Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for the purpose of representing complex probability distributions. While such models have achieved enormous success in machine learning, particularly for generative modeling and density estimation, little is known about their statistical properties. This work establishes the first general nonparametric statistical convergence analysis for distribution learning via ODE models trained through likelihood maximization. We first prove a convergence theorem applicable to arbitrary velocity field classes $\mathcal{F}$ satisfying certain simple boundary constraints. This general result captures the trade-off between approximation error (`bias') and the complexity of the ODE model (`variance'). We show that the latter can be quantified via the $C^1$-metric entropy of the class $\mathcal F$. We then apply this general framework to the setting of $C^k$-smooth target densities, and establish nearly minimax-optimal convergence rates for two relevant velocity field classes $\mathcal F$: $C^k$ functions and neural networks. The latter is the practically important case of neural ODEs. Our proof techniques require a careful synthesis of (i) analytical stability results for ODEs, (ii) classical theory for sieved M-estimators, and (iii) recent results on approximation rates and metric entropies of neural network classes. The results also provide theoretical insight on how the choice of velocity field class, and the dependence of this choice on sample size $n$ (e.g., the scaling of width, depth, and sparsity of neural network classes), impacts statistical performance.

Dictionary learning methods continue to gain popularity for the solution of challenging inverse problems. In the dictionary learning approach, the computational forward model is replaced by a large dictionary of possible outcomes, and the problem is to identify the dictionary entries that best match the data, akin to traditional query matching in search engines. Sparse coding techniques are used to guarantee that the dictionary matching identifies only few of the dictionary entries, and dictionary compression methods are used to reduce the complexity of the matching problem. In this article, we propose a work flow to facilitate the dictionary matching process. First, the full dictionary is divided into subdictionaries that are separately compressed. The error introduced by the dictionary compression is handled in the Bayesian framework as a modeling error. Furthermore, we propose a new Bayesian data-driven group sparsity coding method to help identify subdictionaries that are not relevant for the dictionary matching. After discarding irrelevant subdictionaries, the dictionary matching is addressed as a deflated problem using sparse coding. The compression and deflation steps can lead to substantial decreases of the computational complexity. The effectiveness of compensating for the dictionary compression error and using the novel group sparsity promotion to deflate the original dictionary are illustrated by applying the methodology to real world problems, the glitch detection in the LIGO experiment and hyperspectral remote sensing.

While Evidence Theory (Demster-Shafer Theory, Belief Functions Theory) is being increasingly used in data fusion, its potentialities in the Social and Life Sciences are often obscured by lack of awareness of its distinctive features. With this paper we stress that Evidence Theory can express the uncertainty deriving from the fear that events may materialize, that one has not been able to figure out. By contrast, Probability Theory must limit itself to the possibilities that a decision-maker is currently envisaging. Subsequently, we illustrate how Dempster-Shafer's combination rule relates to Bayes' Theorem for various versions of Probability Theory and discuss which applications of Information Theory can be enhanced by Evidence Theory. Finally, we illustrate our claims with an example where Evidence Theory is used to make sense of the partially overlapping, partially contradictory solutions that appear in an auditing exercise.

This paper studies the cosine as basis function for the approximation of univariate and continuous functions without memory. This work studies a supervised learning to obtain the approximation coefficients, instead of using the Discrete Cosine Transform (DCT). Due to the finite dynamics and orthogonality of the cosine basis functions, simple gradient algorithms, such as the Normalized Least Mean Squares (NLMS), can benefit from it and present a controlled and predictable convergence time and error misadjustment. Due to its simplicity, the proposed technique ranks as the best in terms of learning quality versus complexity, and it is presented as an attractive technique to be used in more complex supervised learning systems. Simulations illustrate the performance of the approach. This paper celebrates the 50th anniversary of the publication of the DCT by Nasir Ahmed in 1973.

We outline how to create a mechanism that provides an optimal way to elicit, from an arbitrary group of experts, the probability of the truth of an arbitrary logical proposition together with collective information that has an explicit form and interprets this probability. Namely, we provide strong arguments for the possibility of the development of a self-resolving prediction market with play money that incentivizes direct information exchange between experts. Such a system could, in particular, motivate simultaneously many experts to collectively solve scientific or medical problems in a very efficient manner. We also note that in our considerations, experts are not assumed to be Bayesian.

