There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample $x$, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in $ [0,1]$ acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near $1$, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near $0$, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.
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Current person image retrieval methods have achieved great improvements in accuracy metrics. However, they rarely describe the reliability of the prediction. In this paper, we propose an Uncertainty-Aware Learning (UAL) method to remedy this issue. UAL aims at providing reliability-aware predictions by considering data uncertainty and model uncertainty simultaneously. Data uncertainty captures the ``noise" inherent in the sample, while model uncertainty depicts the model's confidence in the sample's prediction. Specifically, in UAL, (1) we propose a sampling-free data uncertainty learning method to adaptively assign weights to different samples during training, down-weighting the low-quality ambiguous samples. (2) we leverage the Bayesian framework to model the model uncertainty by assuming the parameters of the network follow a Bernoulli distribution. (3) the data uncertainty and the model uncertainty are jointly learned in a unified network, and they serve as two fundamental criteria for the reliability assessment: if a probe is high-quality (low data uncertainty) and the model is confident in the prediction of the probe (low model uncertainty), the final ranking will be assessed as reliable. Experiments under the risk-controlled settings and the multi-query settings show the proposed reliability assessment is effective. Our method also shows superior performance on three challenging benchmarks under the vanilla single query settings.
This paper presents a computationally feasible method to compute rigorous bounds on the interval-generalisation of regression analysis to account for epistemic uncertainty in the output variables. The new iterative method uses machine learning algorithms to fit an imprecise regression model to data that consist of intervals rather than point values. The method is based on a single-layer interval neural network which can be trained to produce an interval prediction. It seeks parameters for the optimal model that minimizes the mean squared error between the actual and predicted interval values of the dependent variable using a first-order gradient-based optimization and interval analysis computations to model the measurement imprecision of the data. An additional extension to a multi-layer neural network is also presented. We consider the explanatory variables to be precise point values, but the measured dependent values are characterized by interval bounds without any probabilistic information. The proposed iterative method estimates the lower and upper bounds of the expectation region, which is an envelope of all possible precise regression lines obtained by ordinary regression analysis based on any configuration of real-valued points from the respective y-intervals and their x-values.
The problem of robust binary hypothesis testing is studied. Under both hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through moments; in particular, the sets contain distributions whose moments are centered around the empirical moments obtained from training samples. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., minimize the worst-case error probability over the uncertainty sets. In the finite-alphabet case, the optimal test is obtained. In the infinite-alphabet case, a tractable approximation to the worst-case error is derived that converges to the optimal value using finite samples from the alphabet. A test is further constructed to generalize to the entire alphabet. An exponentially consistent test for testing batch samples is also proposed. Numerical results are provided to demonstrate the performance of the proposed robust tests.
Reliably estimating the uncertainty of a prediction throughout the model lifecycle is crucial in many safety-critical applications. The most common way to measure this uncertainty is via the predicted confidence. While this tends to work well for in-domain samples, these estimates are unreliable under domain drift. Alternatively, a bias-variance decomposition allows to directly measure the predictive uncertainty across the entire input space. But, such a decomposition for proper scores does not exist in current literature, and for exponential families it is convoluted. In this work, we introduce a general bias-variance decomposition for proper scores and reformulate the exponential family case, giving rise to the Bregman Information as the variance term in both cases. This allows us to prove that the Bregman Information for classification measures the uncertainty in the logit space. We showcase the practical relevance of this decomposition on two downstream tasks. First, we show how to construct confidence intervals for predictions on the instance-level based on the Bregman Information. Second, we demonstrate how different approximations of the instance-level Bregman Information allow reliable out-of-distribution detection for all degrees of domain drift.
We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.
