The identification of the governing equations of chaotic dynamical systems from data has recently emerged as a hot topic. While the seminal work by Brunton et al. reported proof-of-concepts for idealized observation setting for fully-observed systems, {\em i.e.} large signal-to-noise ratios and high-frequency sampling of all system variables, we here address the learning of data-driven representations of chaotic dynamics for partially-observed systems, including significant noise patterns and possibly lower and irregular sampling setting. Instead of considering training losses based on short-term prediction error like state-of-the-art learning-based schemes, we adopt a Bayesian formulation and state this issue as a data assimilation problem with unknown model parameters. To solve for the joint inference of the hidden dynamics and of model parameters, we combine neural-network representations and state-of-the-art assimilation schemes. Using iterative Expectation-Maximization (EM)-like procedures, the key feature of the proposed inference schemes is the derivation of the posterior of the hidden dynamics. Using a neural-network-based Ordinary Differential Equation (ODE) representation of these dynamics, we investigate two strategies: their combination to Ensemble Kalman Smoothers and Long Short-Term Memory (LSTM)-based variational approximations of the posterior. Through numerical experiments on the Lorenz-63 system with different noise and time sampling settings, we demonstrate the ability of the proposed schemes to recover and reproduce the hidden chaotic dynamics, including their Lyapunov characteristic exponents, when classic machine learning approaches fail.

In this paper we contribute a novel algorithm family, which generalizes many unsupervised techniques including unnormalized and energy models, and allows to infer different statistical modalities (e.g.~data likelihood and ratio between densities) from data samples. The proposed unsupervised technique Probabilistic Surface Optimization (PSO) views a neural network (NN) as a flexible surface which can be pushed according to loss-specific virtual stochastic forces, where a dynamical equilibrium is achieved when the point-wise forces on the surface become equal. Concretely, the surface is pushed up and down at points sampled from two different distributions, with overall up and down forces becoming functions of these two distribution densities and of force intensity magnitudes defined by loss of a particular PSO instance. The eventual force equilibrium upon convergence enforces the NN to be equal to various statistical functions depending on the used magnitude functions, such as data density. Furthermore, this dynamical-statistical equilibrium is extremely intuitive and useful, providing many implications and possible usages in probabilistic inference. Further, we provide new PSO-based approaches as demonstration of PSO exceptional usability. We also analyze PSO convergence and optimization stability, and relate them to the gradient similarity function over NN input space. Further, we propose new ways to improve the above stability. Finally, we present new instances of PSO, termed PSO-LDE, for data density estimation on logarithmic scale and also provide a new NN block-diagonal architecture for increased surface flexibility, which significantly improves estimation accuracy. Both PSO-LDE and the new architecture are combined together as a new density estimation technique. In our experiments we demonstrate this technique to produce highly accurate density estimation for 20D data.

Complex networks can be modeled as a probabilistic graphical model, where the interactions between binary variables, "spins", on nodes are described by a coupling matrix that is inferred from observations. The inverse statistical problem of finding direct interactions is difficult, especially for large systems, because of the exponential growth in the possible number of states and the possible number of networks. In the context of the experimental sciences, well-controlled perturbations can be applied to a system, shedding light on the internal structure of the network. Therefore, we propose a method to improve the accuracy and efficiency of inference by iteratively applying perturbations to a network that are advantageous under a Bayesian framework. The spectrum of the empirical Fisher information can be used as a measure for the difficulty of the inference during the training process. We significantly improve the accuracy and efficiency of inference in medium-sized networks based on this strategy with a reasonable number of experimental queries. Our method could be powerful in the analysis of complex networks as well as in the rational design of experiments.

In 1891, when the German biologist Hans Driesch split two-cell sea urchin embryos in half, he found that each of the separated cells then gave rise to its own complete, albeit smaller, larva. Somehow, the halves "knew" to change their entire developmental program: At that stage, the blueprint for what they would become had apparently not yet been drawn out, at least not in ink. Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences. Since then, scientists have been trying to understand what goes into making this blueprint, and how instructive it is. It's now known that some form of positional information makes genes variously switch on and off throughout the embryo, giving cells distinct identities based on their location.