Devising deep latent variable models for multi-modal data has been a long-standing theme in machine learning research. Multi-modal Variational Autoencoders (VAEs) have been a popular generative model class that learns latent representations which jointly explain multiple modalities. Various objective functions for such models have been suggested, often motivated as lower bounds on the multi-modal data log-likelihood or from information-theoretic considerations. In order to encode latent variables from different modality subsets, Product-of-Experts (PoE) or Mixture-of-Experts (MoE) aggregation schemes have been routinely used and shown to yield different trade-offs, for instance, regarding their generative quality or consistency across multiple modalities. In this work, we consider a variational bound that can tightly lower bound the data log-likelihood. We develop more flexible aggregation schemes that generalise PoE or MoE approaches by combining encoded features from different modalities based on permutation-invariant neural networks. Our numerical experiments illustrate trade-offs for multi-modal variational bounds and various aggregation schemes. We show that tighter variational bounds and more flexible aggregation models can become beneficial when one wants to approximate the true joint distribution over observed modalities and latent variables in identifiable models.

As the volume and quality of modern galaxy surveys increase, so does the difficulty of measuring the cosmological signal imprinted in galaxy shapes. Weak gravitational lensing sourced by the most massive structures in the Universe generates a slight shearing of galaxy morphologies called cosmic shear, key probe for cosmological models. Modern techniques of shear estimation based on statistics of ellipticity measurements suffer from the fact that the ellipticity is not a well-defined quantity for arbitrary galaxy light profiles, biasing the shear estimation. We show that a hybrid physical and deep learning Hierarchical Bayesian Model, where a generative model captures the galaxy morphology, enables us to recover an unbiased estimate of the shear on realistic galaxies, thus solving the model bias.

Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper addresses this computational challenge through the development of a novel generalised Bayesian inference procedure suitable for discrete intractable likelihood. Inspired by recent methodological advances for continuous data, the main idea is to update beliefs about model parameters using a discrete Fisher divergence, in lieu of the problematic intractable likelihood. The result is a generalised posterior that can be sampled from using standard computational tools, such as Markov chain Monte Carlo, circumventing the intractable normalising constant. The statistical properties of the generalised posterior are analysed, with sufficient conditions for posterior consistency and asymptotic normality established. In addition, a novel and general approach to calibration of generalised posteriors is proposed. Applications are presented on lattice models for discrete spatial data and on multivariate models for count data, where in each case the methodology facilitates generalised Bayesian inference at low computational cost.

We propose a new parameter-adaptive uncertainty-penalized Bayesian information criterion (UBIC) to prioritize the parsimonious partial differential equation (PDE) that sufficiently governs noisy spatial-temporal observed data with few reliable terms. Since the naive use of the BIC for model selection has been known to yield an undesirable overfitted PDE, the UBIC penalizes the found PDE not only by its complexity but also the quantified uncertainty, derived from the model supports' coefficient of variation in a probabilistic view. We also introduce physics-informed neural network learning as a simulation-based approach to further validate the selected PDE flexibly against the other discovered PDE. Numerical results affirm the successful application of the UBIC in identifying the true governing PDE. Additionally, we reveal an interesting effect of denoising the observed data on improving the trade-off between the BIC score and model complexity. Code is available at https://github.com/Pongpisit-Thanasutives/UBIC.

Computer games are widespread nowadays and enjoyed by people of all ages. But when it comes to kids, playing these games can be more than just fun, it is a way for them to develop important skills and build emotional intelligence. Facial expressions and sounds that kids produce during gameplay reflect their feelings, thoughts, and moods. In this paper, we propose a novel framework that integrates a fuzzy approach for the recognition of emotions through the analysis of audio and video data. Our focus lies within the specific context of computer games tailored for children, aiming to enhance their overall user experience. We use the FER dataset to detect facial emotions in video frames recorded from the screen during the game. For the audio emotion recognition of sounds a kid produces during the game, we use CREMA-D, TESS, RAVDESS, and Savee datasets. Next, a fuzzy inference system is used for the fusion of results. Besides this, our system can detect emotion stability and emotion diversity during gameplay, which, together with prevailing emotion report, can serve as valuable information for parents worrying about the effect of certain games on their kids. The proposed approach has shown promising results in the preliminary experiments we conducted, involving 3 different video games, namely fighting, racing, and logic games, and providing emotion-tracking results for kids in each game. Our study can contribute to the advancement of child-oriented game development, which is not only engaging but also accounts for children's cognitive and emotional states.