We present a novel extension of the Weak Neural Variational Inference (WNVI) framework for probabilistic material property estimation that explicitly quantifies model errors in PDE-based inverse problems. Traditional approaches assume the correctness of all governing equations, including potentially unreliable constitutive laws, which can lead to biased estimates and misinterpretations. Our proposed framework addresses this limitation by distinguishing between reliable governing equations, such as conservation laws, and uncertain constitutive relationships. By treating all state variables as latent random variables, we enforce these equations through separate sets of residuals, leveraging a virtual likelihood approach with weighted residuals. This formulation not only identifies regions where constitutive laws break down but also improves robustness against model uncertainties without relying on a fully trustworthy forward model. We demonstrate the effectiveness of our approach in the context of elastography, showing that it provides a structured, interpretable, and computationally efficient alternative to traditional model error correction techniques. Our findings suggest that the proposed framework enhances the accuracy and reliability of material property estimation by offering a principled way to incorporate uncertainty in constitutive modeling.

The off-switch problem is a critical challenge in AI control: if an AI system resists being switched off, it poses a significant risk. In this paper, we model the off-switch problem as a signalling game, where a human decision-maker communicates its preferences about some underlying decision problem to an AI agent, which then selects actions to maximise the human's utility. We assume that the human is a bounded rational agent and explore various bounded rationality mechanisms. Using real machine learning models, we reprove prior results and demonstrate that a necessary condition for an AI system to refrain from disabling its off-switch is its uncertainty about the human's utility. We also analyse how message costs influence optimal strategies and extend the analysis to scenarios involving incomparability.

Behavior cloning (BC) traditionally relies on demonstration data, assuming the demonstrated actions are optimal. This can lead to overfitting under noisy data, particularly when expressive models are used (e.g., the energy-based model in Implicit BC). To address this, we extend behavior cloning into an iterative process of optimal action estimation within the Interactive Imitation Learning framework. Specifically, we introduce Contrastive policy Learning from Interactive Corrections (CLIC). CLIC leverages human corrections to estimate a set of desired actions and optimizes the policy to select actions from this set. We provide theoretical guarantees for the convergence of the desired action set to optimal actions in both single and multiple optimal action cases. Extensive simulation and real-robot experiments validate CLIC's advantages over existing state-of-the-art methods, including stable training of energy-based models, robustness to feedback noise, and adaptability to diverse feedback types beyond demonstrations. Our code will be publicly available soon.

We show that variational learning naturally induces an adaptive label smoothing where label noise is specialized for each example. Such label-smoothing is useful to handle examples with labeling errors and distribution shifts, but designing a good adaptivity strategy is not always easy. We propose to skip this step and simply use the natural adaptivity induced during the optimization of a variational objective. We show empirical results where a variational algorithm called IVON outperforms traditional label smoothing and yields adaptivity strategies similar to those of an existing approach. By connecting Bayesian methods to label smoothing, our work provides a new way to handle overconfident predictions.

Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on samples from the two densities. However, existing methods for this problem remain computationally expensive and can yield inaccurate estimates. Inspired by recent advances in generative modeling, we introduce a novel framework for time score estimation, based on a conditioning variable. Choosing the conditioning variable judiciously enables a closed-form objective function. We demonstrate that, compared to previous approaches, our approach results in faster learning of the time score and competitive or better estimation accuracies of the density ratio on challenging tasks. Furthermore, we establish theoretical guarantees on the error of the estimated density ratio.

Monte Carlo (MC) methods are powerful tools for numerical inference and optimization widely employed in statistics, signal processing and machine learning Liu (2004); Robert and Casella (2004). They are mainly used for computing approximately the solution of definite integrals, and by extension, of differential equations (for this reason, MC schemes can be considered stochastic quadrature rules). Although exact analytical solutions to integrals are always desirable, such unicorns are rarely available, specially in real-world systems. Many applications inevitably require the approximation of intractable integrals. Specifically, Bayesian methods need the computation of expectations with respect to posterior probability density function (pdf) which, generally, are analytically intractable Gelman et al. (2013). The MC methods can be divided in four main families: direct methods (based on transformations or random variables), accept-reject techniques, Markov chain Monte Carlo (MCMC) algorithms, and importance sampling (IS) schemes Luengo et al. (2020); Martino et al. (2018). The last two families are the most popular for the facility and universality of their possible application Liang et al. (2010); Liu (2004); Robert and Casella (2004). All the MC methods require the choice of a suitable proposal density that is crucial for their performance Luengo et al. (2020); Robert and Casella (2004).

Singular learning models with non-positive Fisher information matrices include neural networks, reduced-rank regression, Boltzmann machines, normal mixture models, and others. These models have been widely used in the development of learning machines. However, theoretical analysis is still in its early stages. In this paper, we examine learning coefficients, which indicate the general learning efficiency of deep linear learning models and three-layer neural network models with ReLU units. Finally, we extend the results to include the case of the Softmax function.

We derive a novel generative model from the simple act of Gaussian posterior inference. Treating the generated sample as an unknown variable to infer lets us formulate the sampling process in the language of Bayesian probability. Our model uses a sequence of prediction and posterior update steps to narrow down the unknown sample from a broad initial belief. In addition to a rigorous theoretical analysis, we establish a connection between our model and diffusion models and show that it includes Bayesian Flow Networks (BFNs) as a special case. In our experiments, we demonstrate improved performance over both BFNs and Variational Diffusion Models, achieving competitive likelihood scores on CIFAR10 and ImageNet.

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with $\varepsilon$-accuracy requires $O(\log(1/\varepsilon))$ iterations in Kullback-Leibler divergence assuming access to exact oracles and $O(\log^2(1/\varepsilon))$ iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan's asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology.

Neural amortized Bayesian inference (ABI) can solve probabilistic inverse problems orders of magnitude faster than classical methods. However, neural ABI is not yet sufficiently robust for widespread and safe applicability. In particular, when performing inference on observations outside of the scope of the simulated data seen during training, for example, because of model misspecification, the posterior approximations are likely to become highly biased. Due to the bad pre-asymptotic behavior of current neural posterior estimators in the out-of-simulation regime, the resulting estimation biases cannot be fixed in acceptable time by just simulating more training data. In this proof-of-concept paper, we propose a semi-supervised approach that enables training not only on (labeled) simulated data generated from the model, but also on unlabeled data originating from any source, including real-world data. To achieve the latter, we exploit Bayesian self-consistency properties that can be transformed into strictly proper losses without requiring knowledge of true parameter values, that is, without requiring data labels. The results of our initial experiments show remarkable improvements in the robustness of ABI on out-of-simulation data. Even if the observed data is far away from both labeled and unlabeled training data, inference remains highly accurate. If our findings also generalize to other scenarios and model classes, we believe that our new method represents a major breakthrough in neural ABI.