Knowledge of biological processes captured in such equations, when solutions to them match measurements made from the system of interest, help confirm our understanding of systems level function. Examples of such models include cell cycle progression (Chen et al., 2000), integrate and fire generation of heart pacemaker pulses (Zhang et al., 2000) and cellular behavior in synchrony with the circadian cycle (Leloup and Goldbeter, 2003). A particular appeal of modeling is that models can be interrogated with what if type questions to improve our understanding of the system, or be used to make quantitative predictions in domains in which measurements are unavailable. A central issue in developing computational models of biological systems is setting parameters such as rate constants of biochemical reactions, synthesis and decay rates of macromolecules, delays incurred in transcription of genes and translation of proteins, and sharpness of nonlinear effects (Hill coefficient) are examples of such parameters. Parameter values are usually determined by conducting in vitro experiments (e.g.

The task of learning a BN can be divided into two subtasks: (1) structural learning, i.e., identification of the topology of the BN, and (2) parametric learning, i.e., estimation of the numerical parameters (conditional probabilities) for a given network topology. In particular, the most challenging task of the two is the one of learning the structure of a BN. Different methods have been proposed to face this problem, and they can be classified into two categories [4, 5]: (1) methods based on detecting conditional independencies, also known as constraint-based methods, and (2) score search methods, also known as score-based approaches. As discussed in [6], the input of the former algorithms is a set of conditional independence relations between subsets of variables, which are used to build a BN that represents a large percentage (and, whenever possible, all) of these relations [7]. However, the number of conditional independence tests that such methods should perform is exponential and, thus, approximation techniques are required.

Video analytics requires operating with large amounts of data. Compressive sensing allows to reduce the number of measurements required to represent the video using the prior knowledge of sparsity of the original signal, but it imposes certain conditions on the design matrix. The Bayesian compressive sensing approach relaxes the limitations of the conventional approach using the probabilistic reasoning and allows to include different prior knowledge about the signal structure. This paper presents two Bayesian compressive sensing methods for autonomous object detection in a video sequence from a static camera. Their performance is compared on the real datasets with the non-Bayesian greedy algorithm. It is shown that the Bayesian methods can provide the same accuracy as the greedy algorithm but much faster; or if the computational time is not critical they can provide more accurate results.

Learning DAG or Bayesian network models is an important problem in multi-variate causal inference. However, a number of challenges arises in learning large-scale DAG models including model identifiability and computational complexity since the space of directed graphs is huge. In this paper, we address these issues in a number of steps for a broad class of DAG models where the noise or variance is signal-dependent. Firstly we introduce a new class of identifiable DAG models, where each node has a distribution where the variance is a quadratic function of the mean (QVF DAG models). Our QVF DAG models include many interesting classes of distributions such as Poisson, Binomial, Geometric, Exponential, Gamma and many other distributions in which the noise variance depends on the mean. We prove that this class of QVF DAG models is identifiable, and introduce a new algorithm, the OverDispersion Scoring (ODS) algorithm, for learning large-scale QVF DAG models. Our algorithm is based on firstly learning the moralized or undirected graphical model representation of the DAG to reduce the DAG search-space, and then exploiting the quadratic variance property to learn the causal ordering. We show through theoretical results and simulations that our algorithm is statistically consistent in the high-dimensional p>n setting provided that the degree of the moralized graph is bounded and performs well compared to state-of-the-art DAG-learning algorithms.

Sparse regression problems arise often in various applications, e.g., model selection, compressive sensing, EEG source localisation and gene modelling [1], [2]. One of the Bayesian approaches to force the coefficients being zeros is the spike and slab prior [3]: each component is modelled as a mixture of spike, that is the delta-function in zero, and slab, that is some vague distribution. Following the Bayesian approach, latent variables that are indicators of spikes are added to the model [4] and the relevant distribution is placed over them [5]. In this model each component is modelled to be spike or slab independently. However, in many applications nonzero elements tend to appear in groups forming an unknown structure: wavelet coefficients of images are usually organised in trees [6], chromosomes have a spatial structure along the genome [2]. We propose an extension of the spike and slab model by imposing a hierarchical Gaussian process (GP) prior on the latent variables. Such hierarchical prior allows to model spatial structural dependencies for coefficients that can evolve in time. The new model is flexible as spatial and temporal dependencies are decoupled by different levels of the hierarchical GP prior.

There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster.

We introduce Fisher consistency in the sense of unbiasedness as a desirable property for estimators of class prior probabilities. Lack of Fisher consistency could be used as a criterion to dismiss estimators that are unlikely to deliver precise estimates in test datasets under prior probability and more general dataset shift. The usefulness of this unbiasedness concept is demonstrated with three examples of classifiers used for quantification: Adjusted Classify & Count, EM-algorithm and CDE-Iterate. We find that Adjusted Classify & Count and EM-algorithm are Fisher consistent. A counter-example shows that CDE-Iterate is not Fisher consistent and, therefore, cannot be trusted to deliver reliable estimates of class probabilities.

The scalable calculation of matrix determinants has been a bottleneck to the widespread application of many machine learning methods such as determinantal point processes, Gaussian processes, generalised Markov random fields, graph models and many others. In this work, we estimate log determinants under the framework of maximum entropy, given information in the form of moment constraints from stochastic trace estimation. The estimates demonstrate a significant improvement on state-of-the-art alternative methods, as shown on a wide variety of UFL sparse matrices. By taking the example of a general Markov random field, we also demonstrate how this approach can significantly accelerate inference in large-scale learning methods involving the log determinant.

Written by Chris Fonnesbeck, Assistant Professor of Biostatistics, Vanderbilt University Medical Center. You can view, fork, and play with this project on the Domino data science platform. A common applied statistics task involves building regression models to characterize non-linear relationships between variables. It is possible to fit such models by assuming a particular non-linear functional form, such as a sinusoidal, exponential, or polynomial function, to describe one variable's response to the variation in another. Unless this relationship is obvious from the outset, however, it involves possibly extensive model selection procedures to ensure the most appropriate model is retained. Alternatively, a non-parametric approach can be adopted by defining a set of knots across the variable space and use a spline or kernel regression to describe arbitrary non-linear relationships.

Probabilistic models with discrete latent variables naturally capture datasets composed of discrete classes. However, they are difficult to train efficiently, since backpropagation through discrete variables is generally not possible. We present a novel method to train a class of probabilistic models with discrete latent variables using the variational autoencoder framework, including backpropagation through the discrete latent variables. The associated class of probabilistic models comprises an undirected discrete component and a directed hierarchical continuous component. The discrete component captures the distribution over the disconnected smooth manifolds induced by the continuous component. As a result, this class of models efficiently learns both the class of objects in an image, and their specific realization in pixels, from unsupervised data, and outperforms state-of-the-art methods on the permutation-invariant MNIST, Omniglot, and Caltech-101 Silhouettes datasets.