We propose a Bayesian tensor-on-tensor regression approach to predict a multidimensional array (tensor) of arbitrary dimensions from another tensor of arbitrary dimensions, building upon the Tucker decomposition of the regression coefficient tensor. Traditional tensor regression methods making use of the Tucker decomposition either assume the dimension of the core tensor to be known or estimate it via cross-validation or some model selection criteria. However, no existing method can simultaneously estimate the model dimension (the dimension of the core tensor) and other model parameters. To fill this gap, we develop an efficient Markov Chain Monte Carlo (MCMC) algorithm to estimate both the model dimension and parameters for posterior inference. Besides the MCMC sampler, we also develop an ultra-fast optimization-based computing algorithm wherein the maximum a posteriori estimators for parameters are computed, and the model dimension is optimized via a simulated annealing algorithm. The proposed Bayesian framework provides a natural way for uncertainty quantification. Through extensive simulation studies, we evaluate the proposed Bayesian tensor-on-tensor regression model and show its superior performance compared to alternative methods. We also demonstrate its practical effectiveness by applying it to two real-world datasets, including facial imaging data and 3D motion data.
A simple generative model based on a continuous-time normalizing flow between any pair of base and target probability densities is proposed. The velocity field of this flow is inferred from the probability current of a time-dependent density that interpolates between the base and the target in finite time. Unlike conventional normalizing flow inference methods based the maximum likelihood principle, which require costly backpropagation through ODE solvers, our interpolant approach leads to a simple quadratic loss for the velocity itself which is expressed in terms of expectations that are readily amenable to empirical estimation. The flow can be used to generate samples from either the base or target, and to estimate the likelihood at any time along the interpolant. In addition, the flow can be optimized to minimize the path length of the interpolant density, thereby paving the way for building optimal transport maps. The approach is also contextualized in its relation to diffusions. In particular, in situations where the base is a Gaussian density, we show that the velocity of our normalizing flow can also be used to construct a diffusion model to sample the target as well as estimating its score. This allows one to map methods based on stochastic differential equations to those using ordinary differential equations, simplifying the mechanics of the model, but capturing equivalent dynamics. Benchmarking on density estimation tasks illustrates that the learned flow can match and surpass maximum likelihood continuous flows at a fraction of the conventional ODE training costs.
In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases, statistical measures have been constructed, such as cross validation, information criteria, and marginal likelihood, however, their mathematical properties have not yet been completely clarified when statistical models are under- and over- parametrized. We introduce a place of mathematical theory of Bayesian statistics for unknown uncertainty, which clarifies general properties of cross validation, information criteria, and marginal likelihood, even if an unknown data-generating process is unrealizable by a model or even if the posterior distribution cannot be approximated by any normal distribution. Hence it gives a helpful standpoint for a person who cannot believe in any specific model and prior. This paper consists of three parts. The first is a new result, whereas the second and third are well-known previous results with new experiments. We show there exists a more precise estimator of the generalization loss than leave-one-out cross validation, there exists a more accurate approximation of marginal likelihood than BIC, and the optimal hyperparameters for generalization loss and marginal likelihood are different.
This report describes our approach to design and evaluate a software stack for a race car capable of achieving competitive driving performance in the different disciplines of the Formula Student Driverless. By using a 360{\deg} LiDAR and optionally three cameras, we reliably recognize the plastic cones that mark the track boundaries at distances of around 35 m, enabling us to drive at the physical limits of the car. Using a GraphSLAM algorithm, we are able to map these cones with a root-mean-square error of less than 15 cm while driving at speeds of over 70 kph on a narrow track. The high-precision map is used in the trajectory planning to detect the lane boundaries using Delaunay triangulation and a parametric cubic spline. We calculate an optimized trajectory using a minimum curvature approach together with a GGS-diagram that takes the aerodynamics at different velocities into account. To track the target path with accelerations of up to 1.6 g, the control system is split into a PI controller for longitudinal control and model predictive controller for lateral control. Additionally, a low-level optimal control allocation is used. The software is realized in ROS C++ and tested in a custom simulation, as well as on the actual race track.
Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.
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