Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for $\ell_1$-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian sparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with a theoretically guaranteed uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochatic gradient Langevin dynamics (SGLD) which gives a computationally tractable way of Bayesian posterior sampling within our framework. Bayesian SINDy autoencoder achieves better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification suggested by the experimental studies. The Bayesian SINDy autoencoder can be applied to real video data, with accurate physics discovery which correctly identifies the governing equation and provides a close estimate for standard physics constants like gravity $g$, for example, in videos of a pendulum.

Probabilistic models for sequential data are the basis for a variety of applications concerned with processing timely ordered information. The predominant approach in this domain is given by recurrent neural networks, implementing either an approximate Bayesian approach (e.g. Variational Autoencoders or Generative Adversarial Networks) or a regression-based approach, i.e. variations of Mixture Density networks (MDN). In this paper, we focus on the $\mathcal{N}$-MDN variant, which parameterizes (mixtures of) probabilistic B\'ezier curves ($\mathcal{N}$-Curves) for modeling stochastic processes. While in favor in terms of computational cost and stability, MDNs generally fall behind approximate Bayesian approaches in terms of expressiveness. Towards this end, we present an approach for closing this gap by enabling full Bayesian inference on top of $\mathcal{N}$-MDNs. For this, we show that $\mathcal{N}$-Curves are a special case of Gaussian processes (denoted as $\mathcal{N}$-GP) and then derive corresponding mean and kernel functions for different modalities. Following this, we propose the use of the $\mathcal{N}$-MDN as a data-dependent generator for $\mathcal{N}$-GP prior distributions. We show the advantages granted by this combined model in an application context, using human trajectory prediction as an example.

Desirability can be understood as an extension of Anscombe and Aumann's Bayesian decision theory to sets of expected utilities. At the core of desirability lies an assumption of linearity of the scale in which rewards are measured. It is a traditional assumption used to derive the expected utility model, which clashes with a general representation of rational decision making, though. Allais has, in particular, pointed this out in 1953 with his famous paradox. We note that the utility scale plays the role of a closure operator when we regard desirability as a logical theory. This observation enables us to extend desirability to the nonlinear case by letting the utility scale be represented via a general closure operator. The new theory directly expresses rewards in actual nonlinear currency (money), much in Savage's spirit, while arguably weakening the founding assumptions to a minimum. We characterise the main properties of the new theory both from the perspective of sets of gambles and of their lower and upper prices (previsions). We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.

This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statistical model is not one-to-one. In nonidentifiable models, both the likelihood function and the posterior distribution have singularities in general, hence it was difficult to analyze their statistical properties. However, from the end of the 20th century, new theory and methodology based on algebraic geometry have been established which enables us to investigate such models and machines in the real world. In this article, the following results in recent advances are reported. First, we explain the framework of Bayesian statistics and introduce a new perspective from the birational geometry. Second, two mathematical solutions are derived based on algebraic geometry. An appropriate parameter space can be found by a resolution map, which makes the posterior distribution be normal crossing and the log likelihood ratio function be well-defined. Third, three applications to statistics are introduced. The posterior distribution is represented by the renormalized form, the asymptotic free energy is derived, and the universal formula among the generalization loss, the cross validation, and the information criterion is established. Two mathematical solutions and three applications to statistics based on algebraic geometry reported in this article are now being used in many practical fields in data science and artificial intelligence.

Through ML we can address SE problems that cannot be completely algorithmically modeled, or for which existing solutions do not provide satisfactory results yet (e.g., defect/fault detection [16, 165, 180]). In addition, ML finds application in SE tasks where data cannot be easily analyzed with other algorithms (e.g., software requirements, code comments, code reviews, issues [9, 91, 174]). Another important aspect of ML is that it can significantly reduce manual effort in common SE tasks (e.g., automatic program repair [157], code suggestion [61], defect prediction [19], malware detection [147], feature location [40]) with great accuracy results [146, 164]. In fields such as health informatics ML and SE are considered complementary disciplines, since the growing scale and complexity of healthcare datasets have posed a challenge for clinical practice and medical research, requiring new engineering approaches from both fields [38]. In the early nineties, Huff and Selfridge [68] recognized the need for creating software systems that partially take some responsibility for their own evolution, offering the ability to implement, measure, and assess changes easily. These changes should also contribute to the overall improvement of the corresponding systems [142].

