The vast majority of current machine learning algorithms are designed to predict single responses or a vector of responses, yet many types of response are more naturally organized as matrices or higher-order tensor objects where characteristics are shared across modes. We present a new machine learning algorithm BaTFLED (Bayesian Tensor Factorization Linked to External Data) that predicts values in a three-dimensional response tensor using input features for each of the dimensions. BaTFLED uses a probabilistic Bayesian framework to learn projection matrices mapping input features for each mode into latent representations that multiply to form the response tensor. By utilizing a Tucker decomposition, the model can capture weights for interactions between latent factors for each mode in a small core tensor. Priors that encourage sparsity in the projection matrices and core tensor allow for feature selection and model regularization. This method is shown to far outperform elastic net and neural net models on 'cold start' tasks from data simulated in a three-mode structure. Additionally, we apply the model to predict dose-response curves in a panel of breast cancer cell lines treated with drug compounds that was used as a Dialogue for Reverse Engineering Assessments and Methods (DREAM) challenge.

We evaluate the uncertainty quality in neural networks using anomaly detection. We extract uncertainty measures (e.g. entropy) from the predictions of candidate models, use those measures as features for an anomaly detector, and gauge how well the detector differentiates known from unknown classes. We assign higher uncertainty quality to candidate models that lead to better detectors. We also propose a novel method for sampling a variational approximation of a Bayesian neural network, called One-Sample Bayesian Approximation (OSBA). We experiment on two datasets, MNIST and CIFAR10. We compare the following candidate neural network models: Maximum Likelihood, Bayesian Dropout, OSBA, and --- for MNIST --- the standard variational approximation. We show that Bayesian Dropout and OSBA provide better uncertainty information than Maximum Likelihood, and are essentially equivalent to the standard variational approximation, but much faster.

The computational properties of neural systems are often thought to be implemented in terms of their network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single-unit (MSU) recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a state space representation of the dynamics, but would wish to have access to its governing equations for in-depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi-analytical maximum-likelihood estimation scheme for piecewise-linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation-Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, the approach is applied to MSU recordings from the rodent anterior cingulate cortex obtained during performance of a classical working memory task, delayed alternation. A model with 5 states turned out to be sufficient to capture the essential computational dynamics underlying task performance, including stimulus-selective delay activity. The estimated models were rarely multi-stable, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi-analytical (thus reasonably fast) maximum-likelihood estimation framework for PLRNNs that may enable to recover the relevant dynamics underlying observed neuronal time series, and directly link them to computational properties.

In our previous post, we showed the examples of using linear models and machine learning approach for forecasting sales time series. Sometimes we need to forecast not only more probable values of sales but also their distribution. Especially we need it in the risk analysis for assessing different risks related to sales dynamics. In this case we need to take into account sales distributions and dependencies between sales time series features (e.g. One can consider sales as a stochastic variable with some marginal distributions.

Rank aggregation based on pairwise comparisons over a set of items has a wide range of applications. Although considerable research has been devoted to the development of rank aggregation algorithms, one basic question is how to efficiently collect a large amount of high-quality pairwise comparisons for the ranking purpose. Because of the advent of many crowdsourcing services, a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare. Since different workers have different levels of reliability and different pairs have different levels of ambiguity, it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results. To this end, we model the active sampling problem in crowdsourced ranking as a Bayesian Markov decision process, which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint. We further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation. Experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost.

Bayesian Reasoning and Machine Learning by David Barber has a chapter on Approximate Sampling Christophe Andrieu et al. have written an introductory tutorial (pdf) on MCMC methods that covers most of the MCMC algorithms Dr. Daphne Koller offers an online course on Coursera, Probabilistic Graphical Models, which also covers the Gibbs Sampler and the Metropolis-Hastings Algorithm Dr. A. Taylan Cemgil has prepared very useful lecture notes (pdf) for his Monte Carlo methods course

Presbyterian reverend Thomas Bayes had no reason to suspect he'd make any lasting contribution to humankind. Born in England at the beginning of the 18th century, Bayes was a quiet and questioning man. He published only two works in his lifetime. In 1731, he wrote a defense of God's--and the British monarchy's--"divine benevolence," and in 1736, an anonymous defense of the logic of Isaac Newton's calculus. Yet an argument he wrote before his death in 1761 would shape the course of history.

The Perona-Malik model has been very successful at restoring images from noisy input. In this paper, we reinterpret the Perona-Malik model in the language of Gaussian scale mixtures and derive some extensions of the model. Specifically, we show that the expectation-maximization (EM) algorithm applied to Gaussian scale mixtures leads to the lagged-diffusivity algorithm for computing stationary points of the Perona-Malik diffusion equations. Moreover, we show how mean field approximations to these Gaussian scale mixtures lead to a modification of the lagged-diffusivity algorithm that better captures the uncertainties in the restoration. Since this modification can be hard to compute in practice we propose relaxations to the mean field objective to make the algorithm computationally feasible. Our numerical experiments show that this modified lagged-diffusivity algorithm often performs better at restoring textured areas and fuzzy edges than the unmodified algorithm. As a second application of the Gaussian scale mixture framework, we show how an efficient sampling procedure can be obtained for the probabilistic model, making the computation of the conditional mean and other expectations algorithmically feasible. Again, the resulting algorithm has a strong resemblance to the lagged-diffusivity algorithm. Finally, we show that a probabilistic version of the Mumford-Shah segementation model can be obtained in the same framework with a discrete edge-prior.

We introduce a class of network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by permitting the location of a new edge to explicitly depend on the structure of the graph, but being nonetheless statistically and computationally tractable. In the limit of infinite walk length, the model converges to an extension of the preferential attachment model---in this sense, it can be motivated alternatively by asking what preferential attachment is an approximation to. Theoretical properties, including the limiting degree sequence, are studied analytically. If the entire history of the graph is observed, parameters can be estimated by maximum likelihood. If only the final graph is available, its history can be imputed using MCMC. We develop a class of sequential Monte Carlo algorithms that are more generally applicable to sequential random graph models, and may be of interest in their own right. The model parameters can be recovered from a single graph generated by the model. Applications to data clarify the role of the random walk length as a length scale of interactions within the graph.

What is normal for a machine (or the System under observation) is environment-dependent and ever-changing in real life!