Although deep neural networks have been widely applied in various domains [19, 25, 27], usually its parameters are learned via the principle of maximum likelihood, hence its success crucially hinges on the availability of large scale datasets. When training rich models on small datasets, explicit regularization techniques are crucial to alleviate overfitting. Previous works have explored various regularization [39] and data augmentation [19, 38] techniques to learn diversified representations. In this paper, we look into an alternative direction by proposing an adaptive and data-dependent regularization method to encourage neurons of the same layer to share statistical strength. The goal of our method is to prevent overfitting when training (large) networks on small dataset. Our key insight stems from the famous argument by Efron [8] in the literature of the empirical Bayes method: It is beneficial to learn from the experience of others. From an algorithmic perspective, we argue that the connection weights of neurons in the same layer (row/column vectors of the weight matrix) will be correlated with each other through the backpropagation learning. Hence, by learning the correlations of the weight matrix, a neuron can "borrow statistical strength" from other neurons in the same layer.

We present new techniques for automatically constructing probabilistic programs for data analysis, interpretation, and prediction. These techniques work with probabilistic domain-specific data modeling languages that capture key properties of a broad class of data generating processes, using Bayesian inference to synthesize probabilistic programs in these modeling languages given observed data. We provide a precise formulation of Bayesian synthesis for automatic data modeling that identifies sufficient conditions for the resulting synthesis procedure to be sound. We also derive a general class of synthesis algorithms for domain-specific languages specified by probabilistic context-free grammars and establish the soundness of our approach for these languages. We apply the techniques to automatically synthesize probabilistic programs for time series data and multivariate tabular data. We show how to analyze the structure of the synthesized programs to compute, for key qualitative properties of interest, the probability that the underlying data generating process exhibits each of these properties. Second, we translate probabilistic programs in the domain-specific language into probabilistic programs in Venture, a general-purpose probabilistic programming system. The translated Venture programs are then executed to obtain predictions of new time series data and new multivariate data records. Experimental results show that our techniques can accurately infer qualitative structure in multiple real-world data sets and outperform standard data analysis methods in forecasting and predicting new data.

Gaussian processes are used in machine learning to learn input-output mappings from observed data. Gaussian process regression is based on imposing a Gaussian process prior on the unknown regressor function and statistically conditioning it on the observed data. In system identification, Gaussian processes are used to form time series prediction models such as non-linear finite-impulse response (NFIR) models as well as non-linear autoregressive (NARX) models. Gaussian process state-space models (GPSS) can be used to learn the dynamic and measurement models for a state-space representation of the input-output data. Temporal and spatio-temporal Gaussian processes can be directly used to form regressor on the data in the time domain. The aim of this article is to briefly outline the main directions in system identification methods using Gaussian processes.

The Gaussian process (GP) is a nonparametric prior distribution over functions indexed by time, space, or other high-dimensional index set. The GP is a flexible model yet its limitation is given by its very nature: it can only model Gaussian marginal distributions. To model non-Gaussian data, a GP can be warped by a nonlinear transformation (or warping) as performed by warped GPs (WGPs) and more computationally-demanding alternatives such as Bayesian WGPs and deep GPs. However, the WGP requires a numerical approximation of the inverse warping for prediction, which increases the computational complexity in practice. To sidestep this issue, we construct a novel class of warpings consisting of compositions of multiple elementary functions, for which the inverse is known explicitly. We then propose the compositionally-warped GP (CWGP), a non-Gaussian generative model whose expressiveness follows from its deep compositional architecture, and its computational efficiency is guaranteed by the analytical inverse warping. Experimental validation using synthetic and real-world datasets confirms that the proposed CWGP is robust to the choice of warpings and provides more accurate point predictions, better trained models and shorter computation times than WGP.

