Probabilistic regression models trained with maximum likelihood estimation (MLE), can sometimes overestimate variance to an unacceptable degree. This is mostly problematic in the multivariate domain. While univariate models often optimize the popular Continuous Ranked Probability Score (CRPS), in the multivariate domain, no such alternative to MLE has yet been widely accepted. The Energy Score - the most investigated alternative - notoriously lacks closed-form expressions and sensitivity to the correlation between target variables. In this paper, we propose Conditional CRPS: a multivariate strictly proper scoring rule that extends CRPS. We show that closed-form expressions exist for popular distributions and illustrate their sensitivity to correlation. We then show in a variety of experiments on both synthetic and real data, that Conditional CRPS often outperforms MLE, and produces results comparable to state-of-the-art non-parametric models, such as Distributional Random Forest (DRF).

Learning Bayesian Networks (BNs) from high-dimensional data is a complex and time-consuming task. Although there are approaches based on horizontal (instances) or vertical (variables) partitioning in the literature, none can guarantee the same theoretical properties as the Greedy Equivalence Search (GES) algorithm, except those based on the GES algorithm itself. In this paper, we propose a directed ring-based distributed method that uses GES as the local learning algorithm, ensuring the same theoretical properties as GES but requiring less CPU time. The method involves partitioning the set of possible edges and constraining each processor in the ring to work only with its received subset. The global learning process is an iterative algorithm that carries out several rounds until a convergence criterion is met. In each round, each processor receives a BN from its predecessor in the ring, fuses it with its own BN model, and uses the result as the starting solution for a local learning process constrained to its set of edges. Subsequently, it sends the model obtained to its successor in the ring. Experiments were carried out on three large domains (400-1000 variables), demonstrating our proposal's effectiveness compared to GES and its fast version (fGES).

The design of food diets in the context of animal nutrition is a complex problem that aims to develop cost-effective formulations while balancing minimum nutritional content. Traditional approaches based on theoretical models of metabolic responses and concentrations of digestible energy in raw materials face limitations in incorporating zootechnical or environmental variables affecting the performance of animals and including multiple objectives aligned with sustainable development policies. Recently, multi-objective Bayesian optimization has been proposed as a promising heuristic alternative able to deal with the combination of multiple sources of information, multiple and diverse objectives, and with an intrinsic capacity to deal with uncertainty in the measurements that could be related to variability in the nutritional content of raw materials. However, Bayesian optimization encounters difficulties in high-dimensional search spaces, leading to exploration predominantly at the boundaries. This work analyses a strategy to split the search space into regions that provide local candidates termed multi-objective regionalized Bayesian optimization as an alternative to improve the quality of the Pareto set and Pareto front approximation provided by BO in the context of swine diet design. Results indicate that this regionalized approach produces more diverse non-dominated solutions compared to the standard multi-objective Bayesian optimization. Besides, the regionalized strategy was four times more effective in finding solutions that outperform those identified by a stochastic programming approach referenced in the literature. Experiments using batches of query candidate solutions per iteration show that the optimization process can also be accelerated without compromising the quality of the Pareto set approximation during the initial, most critical phase of optimization.

Fairness metrics are used to assess discrimination and disparity of the chances between yellow and blue candidates of getting bias in decision-making processes across various domains, including accepted. Intuitively, we are more certain about the decisions machine learning models and human decision-makers in real-world being made by company A than company B. In the case of company applications. This involves calculating the disparities between probabilistic B, the rejection of blue candidates can be attributed to random outcomes among social groups, such as acceptance rates circumstances. In this case, we would judge company A as more discriminatory between male and female applicants. However, traditional fairness than company B because we are more certain that A metrics do not account for the uncertainty in these processes and is unfair and very uncertain about the unfairness of B. But if both lack of comparability when two decision-makers exhibit the same companies accepted all applicants, the disparity would be 0%, and disparity. Using Bayesian statistics, we quantify the uncertainty of we would conversely judge B as more discriminatory than A. This is the disparity to enhance discrimination assessments. We represent because we are certain that A is fair, while we are uncertain about the each decision-maker, whether a machine learning model or a human, fairness of B. Lastly, when comparing between uncertain fair and uncertain by its disparity and the corresponding uncertainty in that disparity.

Much of Bayesian inference centers around the design of estimators for inverse problems which are optimal assuming the data comes from a known prior. But what do these optimality guarantees mean if the prior is unknown? In recent years, algorithm unrolling has emerged as deep learning's answer to this age-old question: design a neural network whose layers can in principle simulate iterations of inference algorithms and train on data generated by the unknown prior. Despite its empirical success, however, it has remained unclear whether this method can provably recover the performance of its optimal, prior-aware counterparts. In this work, we prove the first rigorous learning guarantees for neural networks based on unrolling approximate message passing (AMP). For compressed sensing, we prove that when trained on data drawn from a product prior, the layers of the network approximately converge to the same denoisers used in Bayes AMP. We also provide extensive numerical experiments for compressed sensing and rank-one matrix estimation demonstrating the advantages of our unrolled architecture - in addition to being able to obliviously adapt to general priors, it exhibits improvements over Bayes AMP in more general settings of low dimensions, non-Gaussian designs, and non-product priors.

