then searches for a network structure satisfying the DAG constraint in the second stage. Although this approach is effective in a lowdimensional setting, it is difficult to ensure that the correct network structure is not pruned in the first stage in a high-dimensional setting. In this paper, we propose a single-stage method, called A* lasso, that recovers the optimal sparse Bayesian network structure by solving a single optimization problem with A* search algorithm that uses lasso in its scoring system. Our approach substantially improves the computational efficiency of the well-known exact methods based on dynamic programming. We also present a heuristic scheme that further improves the efficiency of A* lasso without significantly compromising the quality of solutions. We demonstrate our approach on data simulated from benchmark Bayesian networks and real data.