This paper considers the problem of estimating a low-rank matrix from the observation of all, or a subset, of its entries in the presence of Poisson noise. When we observe all the entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

We propose a new framework of XGBoost that predicts the entire conditional distribution of a univariate response variable. In particular, XGBoostLSS models all moments of a parametric distribution, i.e., mean, location, scale and shape (LSS), instead of the conditional mean only. Choosing from a wide range of continuous, discrete and mixed discrete-continuous distribution, modelling and predicting the entire conditional distribution greatly enhances the flexibility of XGBoost, as it allows to gain additional insight into the data generating process, as well as to create probabilistic forecasts from which prediction intervals and quantiles of interest can be derived. We present both a simulation study and real world examples that demonstrate the benefits of our approach.

We derive the functional form of mutual information (MI) from a set of design criteria and a principle of maximal sufficiency. The (MI) between two sets of propositions is a global quantifier of correlations and is implemented as a tool for ranking joint probability distributions with respect to said correlations. The derivation parallels the derivations of relative entropy with an emphasis on the behavior of independent variables. By constraining the functional $I$ according to special cases, we arrive at its general functional form and hence establish a clear meaning behind its definition. We also discuss the notion of sufficiency and offer a new definition which broadens its applicability.

The time series cluster kernel (TCK) provides a powerful tool for analysing multivariate time series subject to missing data. TCK is designed using an ensemble learning approach in which Bayesian mixture models form the base models. Because of the Bayesian approach, TCK can naturally deal with missing values without resorting to imputation and the ensemble strategy ensures robustness to hyperparameters, making it particularly well suited for unsupervised learning. However, TCK assumes missing at random and that the underlying missingness mechanism is ignorable, i.e. uninformative, an assumption that does not hold in many real-world applications, such as e.g. medicine. To overcome this limitation, we present a kernel capable of exploiting the potentially rich information in the missing values and patterns, as well as the information from the observed data. In our approach, we create a representation of the missing pattern, which is incorporated into mixed mode mixture models in such a way that the information provided by the missing patterns is effectively exploited. Moreover, we also propose a semi-supervised kernel, capable of taking advantage of incomplete label information to learn more accurate similarities. Experiments on benchmark data, as well as a real-world case study of patients described by longitudinal electronic health record data who potentially suffer from hospital-acquired infections, demonstrate the effectiveness of the proposed methods.

The framework of variational autoencoders allows us to efficiently learn deep latent-variable models, such that the model's marginal distribution over observed variables fits the data. Often, we're interested in going a step further, and want to approximate the true joint distribution over observed and latent variables, including the true prior and posterior distributions over latent variables. This is known to be generally impossible due to unidentifiability of the model. We address this issue by showing that for a broad family of deep latent-variable models, identification of the true joint distribution over observed and latent variables is actually possible up to a simple transformation, thus achieving a principled and powerful form of disentanglement. Our result requires a factorized prior distribution over the latent variables that is conditioned on an additionally observed variable, such as a class label or almost any other observation. We build on recent developments in nonlinear ICA, which we extend to the case with noisy, undercomplete or discrete observations, integrated in a maximum likelihood framework. The result also trivially contains identifiable flow-based generative models as a special case.

We consider probabilistic programming for birth-death models of evolution and introduce a new widely-applicable inference method that combines an extension of the alive particle filter (APF) with automatic Rao-Blackwellization via delayed sampling. Birth-death models of evolution are an important family of phylogenetic models of the diversification processes that lead to evolutionary trees. Probabilistic programming languages (PPLs) give phylogeneticists a new and exciting tool: their models can be implemented as probabilistic programs with just a basic knowledge of programming. The general inference methods in PPLs reduce the need for external experts, allow quick prototyping and testing, and accelerate the development and deployment of new models. We show how these birth-death models can be implemented as simple programs in existing PPLs, and demonstrate the usefulness of the proposed inference method for such models. For the popular BiSSE model the method yields an increase of the effective sample size and the conditional acceptance rate by a factor of 30 in comparison with a standard bootstrap particle filter. Although concentrating on phylogenetics, the extended APF is a general inference method that shows its strength in situations where particles are often assigned zero weight. In the case when the weights are always positive, the extra cost of using the APF rather than the bootstrap particle filter is negligible, making our method a suitable drop-in replacement for the bootstrap particle filter in probabilistic programming inference.