There are several reasons why everyone isn't using Bayesian methods for regression modeling. One reason is that Bayesian modeling requires more thought: you need pesky things like priors, and you can't assume that if a procedure runs without throwing an error that the answers are valid. A second reason is that MCMC sampling -- the bedrock of practical Bayesian modeling -- can be slow compared to closed-form or MLE procedures. A third reason is that existing Bayesian solutions have either been highly-specialized (and thus inflexible), or have required knowing how to use a generalized tool like BUGS, JAGS, or Stan. This third reason has recently been shattered in the R world by not one but two packages: brms and rstanarm.

We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

Independent component analysis (ICA), as an approach to the blind source-separation (BSS) problem, has become the de-facto standard in many medical imaging settings. Despite successes and a large ongoing research effort, the limitation of ICA to square linear transformations have not been overcome, so that general INFOMAX is still far from being realized. As an alternative, we present feature analysis in medical imaging as a problem solved by Helmholtz machines, which include dimensionality reduction and reconstruction of the raw data under the same objective, and which recently have overcome major difficulties in inference and learning with deep and nonlinear configurations. We demonstrate one approach to training Helmholtz machines, variational auto-encoders (VAE), as a viable approach toward feature extraction with magnetic resonance imaging (MRI) data.

If a document is about travel, we may expect that short snippets of the document should also be about travel. We introduce a general framework for incorporating these types of invariances into a discriminative classifier. The framework imagines data as being drawn from a slice of a Levy process. If we slice the Levy process at an earlier point in time, we obtain additional pseudo-examples, which can be used to train the classifier. We show that this scheme has two desirable properties: it preserves the Bayes decision boundary, and it is equivalent to fitting a generative model in the limit where we rewind time back to 0. Our construction captures popular schemes such as Gaussian feature noising and dropout training, as well as admitting new generalizations.

The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make `helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these `helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.

Markov chains, named after Andrey Markov, are mathematical systems that hop from one "state" (a situation or set of values) to another. For example, if you made a Markov chain model of a baby's behavior, you might include "playing," "eating", "sleeping," and "crying" as states, which together with other behaviors could form a'state space': a list of all possible states. In addition, on top of the state space, a Markov chain tells you the probabilitiy of hopping, or "transitioning," from one state to any other state---e.g., the chance that a baby currently playing will fall asleep in the next five minutes without crying first. With two states (A and B) in our state space, there are 4 possible transitions (not 2, because a state can transition back into itself). In this two state diagram, the probability of transitioning from any state to any other state is 0.5.

This paper considers the computational power of constant size, dynamic Bayesian networks. Although discrete dynamic Bayesian networks are no more powerful than hidden Markov models, dynamic Bayesian networks with continuous random variables and discrete children of continuous parents are capable of performing Turing-complete computation. With modified versions of existing algorithms for belief propagation, such a simulation can be carried out in real time. This result suggests that dynamic Bayesian networks may be more powerful than previously considered. Relationships to causal models and recurrent neural networks are also discussed.

Fault detection in industrial plants is a hot research area as more and more sensor data are being collected throughout the industrial process. Automatic data-driven approaches are widely needed and seen as a promising area of investment. This paper proposes an effective machine learning algorithm to predict industrial plant faults based on classification methods such as penalized logistic regression, random forest and gradient boosted tree. A fault's start time and end time are predicted sequentially in two steps by formulating the original prediction problems as classification problems. The algorithms described in this paper won first place in the Prognostics and Health Management Society 2015 Data Challenge.

A thesis submitted for the degree of Doctor of Philosophy of The Australian National University. In this work we introduce several new optimisation methods for problems in machine learning. Our algorithms broadly fall into two categories: optimisation of finite sums and of graph structured objectives. The finite sum problem is simply the minimisation of objective functions that are naturally expressed as a summation over a large number of terms, where each term has a similar or identical weight. Such objectives most often appear in machine learning in the empirical risk minimisation framework in the non-online learning setting. The second category, that of graph structured objectives, consists of objectives that result from applying maximum likelihood to Markov random field models. Unlike the finite sum case, all the non-linearity is contained within a partition function term, which does not readily decompose into a summation. For the finite sum problem, we introduce the Finito and SAGA algorithms, as well as variants of each. For graph-structured problems, we take three complementary approaches. We look at learning the parameters for a fixed structure, learning the structure independently, and learning both simultaneously. Specifically, for the combined approach, we introduce a new method for encouraging graph structures with the "scale-free" property. For the structure learning problem, we establish SHORTCUT, a O(n^{2.5}) expected time approximate structure learning method for Gaussian graphical models. For problems where the structure is known but the parameters unknown, we introduce an approximate maximum likelihood learning algorithm that is capable of learning a useful subclass of Gaussian graphical models.

A Gaussian restricted Boltzmann machine (GRBM) is a Boltzmann machine defined on a bipartite graph and is an extension of usual restricted Boltzmann machines. A GRBM consists of two different layers: a visible layer composed of continuous visible variables and a hidden layer composed of discrete hidden variables. In this paper, we derive two different inference algorithms for GRBMs based on the naive mean-field approximation (NMFA). One is an inference algorithm for whole variables in a GRBM, and the other is an inference algorithm for partial variables in a GBRBM. We compare the two methods analytically and numerically and show that the latter method is better.