Decision making under uncertainty is critical to real-world, autonomous systems. Model Predictive Control (MPC) methods have demonstrated favorable performance in practice, but remain limited when dealing with complex probability distributions. In this paper, we propose a generalization of MPC that represents a multitude of solutions as posterior distributions. By casting MPC as a Bayesian inference problem, we employ variational methods for posterior computation, naturally encoding the complexity and multi-modality of the decision making problem. We present a Stein variational gradient descent method to estimate the posterior directly over control parameters, given a cost function and observed state trajectories. We show that this framework leads to successful planning in challenging, non-convex optimal control problems.
Quasi-Monte Carlo (QMC) is an essential tool for integral approximation, Bayesian inference, and sampling for simulation in science, etc. In the QMC area, the rank-1 lattice is important due to its simple operation, and nice properties for point set construction. However, the construction of the generating vector of the rank-1 lattice is usually time-consuming because of an exhaustive computer search. To address this issue, we propose a simple closed-form rank-1 lattice construction method based on group theory. Our method reduces the number of distinct pairwise distance values to generate a more regular lattice. We theoretically prove a lower and an upper bound of the minimum pairwise distance of any non-degenerate rank-1 lattice. Empirically, our methods can generate a near-optimal rank-1 lattice compared with the Korobov exhaustive search regarding the $l_1$-norm and $l_2$-norm minimum distance. Moreover, experimental results show that our method achieves superior approximation performance on benchmark integration test problems and kernel approximation problems.
An established way to improve the transferability of black-box evasion attacks is to craft the adversarial examples on an ensemble-based surrogate to increase diversity. We argue that transferability is fundamentally related to uncertainty. Based on a state-of-the-art Bayesian Deep Learning technique, we propose a new method to efficiently build a surrogate by sampling approximately from the posterior distribution of neural network weights, which represents the belief about the value of each parameter. Our extensive experiments on ImageNet, CIFAR-10 and MNIST show that our approach improves the success rates of four state-of-the-art attacks significantly (up to 83.2 percentage points), in both intra-architecture and inter-architecture transferability. On ImageNet, our approach can reach 94% of success rate while reducing training computations from 11.6 to 2.4 exaflops, compared to an ensemble of independently trained DNNs. Our vanilla surrogate achieves 87.5% of the time higher transferability than three test-time techniques designed for this purpose. Our work demonstrates that the way to train a surrogate has been overlooked, although it is an important element of transfer-based attacks. We are, therefore, the first to review the effectiveness of several training methods in increasing transferability. We provide new directions to better understand the transferability phenomenon and offer a simple but strong baseline for future work.
Bayesian inference is a principled framework for dealing with uncertainty. The practitioner can perform an initial assumption for the physical phenomenon they want to model (prior belief), collect some data and then adjust the initial assumption in the light of the new evidence (posterior belief). Approximate Bayesian Computation (ABC) methods, also known as likelihood-free inference techniques, are a class of models used for performing inference when the likelihood is intractable. The unique requirement of these models is a black-box sampling machine. Due to the modelling-freedom they provide these approaches are particularly captivating. Robust Optimisation Monte Carlo (ROMC) is one of the most recent techniques of the specific domain. It approximates the posterior distribution by solving independent optimisation problems. This dissertation focuses on the implementation of the ROMC method in the software package Engine for Likelihood-Free Inference (ELFI). In the first chapters, we provide the mathematical formulation and the algorithmic description of the ROMC approach. In the following chapters, we describe our implementation; (a) we present all the functionalities provided to the user and (b) we demonstrate how to perform inference on some real examples. Our implementation provides a robust and efficient solution to a practitioner who wants to perform inference on a simulator-based model. Furthermore, it exploits parallel processing for accelerating the inference wherever it is possible. Finally, it has been designed to serve extensibility; the user can easily replace specific subparts of the method without significant overhead on the development side. Therefore, it can be used by a researcher for further experimentation.
While deep neural networks provide good performance for a range of challenging tasks, calibration and uncertainty estimation remain major challenges, especially under distribution shift. In this paper, we propose the amortized conditional normalized maximum likelihood (ACNML) method as a scalable general-purpose approach for uncertainty estimation, calibration, and out-of-distribution robustness with deep networks. Our algorithm builds on the conditional normalized maximum likelihood (CNML) coding scheme, which has minimax optimal properties according to the minimum description length principle, but is computationally intractable to evaluate exactly for all but the simplest of model classes. We propose to use approximate Bayesian inference technqiues to produce a tractable approximation to the CNML distribution. Our approach can be combined with any approximate inference algorithm that provides tractable posterior densities over model parameters. We demonstrate that ACNML compares favorably to a number of prior techniques for uncertainty estimation in terms of calibration on out-of-distribution inputs.
Recent advancements in deep generative modeling make it possible to learn prior distributions from complex data that subsequently can be used for Bayesian inference. However, we find that distributions learned by deep generative models for audio signals do not exhibit the right properties that are necessary for tasks like audio source separation using a probabilistic approach. We observe that the learned prior distributions are either discriminative and extremely peaked or smooth and non-discriminative. We quantify this behavior for two types of deep generative models on two audio datasets.
