We consider the problem of estimating probability density functions based on sample data, using a finite mixture of densities from some component class. To this end, we introduce the $h$-lifted Kullback--Leibler (KL) divergence as a generalization of the standard KL divergence and a criterion for conducting risk minimization. Under a compact support assumption, we prove an $\mc{O}(1/{\sqrt{n}})$ bound on the expected estimation error when using the $h$-lifted KL divergence, which extends the results of Rakhlin et al. (2005, ESAIM: Probability and Statistics, Vol. 9) and Li and Barron (1999, Advances in Neural Information ProcessingSystems, Vol. 12) to permit the risk bounding of density functions that are not strictly positive. We develop a procedure for the computation of the corresponding maximum $h$-lifted likelihood estimators ($h$-MLLEs) using the Majorization-Maximization framework and provide experimental results in support of our theoretical bounds.

Large language models primarily rely on inductive reasoning for decision making. This results in unreliable decisions when applied to real-world tasks that often present incomplete contexts and conditions. Thus, accurate probability estimation and appropriate interpretations are required to enhance decision-making reliability. In this paper, we propose a Bayesian inference framework called BIRD for large language models. BIRD provides controllable and interpretable probability estimation for model decisions, based on abductive factors, LLM entailment, as well as learnable deductive Bayesian modeling. Experiments show that BIRD produces probability estimations that align with human judgments over 65% of the time using open-sourced Llama models, outperforming the state-of-the-art GPT-4 by 35%. We also show that BIRD can be directly used for trustworthy decision making on many real-world applications.

The purpose of this paper is twofold. First, we propose a novel algorithm for estimating parameters in one-dimensional Gaussian mixture models (GMMs). The algorithm takes advantage of the Hankel structure inherent in the Fourier data obtained from independent and identically distributed (i.i.d) samples of the mixture. For GMMs with a unified variance, a singular value ratio functional using the Fourier data is introduced and used to resolve the variance and component number simultaneously. The consistency of the estimator is derived. Compared to classic algorithms such as the method of moments and the maximum likelihood method, the proposed algorithm does not require prior knowledge of the number of Gaussian components or good initial guesses. Numerical experiments demonstrate its superior performance in estimation accuracy and computational cost. Second, we reveal that there exists a fundamental limit to the problem of estimating the number of Gaussian components or model order in the mixture model if the number of i.i.d samples is finite. For the case of a single variance, we show that the model order can be successfully estimated only if the minimum separation distance between the component means exceeds a certain threshold value and can fail if below. We derive a lower bound for this threshold value, referred to as the computational resolution limit, in terms of the number of i.i.d samples, the variance, and the number of Gaussian components. Numerical experiments confirm this phase transition phenomenon in estimating the model order. Moreover, we demonstrate that our algorithm achieves better scores in likelihood, AIC, and BIC when compared to the EM algorithm.

We propose a novel method (floZ), based on normalizing flows, for estimating the Bayesian evidence (and its numerical uncertainty) from a set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a k-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. It has wide applicability, e.g., to estimate the evidence from variational inference, Markov-chain Monte Carlo samples, or any other method that delivers samples from the unnormalized posterior density.

Parallelisation in Bayesian optimisation is a common strategy but faces several challenges: the need for flexibility in acquisition functions and kernel choices, flexibility dealing with discrete and continuous variables simultaneously, model misspecification, and lastly fast massive parallelisation. To address these challenges, we introduce a versatile and modular framework for batch Bayesian optimisation via probabilistic lifting with kernel quadrature, called SOBER, which we present as a Python library based on GPyTorch/BoTorch. Our framework offers the following unique benefits: (1) Versatility in downstream tasks under a unified approach.

Uncertainty representation and quantification are paramount in machine learning and constitute an important prerequisite for safety-critical applications. In this paper, we propose novel measures for the quantification of aleatoric and epistemic uncertainty based on proper scoring rules, which are loss functions with the meaningful property that they incentivize the learner to predict ground-truth (conditional) probabilities. We assume two common representations of (epistemic) uncertainty, namely, in terms of a credal set, i.e. a set of probability distributions, or a second-order distribution, i.e., a distribution over probability distributions. Our framework establishes a natural bridge between these representations. We provide a formal justification of our approach and introduce new measures of epistemic and aleatoric uncertainty as concrete instantiations.

