Mixture Density Networks (MDNs) allow to model arbitrarily complex mappings between inputs and mixture densities, enabling flexible conditional density estimation, at the risk of severe overfitting. A Bayesian approach can alleviate this problem by specifying a prior over the parameters of the neural network. However, these priors can be difficult to specify due to the lack of interpretability. We propose a novel neural network construction for conditional mixture densities that allows one to specify the prior in the predictive distribution domain. The construction is based on mapping the targets to the unit hypercube via a diffeomorphism, enabling the use of mixtures of Beta distributions. We prove that the prior predictive distributions are calibrated in the sense that they are equal to the unconditional density function defined by the diffeomorphism. Contrary to Bayesian Gaussian MDNs, which exhibit tied functional and distributional complexity, we show that our construction allows to decouple them. We propose an extension allowing to model correlations in the covariates via Gaussian copulas, potentially reducing the necessary number of mixture components. Our experiments show competitive performance on standard benchmarks with respect to the state of the art.

Binary Neural Networks (BiNNs), which employ single-bit precision weights, have emerged as a promising solution to reduce memory usage and power consumption while maintaining competitive performance in large-scale systems. However, training BiNNs remains a significant challenge due to the limitations of conventional training algorithms. Quantum HyperNetworks offer a novel paradigm for enhancing the optimization of BiNN by leveraging quantum computing. Specifically, a Variational Quantum Algorithm is employed to generate binary weights through quantum circuit measurements, while key quantum phenomena such as superposition and entanglement facilitate the exploration of a broader solution space. In this work, we establish a connection between this approach and Bayesian inference by deriving the Evidence Lower Bound (ELBO), when direct access to the output distribution is available (i.e., in simulations), and introducing a surrogate ELBO based on the Maximum Mean Discrepancy (MMD) metric for scenarios involving implicit distributions, as commonly encountered in practice. Our experimental results demonstrate that the proposed methods outperform standard Maximum Likelihood Estimation (MLE), improving trainability and generalization.

Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR)-based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the quantiles have been shown to improve the performance of QR-DQN for uncertainty-based exploration strategies. The contribution of this work is in the setting of fixed quantile levels and is twofold. First, we prove that the Cramer distance yields a projection that coincides with the 1-Wasserstein one and that, under non-crossing constraints, the squared Cramer and the quantile regression losses yield collinear gradients, shedding light on the connection between these important elements of DRL. Second, we propose a low complexity algorithm to compute the Cramer distance.

We propose a novel nonparametric online predictor for discrete labels conditioned on multivariate continuous features. The predictor is based on a feature space discretization induced by a full-fledged k-d tree with randomly picked directions and a recursive Bayesian distribution, which allows to automatically learn the most relevant feature scales characterizing the conditional distribution. We prove its pointwise universality, i.e., it achieves a normalized log loss performance asymptotically as good as the true conditional entropy of the labels given the features. The time complexity to process the n-th sample point is O(log n) in probability with respect to the distribution generating the data points, whereas other exact nonparametric methods require to process all past observations. Experiments on challenging datasets show the computational and statistical efficiency of our algorithm in comparison to standard and state-of-the-art methods.

Given samples from two distributions, a non-parametric two-sample test aims at determining whether the two distributions are equal or not, based on a test statistic. Classically, this statistic is computed on the whole data set, or is computed on a subset of the data set by a function trained on its complement. We consider methods in a third tier, so as to deal with large (possibly infinite) data sets, and to automatically determine the most relevant scales to work at, making two contributions. First, we develop a generic sequential non-parametric testing framework, in which the sample size need not be fixed in advance. This makes our test a truly sequential non-parametric multivariate two-sample test. Under information theoretic conditions qualifying the difference between the tested distributions, consistency of the two-sample test is established. Second, we instantiate our framework using nearest neighbor regressors, and show how the power of the resulting two-sample test can be improved using Bayesian mixtures and switch distributions. This combination of techniques yields automatic scale selection, and experiments performed on challenging data sets show that our sequential tests exhibit comparable performances to those of state-of-the-art non-sequential tests.

Comparing two sets of multivariate samples is a central problem in data analysis. From a statistical standpoint, the simplest way to perform such a comparison is to resort to a non-parametric two-sample test (TST), which checks whether the two sets can be seen as i.i.d. samples of an identical unknown distribution (the null hypothesis). If the null is rejected, one wishes to identify regions accounting for this difference. This paper presents a two-stage method providing feedback on this difference, based upon a combination of statistical learning (regression) and computational topology methods. Consider two populations, each given as a point cloud in Rd. In the first step, we assign a label to each set and we compute, for each sample point, a discrepancy measure based on comparing an estimate of the conditional probability distribution of the label given a position versus the global unconditional label distribution. In the second step, we study the height function defined at each point by the aforementioned estimated discrepancy. Topological persistence is used to identify persistent local minima of this height function, their basins defining regions of points with high discrepancy and in spatial proximity. Experiments are reported both on synthetic and real data (satellite images and handwritten digit images), ranging in dimension from d = 2 to d = 784, illustrating the ability of our method to localize discrepancies. On a general perspective, the ability to provide feedback downstream TST may prove of ubiquitous interest in exploratory statistics and data science.