Epistemic emotions, such as curiosity and interest, drive the inquiry process. This study proposes a novel formulation of epistemic emotions such as curiosity and interest using two types of information gain generated by the principle of free energy minimization: Kullback-Leibler divergence(KLD) from Bayesian posterior to prior, which represents free energy reduction in recognition, and Bayesian surprise (BS), which represents the expected information gain by Bayesian prior update. By applying a Gaussian generative model with an additional uniform likelihood, we found that KLD and BS form an upward-convex function of surprise (minimized free energy and prediction error), similar to Berlyne's arousal potential functions, or the Wundt curve. We consider that the alternate maximization of BS and KLD generates an ideal inquiry cycle to approach the optimal arousal level with fluctuations in surprise, and that curiosity and interest drive to facilitate the cyclic process. We exhaustively analyzed the effects of prediction uncertainty (prior variance) and observation uncertainty (likelihood variance) on the peaks of the information gain function as optimal surprises. The results show that greater prediction uncertainty, meaning an open-minded attitude, and less observational uncertainty, meaning precise observation with attention, are expected to provide greater information gains through a greater range of exploration. The proposed mathematical framework unifies the free energy principle of the brain and the arousal potential theory to explain the Wundt curve as an information gain function and suggests an ideal inquiry process driven by epistemic emotions.

Model explanations are very valuable for interpreting and debugging prediction models. We study a specific kind of global explanations called Concept Explanations, where the goal is to interpret a model using human-understandable concepts. Recent advances in multi-modal learning rekindled interest in concept explanations and led to several label-efficient proposals for estimation. However, existing estimation methods are unstable to the choice of concepts or dataset that is used for computing explanations. We observe that instability in explanations is due to high variance in point estimation of importance scores. We propose an uncertainty aware Bayesian estimation method, which readily improved reliability of the concept explanations. We demonstrate with theoretical analysis and empirical evaluation that explanations computed by our method are more reliable while also being label-efficient and faithful.

We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. In particular, we focus on characterizing the variance over values induced by a distribution over MDPs. Previous work upper bounds the posterior variance over values by solving a so-called uncertainty Bellman equation (UBE), but the over-approximation may result in inefficient exploration. We propose a new UBE whose solution converges to the true posterior variance over values and leads to lower regret in tabular exploration problems. We identify challenges to apply the UBE theory beyond tabular problems and propose a suitable approximation. Based on this approximation, we introduce a general-purpose policy optimization algorithm, Q-Uncertainty Soft Actor-Critic (QU-SAC), that can be applied for either risk-seeking or risk-averse policy optimization with minimal changes. Experiments in both online and offline RL demonstrate improved performance compared to other uncertainty estimation methods.

Generative models have gained popularity for their potential applications in imaging science, such as image reconstruction, posterior sampling and data sharing. Flow-based generative models are particularly attractive due to their ability to tractably provide exact density estimates along with fast, inexpensive and diverse samples. Training such models, however, requires a large, high quality dataset of objects. In applications such as computed imaging, it is often difficult to acquire such data due to requirements such as long acquisition time or high radiation dose, while acquiring noisy or partially observed measurements of these objects is more feasible. In this work, we propose AmbientFlow, a framework for learning flow-based generative models directly from noisy and incomplete data. Using variational Bayesian methods, a novel framework for establishing flow-based generative models from noisy, incomplete data is proposed. Extensive numerical studies demonstrate the effectiveness of AmbientFlow in learning the object distribution. The utility of AmbientFlow in a downstream inference task of image reconstruction is demonstrated.

This paper presents the "Uncertainty-aware Mixture of Experts" (uMoE), a novel approach designed to address aleatoric uncertainty in the training of predictive models based on Neural Networks (NNs). While existing methods primarily focus on managing uncertainty during infer-ence, uMoE integrates uncertainty directly into the train-ing process. The uMoE approach adopts a "Divide and Conquer" paradigm to partition the uncertain input space into more manageable subspaces. It consists of Expert components, each trained solely on the portion of input uncertainty corresponding to their subspace. On top of the Experts, a Gating Unit, guided by additional infor-mation about the distribution of uncertain inputs across these subspaces, learns to weight the Experts to minimize deviations from the ground truth. Our results highlight that uMoE significantly outperforms baseline methods in handling data uncertainty. Furthermore, we conducted a robustness analysis, illustrating its capability to adapt to varying levels of uncertainty and suggesting optimal threshold parameters. This innovative approach holds wide applicability across diverse data-driven domains, in-cluding biomedical signal processing, autonomous driv-ing, and production quality control.