This paper considers a distributed adaptive optimization problem, where all agents only have access to their local cost functions with a common unknown parameter, whereas they mean to collaboratively estimate the true parameter and find the optimal solution over a connected network. A general mathematical framework for such a problem has not been studied yet. We aim to provide valuable insights for addressing parameter uncertainty in distributed optimization problems and simultaneously find the optimal solution. Thus, we propose a novel Prediction while Optimization scheme, which utilizes distributed fractional Bayesian learning through weighted averaging on the log-beliefs to update the beliefs of unknown parameters, and distributed gradient descent for renewing the estimation of the optimal solution. Then under suitable assumptions, we prove that all agents' beliefs and decision variables converge almost surely to the true parameter and the optimal solution under the true parameter, respectively. We further establish a sublinear convergence rate for the belief sequence. Finally, numerical experiments are implemented to corroborate the theoretical analysis.

The application of artificial intelligence (AI) models in fields such as engineering is limited by the known difficulty of quantifying the reliability of an AI's decision. A well-calibrated AI model must correctly report its accuracy on in-distribution (ID) inputs, while also enabling the detection of out-of-distribution (OOD) inputs. A conventional approach to improve calibration is the application of Bayesian ensembling. However, owing to computational limitations and model misspecification, practical ensembling strategies do not necessarily enhance calibration. This paper proposes an extension of variational inference (VI)-based Bayesian learning that integrates calibration regularization for improved ID performance, confidence minimization for OOD detection, and selective calibration to ensure a synergistic use of calibration regularization and confidence minimization. The scheme is constructed successively by first introducing calibration-regularized Bayesian learning (CBNN), then incorporating out-of-distribution confidence minimization (OCM) to yield CBNN-OCM, and finally integrating also selective calibration to produce selective CBNN-OCM (SCBNN-OCM). Selective calibration rejects inputs for which the calibration performance is expected to be insufficient. Numerical results illustrate the trade-offs between ID accuracy, ID calibration, and OOD calibration attained by both frequentist and Bayesian learning methods. Among the main conclusions, SCBNN-OCM is seen to achieve best ID and OOD performance as compared to existing state-of-the-art approaches at the cost of rejecting a sufficiently large number of inputs.

We introduce a deterministic variational formulation for training Bayesian last layer neural networks. This yields a sampling-free, single-pass model and loss that effectively improves uncertainty estimation. Our variational Bayesian last layer (VBLL) can be trained and evaluated with only quadratic complexity in last layer width, and is thus (nearly) computationally free to add to standard architectures. We experimentally investigate VBLLs, and show that they improve predictive accuracy, calibration, and out of distribution detection over baselines across both regression and classification. Finally, we investigate combining VBLL layers with variational Bayesian feature learning, yielding a lower variance collapsed variational inference method for Bayesian neural networks. Well-calibrated uncertainty quantification is essential for reliable decision-making with machine learning systems.

One of the ways to make artificial intelligence more natural is to give it some room for doubt. Two main questions should be resolved in that way. First, how to train a model to estimate uncertainties of its own predictions? And then, what to do with the uncertain predictions if they appear? First, we proposed an uncertainty-aware negative log-likelihood loss for the case of N-dimensional multivariate normal distribution with spherical variance matrix to the solution of N-classes classification tasks. The loss is similar to the heteroscedastic regression loss. The proposed model regularizes uncertain predictions, and trains to calculate both the predictions and their uncertainty estimations. The model fits well with the label smoothing technique. Second, we expanded the limits of data augmentation at the training and test stages, and made the trained model to give multiple predictions for a given number of augmented versions of each test sample. Given the multi-view predictions together with their uncertainties and confidences, we proposed several methods to calculate final predictions, including mode values and bin counts with soft and hard weights. For the latter method, we formalized the model tuning task in the form of multimodal optimization with non-differentiable criteria of maximum accuracy, and applied particle swarm optimization to solve the tuning task. The proposed methodology was tested using CIFAR-10 dataset with clean and noisy labels and demonstrated good results in comparison with other uncertainty estimation methods related to sample selection, co-teaching, and label smoothing.