Variational data assimilation and deep learning share many algorithmic aspects in common. While the former focuses on system state estimation, the latter provides great inductive biases to learn complex relationships. We here design a hybrid architecture learning the assimilation task directly from partial and noisy observations, using the mechanistic constraint of the 4DVAR algorithm. Finally, we show in an experiment that the proposed method was able to learn the desired inversion with interesting regularizing properties and that it also has computational interests.

Given an environment (e.g., a simulator) for evaluating samples in a specified design space and a set of weighted evaluation metrics -- one can use Theta-Resonance, a single-step Markov Decision Process (MDP), to train an intelligent agent producing progressively more optimal samples. In Theta-Resonance, a neural network consumes a constant input tensor and produces a policy as a set of conditional probability density functions (PDFs) for sampling each design dimension. We specialize existing policy gradient algorithms in deep reinforcement learning (D-RL) in order to use evaluation feedback (in terms of cost, penalty or reward) to update our policy network with robust algorithmic stability and minimal design evaluations. We study multiple neural architectures (for our policy network) within the context of a simple SoC design space and propose a method of constructing synthetic space exploration problems to compare and improve design space exploration (DSE) algorithms. Although we only present categorical design spaces, we also outline how to use Theta-Resonance in order to explore continuous and mixed continuous-discrete design spaces.

Uncertainty plays a crucial role in the machine learning field. Both model trustworthiness and performance require the understanding of uncertainty, especially for models used in high-stake applications where errors can cause cataclysmic consequences, such as medical diagnosis and autonomous driving. Accordingly, uncertainty decomposition and quantification have attracted more and more attention in recent years. This short report aims to demystify the notion of uncertainty decomposition through an introduction to two types of uncertainty and several decomposition exemplars, including maximum likelihood estimation, Gaussian processes, deep neural network, and ensemble learning. In the end, cross connections to other topics in this seminar and two conclusions are provided.

We describe a general approach for maximum a posteriori (MAP) inference in a class of discrete-continuous factor graphs commonly encountered in robotics applications. While there are openly available tools providing flexible and easy-to-use interfaces for specifying and solving inference problems formulated in terms of either discrete or continuous graphical models, at present, no similarly general tools exist enabling the same functionality for hybrid discrete-continuous problems. We aim to address this problem. In particular, we provide a library, DC-SAM, extending existing tools for inference problems defined in terms of factor graphs to the setting of discrete-continuous models. A key contribution of our work is a novel solver for efficiently recovering approximate solutions to discrete-continuous inference problems. The key insight to our approach is that while joint inference over continuous and discrete state spaces is often hard, many commonly encountered discrete-continuous problems can naturally be split into a "discrete part" and a "continuous part" that can individually be solved easily. Leveraging this structure, we optimize discrete and continuous variables in an alternating fashion. In consequence, our proposed work enables straightforward representation of and approximate inference in discrete-continuous graphical models. We also provide a method to approximate the uncertainty in estimates of both discrete and continuous variables. We demonstrate the versatility of our approach through its application to distinct robot perception applications, including robust pose graph optimization, and object-based mapping and localization.

The study of quasar light curves poses two problems: inference of the power spectrum and interpolation of an irregularly sampled time series. A baseline approach to these tasks is to interpolate a time series with a Damped Random Walk (DRW) model, in which the spectrum is inferred using Maximum Likelihood Estimation (MLE). However, the DRW model does not describe the smoothness of the time series, and MLE faces many problems in terms of optimization and numerical precision. In this work, we introduce a new stochastic model that we call $\textit{Convolved Damped Random Walk}$ (CDRW). This model introduces a concept of smoothness to a DRW, which enables it to describe quasar spectra completely. We also introduce a new method of inference of Gaussian process parameters, which we call $\textit{Neural Inference}$. This method uses the powers of state-of-the-art neural networks to improve the conventional MLE inference technique. In our experiments, the Neural Inference method results in significant improvement over the baseline MLE (RMSE: $0.318 \rightarrow 0.205$, $0.464 \rightarrow 0.444$). Moreover, the combination of both the CDRW model and Neural Inference significantly outperforms the baseline DRW and MLE in interpolating a typical quasar light curve ($\chi^2$: $0.333 \rightarrow 0.998$, $2.695 \rightarrow 0.981$). The code is published on GitHub.