Modern reinforcement learning (RL) can be categorized into online and offline variants. As a pivotal aspect of both online and offline RL, current research on the Bellman equation revolves primarily around optimization techniques and performance enhancement rather than exploring the inherent structural properties of the Bellman error, such as its distribution characteristics. This study investigates the distribution of the Bellman approximation error through iterative exploration of the Bellman equation with the observation that the Bellman error approximately follows the Logistic distribution. Based on this, we proposed the utilization of the Logistic maximum likelihood function (LLoss) as an alternative to the commonly used mean squared error (MSELoss) that assumes a Normal distribution for Bellman errors. We validated the hypotheses through extensive numerical experiments across diverse online and offline environments. In particular, we applied the Logistic correction to loss functions in various RL baseline methods and observed that the results with LLoss consistently outperformed the MSE counterparts. We also conducted the Kolmogorov-Smirnov tests to confirm the reliability of the Logistic distribution. Moreover, our theory connects the Bellman error to the proportional reward scaling phenomenon by providing a distribution-based analysis. Furthermore, we applied the bias-variance decomposition for sampling from the Logistic distribution. The theoretical and empirical insights of this study lay a valuable foundation for future investigations and enhancements centered on the distribution of Bellman error.

We study experiment design for unique identification of the causal graph of a simple SCM, where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design as, unlike acyclic graphs, learning the skeleton of causal graphs with cycles may not be possible from merely the observational distribution. Furthermore, intervening on a variable in such graphs does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for uniquely identifying the causal graph in the worst case.

Proper scoring rules evaluate the quality of probabilistic predictions, playing an essential role in the pursuit of accurate and well-calibrated models. Every proper score decomposes into two fundamental components -- proper calibration error and refinement -- utilizing a Bregman divergence. While uncertainty calibration has gained significant attention, current literature lacks a general estimator for these quantities with known statistical properties. To address this gap, we propose a method that allows consistent, and asymptotically unbiased estimation of all proper calibration errors and refinement terms. In particular, we introduce Kullback--Leibler calibration error, induced by the commonly used cross-entropy loss. As part of our results, we prove the relation between refinement and f-divergences, which implies information monotonicity in neural networks, regardless of which proper scoring rule is optimized. Our experiments validate empirically the claimed properties of the proposed estimator and suggest that the selection of a post-hoc calibration method should be determined by the particular calibration error of interest.

In the standard formulation of the denoising problem, one is given a probabilistic model relating a latent variable $\Theta \in \Omega \subset \mathbb{R}^m \; (m\ge 1)$ and an observation $Z \in \mathbb{R}^d$ according to: $Z \mid \Theta \sim p(\cdot\mid \Theta)$ and $\Theta \sim G^*$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $\Theta$ from $Z$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $Z$, and in general may fail to capture the geometric features of the prior distribution $G^*$ (e.g., low dimensionality, discreteness, sparsity, etc.). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to propose a new OT-based denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, our OT-based denoiser is well-defined and unique, and is closely connected to solutions to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, our OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of a standard multimarginal OT (MOT) problem. In particular, thanks to Tweedie's formula, when the likelihood model $\{ p(\cdot \mid \theta) \}_{\theta \in \Omega}$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $Z$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.

Score matching is an approach to learning probability distributions parametrized up to a constant of proportionality (e.g. Energy-Based Models). The idea is to fit the score of the distribution, rather than the likelihood, thus avoiding the need to evaluate the constant of proportionality. While there's a clear algorithmic benefit, the statistical "cost'' can be steep: recent work by Koehler et al. 2022 showed that for distributions that have poor isoperimetric properties (a large Poincar\'e or log-Sobolev constant), score matching is substantially statistically less efficient than maximum likelihood. However, many natural realistic distributions, e.g. multimodal distributions as simple as a mixture of two Gaussians in one dimension -- have a poor Poincar\'e constant. In this paper, we show a close connection between the mixing time of a broad class of Markov processes with generator $\mathcal{L}$ and an appropriately chosen generalized score matching loss that tries to fit $\frac{\mathcal{O} p}{p}$. This allows us to adapt techniques to speed up Markov chains to construct better score-matching losses. In particular, ``preconditioning'' the diffusion can be translated to an appropriate ``preconditioning'' of the score loss. Lifting the chain by adding a temperature like in simulated tempering can be shown to result in a Gaussian-convolution annealed score matching loss, similar to Song and Ermon, 2019. Moreover, we show that if the distribution being learned is a finite mixture of Gaussians in $d$ dimensions with a shared covariance, the sample complexity of annealed score matching is polynomial in the ambient dimension, the diameter of the means, and the smallest and largest eigenvalues of the covariance -- obviating the Poincar\'e constant-based lower bounds of the basic score matching loss shown in Koehler et al. 2022.