This paper illustrates the central role of loss functions in data-driven decision making, providing a comprehensive survey on their influence in cost-sensitive classification (CSC) and reinforcement learning (RL). We demonstrate how different regression loss functions affect the sample efficiency and adaptivity of value-based decision making algorithms. Across multiple settings, we prove that algorithms using the binary cross-entropy loss achieve first-order bounds scaling with the optimal policy's cost and are much more efficient than the commonly used squared loss. Moreover, we prove that distributional algorithms using the maximum likelihood loss achieve second-order bounds scaling with the policy variance and are even sharper than first-order bounds. This in particular proves the benefits of distributional RL. We hope that this paper serves as a guide analyzing decision making algorithms with varying loss functions, and can inspire the reader to seek out better loss functions to improve any decision making algorithm.

Data augmentation is known to contribute significantly to the robustness of machine learning models. In most instances, data augmentation is utilized during the training phase. Test-Time Augmentation (TTA) is a technique that instead leverages these data augmentations during the testing phase to achieve robust predictions. More precisely, TTA averages the predictions of multiple data augmentations of an instance to produce a final prediction. Although the effectiveness of TTA has been empirically reported, it can be expected that the predictive performance achieved will depend on the set of data augmentation methods used during testing. In particular, the data augmentation methods applied should make different contributions to performance. That is, it is anticipated that there may be differing degrees of contribution in the set of data augmentation methods used for TTA, and these could have a negative impact on prediction performance. In this study, we consider a weighted version of the TTA based on the contribution of each data augmentation. Some variants of TTA can be regarded as considering the problem of determining the appropriate weighting. We demonstrate that the determination of the coefficients of this weighted TTA can be formalized in a variational Bayesian framework. We also show that optimizing the weights to maximize the marginal log-likelihood suppresses candidates of unwanted data augmentations at the test phase.

Background: Advanced methods for causal inference, such as targeted maximum likelihood estimation (TMLE), require certain conditions for statistical inference. However, in situations where there is not differentiability due to data sparsity or near-positivity violations, the Donsker class condition is violated. In such situations, TMLE variance can suffer from inflation of the type I error and poor coverage, leading to conservative confidence intervals. Cross-validation of the TMLE algorithm (CVTMLE) has been suggested to improve on performance compared to TMLE in settings of positivity or Donsker class violations. We aim to investigate the performance of CVTMLE compared to TMLE in various settings. Methods: We utilised the data-generating mechanism as described in Leger et al. (2022) to run a Monte Carlo experiment under different Donsker class violations. Then, we evaluated the respective statistical performances of TMLE and CVTMLE with different super learner libraries, with and without regression tree methods. Results: We found that CVTMLE vastly improves confidence interval coverage without adversely affecting bias, particularly in settings with small sample sizes and near-positivity violations. Furthermore, incorporating regression trees using standard TMLE with ensemble super learner-based initial estimates increases bias and variance leading to invalid statistical inference. Conclusions: It has been shown that when using CVTMLE the Donsker class condition is no longer necessary to obtain valid statistical inference when using regression trees and under either data sparsity or near-positivity violations. We show through simulations that CVTMLE is much less sensitive to the choice of the super learner library and thereby provides better estimation and inference in cases where the super learner library uses more flexible candidates and is prone to overfitting.

Gaussian processes (GPs) are Bayesian nonparametric models for function approximation with principled predictive uncertainty estimates. Deep Gaussian processes (DGPs) are multilayer generalizations of GPs that can represent complex marginal densities as well as complex mappings. As exact inference is either computationally prohibitive or analytically intractable in GPs and extensions thereof, some existing methods resort to variational inference (VI) techniques for tractable approximations. However, the expressivity of conventional approximate GP models critically relies on independent inducing variables that might not be informative enough for some problems. In this work we introduce amortized variational inference for DGPs, which learns an inference function that maps each observation to variational parameters. The resulting method enjoys a more expressive prior conditioned on fewer input dependent inducing variables and a flexible amortized marginal posterior that is able to model more complicated functions. We show with theoretical reasoning and experimental results that our method performs similarly or better than previous approaches at less computational cost.

Denoising, the process of reducing random fluctuations in a signal to emphasize essential patterns, has been a fundamental problem of interest since the dawn of modern scientific inquiry. Recent denoising techniques, particularly in imaging, have achieved remarkable success, nearing theoretical limits by some measures. Yet, despite tens of thousands of research papers, the wide-ranging applications of denoising beyond noise removal have not been fully recognized. This is partly due to the vast and diverse literature, making a clear overview challenging. This paper aims to address this gap. We present a clarifying perspective on denoisers, their structure, and desired properties. We emphasize the increasing importance of denoising and showcase its evolution into an essential building block for complex tasks in imaging, inverse problems, and machine learning. Despite its long history, the community continues to uncover unexpected and groundbreaking uses for denoising, further solidifying its place as a cornerstone of scientific and engineering practice.