Linear regression is the most basic and the most widely used technique in machine learning; yet for all its simplicity, studying it can unlock some of the most important concepts in statistics. If you have a basic undestanding of linear regression expressed as $ \hat{Y} \theta_0 \theta_1X$, but don't have a background in statistics and find statements like "ridge regression is equivalent to the maximum a posteriori (MAP) estimate with a zero-mean Gaussian prior" bewildering, then this post is for you. With a superficial goal of understanding that somewhat obtuse statement, its main objective is to explore the topic, starting from the standard formulation of linear regression, moving on to the probabilistic approach (maximum likelihood formulation) and from there to Bayesian linear regression. I'll use the $\theta$ character throughout to refer to the coefficients (weights) of a regression model, either explicitly broken out as $\theta_0$ and $\theta_1$ for intercept and slope respectively, or just $\theta$ referring to the vector of coefficients. I'll usually use the expression $\theta Tx_i$ for the prediction a model gives at $x_i$, the assumption being that a 1 has been added to the vector of values at $x_i$. 1 In the single predictor case, we know that the least squares fit is the line that minimizes the sum of the squared distances between observed data and predicted values, i.e. it minimizes the Residual Sum of Squares (RSS): These residuals are pretty important in how we reason about our model.

The 2020 Democratic candidates are out of the gate and the pollsters have the call! Bernie Sanders is leading by two lengths with Kamala Harris and Elizabeth Warren right behind, but Cory Booker and Beto O'Rourke are coming on fast! The political horse-race season is upon us and I bet I know what you are thinking: "Stop!" Every election we complain about horse-race coverage and every election we stay glued to it all the same. The problem with this kind of coverage is not that it's unimportant.

Nowadays the world is full of digital data, due to the large deployment of sensors, fast internet and more computational power to generate all such that data. This data is might be very useful to extract information and predict events, allowing us to control or profit from them. In order to achieve such goal, we need fast algorithms that are capable of finding features that could bring useful information. However, this is a nontrivial task, as data is very large and usual simple statistics are slow and inaccurate. Thus, the term data mining appeared to describe the problem of finding useful information in large data sets by integrating methods from many fields, like machine learning, statistics and database systems, spatial or temporal data analysis, pattern recognition, image and signal processing. In recent years many works have been done to use machine learning techniques in order to extract useful information from data.

Scalable Data Augmentation (SDA) provides a framework for training deep neural networks (DNNs). Our methodology exploits auxiliary hidden units which are designed to avoid backtracking and traverse local modes in an efficient way. This allows us to exploit recent advantages in high performance computing such as scalable linear algebra (CUDA, XLA). We show how to implement standard activation and objective functions, including ReLU (Polson and Ročková, 2018), logit (Zhou et al., 2012) and SVM (Mallick et al., 2005) are all available as data augmentation schemes. Data augmentation strategies are commonplace in statistical applications such as EM, ECM and MM algorithms, as they accelerate convergence and can use Nesterov acceleration (Nesterov, 1983).

For autonomous agents to successfully operate in real world, the ability to anticipate future motions of surrounding entities in the scene can greatly enhance their safety levels since potentially dangerous situations could be avoided in advance. While impressive results have been shown on predicting each agent's behavior independently, we argue that it is not valid to consider road entities individually since transitions of vehicle states are highly coupled. Moreover, as the predicted horizon becomes longer, modeling prediction uncertainties and multi-modal distributions over future sequences will turn into a more challenging task. In this paper, we address this challenge by presenting a multi-modal probabilistic prediction approach. The proposed method is based on a generative model and is capable of jointly predicting sequential motions of each pair of interacting agents. Most importantly, our model is interpretable, which can explain the underneath logic as well as obtain more reliability to use in real applications. A complicate real-world roundabout scenario is utilized to implement and examine the proposed method.

This note is concerned with an accurate and computationally efficient variational bayesian treatment of mixed-effects modelling. We focus on group studies, i.e. empirical studies that report multiple measurements acquired in multiple subjects. When approached from a bayesian perspective, such mixed-effects models typically rely upon a hierarchical generative model of the data, whereby both within- and between-subject effects contribute to the overall observed variance. The ensuing VB scheme can be used to assess statistical significance at the group level and/or to capture inter-individual differences. Alternatively, it can be seen as an adaptive regularization procedure, which iteratively learns the corresponding within-subject priors from estimates of the group distribution of effects of interest (cf. so-called "empirical bayes" approaches). We outline the mathematical derivation of the ensuing VB scheme, whose open-source implementation is available as part the VBA toolbox.