The machine learning community adopted the use of null hypothesis significance testing (NHST) in order to ensure the statistical validity of results. Many scientific fields however realized the shortcomings of frequentist reasoning and in the most radical cases even banned its use in publications. We should do the same: just as we have embraced the Bayesian paradigm in the development of new machine learning methods, so we should also use it in the analysis of our own results. We argue for abandonment of NHST by exposing its fallacies and, more importantly, offer better - more sound and useful - alternatives for it.

In this paper, we study the trade-offs of different inference approaches for Bayesian matrix factorisation methods, which are commonly used for predicting missing values, and for finding patterns in the data. In particular, we consider Bayesian nonnegative variants of matrix factorisation and tri-factorisation, and compare non-probabilistic inference, Gibbs sampling, variational Bayesian inference, and a maximum-a-posteriori approach. The variational approach is new for the Bayesian nonnegative models. We compare their convergence, and robustness to noise and sparsity of the data, on both synthetic and real-world datasets. Furthermore, we extend the models with the Bayesian automatic relevance determination prior, allowing the models to perform automatic model selection, and demonstrate its efficiency.

We present the first general purpose framework for marginal maximum a posteriori estimation of probabilistic program variables. By using a series of code transformations, the evidence of any probabilistic program, and therefore of any graphical model, can be optimized with respect to an arbitrary subset of its sampled variables. To carry out this optimization, we develop the first Bayesian optimization package to directly exploit the source code of its target, leading to innovations in problem-independent hyperpriors, unbounded optimization, and implicit constraint satisfaction; delivering significant performance improvements over prominent existing packages.

We study a two-level multiview learning with more than two views under the PAC-Bayesian framework. This approach, sometimes referred as late fusion, consists in learning sequentially multiple view-specific classifiers at the first level, and then combining these view-specific classifiers at the second level. Our main theoretical result is a generalization bound on the risk of the majority vote which exhibits a term of diversity in the predictions of the view-specific classifiers. From this result it comes out that controlling the trade-off between diversity and accuracy is a key element for multiview learning, which complements other results in multiview learning. Finally, we experiment our principle on multiview datasets extracted from the Reuters RCV1/RCV2 collection.

In the last few months, I have had several people contact me about their enthusiasm for venturing into the world of data science and using Machine Learning (ML) techniques to probe statistical regularities and build impeccable data-driven products. However, I've observed that some actually lack the necessary mathematical intuition and framework to get useful results. This is the main reason I decided to write this blog post. Recently, there has been an upsurge in the availability of many easy-to-use machine and deep learning packages such as scikit-learn, Weka, Tensorflow etc. Machine Learning theory is a field that intersects statistical, probabilistic, computer science and algorithmic aspects arising from learning iteratively from data and finding hidden insights which can be used to build intelligent applications. Despite the immense possibilities of Machine and Deep Learning, a thorough mathematical understanding of many of these techniques is necessary for a good grasp of the inner workings of the algorithms and getting good results. There are many reasons why the mathematics of Machine Learning is important and I'll highlight some of them below: What Level of Maths Do You Need?

Mixture-of-experts (MoE) models are a powerful paradigm for modeling of data arising from complex data generating processes (DGPs). In this article, we demonstrate how different MoE models can be constructed to approximate the underlying DGPs of arbitrary types of data. Due to the probabilistic nature of MoE models, we propose the maximum quasi-likelihood (MQL) estimator as a method for estimating MoE model parameters from data, and we provide conditions under which MQL estimators are consistent and asymptotically normal. The blockwise minorization-maximizatoin (blockwise-MM) algorithm framework is proposed as an all-purpose method for constructing algorithms for obtaining MQL estimators. An example derivation of a blockwise-MM algorithm is provided. We then present a method for constructing information criteria for estimating the number of components in MoE models and provide justification for the classic Bayesian information criterion (BIC). We explain how MoE models can be used to conduct classification, clustering, and regression and we illustrate these applications via a pair of worked examples.

While Bayesian methods are praised for their ability to incorporate useful prior knowledge, in practice, convenient priors that allow for computationally cheap or tractable inference are commonly used. In this paper, we investigate the following question: for a given model, is it possible to compute an inference result with any convenient false prior, and afterwards, given any target prior of interest, quickly transform this result into the target posterior? A potential solution is to use importance sampling (IS). However, we demonstrate that IS will fail for many choices of the target prior, depending on its parametric form and similarity to the false prior. Instead, we propose prior swapping, a method that leverages the pre-inferred false posterior to efficiently generate accurate posterior samples under arbitrary target priors. Prior swapping lets us apply less-costly inference algorithms to certain models, and incorporate new or updated prior information "post-inference". We give theoretical guarantees about our method, and demonstrate it empirically on a number of models and priors.

The Ornstein-Uhlenbeck, also called mean-reverting diffusion process, describes a process which evolves following a deterministic linear part with an added Gaussian noise, similarly to a vectorautoregressive process in discrete time. This model is ubiquitous in quantitative finance, for instance the one-dimensional version is used for modeling rates and is called the Vasicek model [Hul09]. In a multidimensional setting, it can be therefore used to describe systems with linear interactions perturbed by Gaussian noise, see Figure 1 below. Among many others, an example of application is inter-bank lending [CFS15, FI13], where lending is a flux of reserves and is proportional to the difference in reserves. A natural question is therefore how to estimate the interaction structure from the observation of the process. Unfortunately, the optimal solution based on the maximum likelihood estimator (MLE) is typically quite inaccurate in high-dimensional settings, because of the well-known curse of dimensionality, see for instance [BvdG11]. However, in real-world applications, the interaction structure is sparse: in the example mentioned above, banks have typically only a few lending partners [GG14, GSV15, BBvL15], as the lending arrangements are typically done on a personal level.

We develop a model in which interactions between nodes of a dynamic network are counted by non homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then a maximum-likelihood estimator is developed to estimate non parametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology.

In this paper we provide a conceptual overview of latent variable models within a probabilistic modeling framework, an overview that emphasizes the compositional nature and the interconnectedness of the seemingly disparate models commonly encountered in statistical practice.