Gaussian Processes (GPs) can be used as flexible, non-parametric function priors. Inspired by the growing body of work on Normalizing Flows, we enlarge this class of priors through a parametric invertible transformation that can be made input-dependent. Doing so also allows us to encode interpretable prior knowledge (e.g., boundedness constraints). We derive a variational approximation to the resulting Bayesian inference problem, which is as fast as stochastic variational GP regression (Hensman et al., 2013; Dezfouli and Bonilla,2015). This makes the model a computationally efficient alternative to other hierarchical extensions of GP priors (Lazaro-Gredilla,2012; Damianou and Lawrence, 2013). The resulting algorithm's computational and inferential performance is excellent, and we demonstrate this on a range of data sets. For example, even with only 5 inducing points and an input-dependent flow, our method is consistently competitive with a standard sparse GP fitted using 100 inducing points.
Generalized linear models (GLMs) such as logistic regression are among the most widely used arms in data analyst's repertoire and often used on sensitive datasets. A large body of prior works that investigate GLMs under differential privacy (DP) constraints provide only private point estimates of the regression coefficients, and are not able to quantify parameter uncertainty. In this work, with logistic and Poisson regression as running examples, we introduce a generic noise-aware DP Bayesian inference method for a GLM at hand, given a noisy sum of summary statistics. Quantifying uncertainty allows us to determine which of the regression coefficients are statistically significantly different from zero. We provide a previously unknown tight privacy analysis and experimentally demonstrate that the posteriors obtained from our model, while adhering to strong privacy guarantees, are close to the non-private posteriors.
The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace as a simple, scalable Bayesian deep learning method: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork using the linearized Laplace approximation. We propose a subnetwork selection strategy that aims to maximally preserve the model's predictive uncertainty. Empirically, our approach compares favorably to ensembles and less expressive posterior approximations over full networks. Our proposed subnetwork (linearized) Laplace method is implemented within the laplace PyTorch library at https://github.com/AlexImmer/Laplace.
Although the expectation maximisation (EM) algorithm was introduced in 1970, it remains somewhat inaccessible to machine learning practitioners due to its obscure notation, terse proofs and lack of concrete links to modern machine learning techniques like autoencoded variational Bayes. This has resulted in gaps in the AI literature concerning the meaning of such concepts like "latent variables" and "variational lower bound," which are frequently used but often not clearly explained. The roots of these ideas lie in the EM algorithm. We first give a tutorial presentation of the EM algorithm for estimating the parameters of a $K$-component mixture density. The Gaussian mixture case is presented in detail using $K$-ary scalar hidden (or latent) variables rather than the more traditional binary valued $K$-dimenional vectors. This presentation is motivated by mixture modelling from the target tracking literature. In a similar style to Bishop's 2009 book, we present variational Bayesian inference as a generalised EM algorithm stemming from the variational (or evidential) lower bound, as well as the technique of mean field approximation (or product density transform). We continue the evolution from EM to variational autoencoders, developed by Kingma & Welling in 2014. In so doing, we establish clear links between the EM algorithm and its variational counterparts, hence clarifying the meaning of "latent variables." We provide a detailed coverage of the "reparametrisation trick" and focus on how the AEVB differs from conventional variational Bayesian inference. Throughout the tutorial, consistent notational conventions are used. This unifies the narrative and clarifies the concepts. Some numerical examples are given to further illustrate the algorithms.
We present a novel approach to Bayesian inference and general Bayesian computation that is defined through a sequential decision loop. Our method defines a recursive partitioning of the sample space. It neither relies on gradients nor requires any problem-specific tuning, and is asymptotically exact for any density function with a bounded domain. The output is an approximation to the whole density function including the normalisation constant, via partitions organised in efficient data structures. Such approximations may be used for evidence estimation or fast posterior sampling, but also as building blocks to treat a larger class of estimation problems. The algorithm shows competitive performance to recent state-of-the-art methods on synthetic and real-world problems including parameter inference for gravitational-wave physics.
Models of discrete-valued outcomes are easily misspecified if the data exhibit zero-inflation, overdispersion or contamination. Without additional knowledge about the existence and nature of this misspecification, model inference and prediction are adversely affected. Here, we introduce a robust discrepancy-based Bayesian approach using the Total Variation Distance (TVD). In the process, we address and resolve two challenges: First, we study convergence and robustness properties of a computationally efficient estimator for the TVD between a parametric model and the data-generating mechanism. Second, we provide an efficient inference method adapted from Lyddon et al. (2019) which corresponds to formulating an uninformative nonparametric prior directly over the data-generating mechanism. Lastly, we empirically demonstrate that our approach is robust and significantly improves predictive performance on a range of simulated and real world data.
We introduce a simple and intuitive framework that provides quantitative explanations of statistical models through the probabilistic assessment of input feature importance. The core idea comes from utilizing the Dirichlet distribution to define the importance of input features and learning it via approximate Bayesian inference. The learned importance has probabilistic interpretation and provides the relative significance of each input feature to a model's output, additionally assessing confidence about its importance quantification. As a consequence of using the Dirichlet distribution over the explanations, we can define a closed-form divergence to gauge the similarity between learned importance under different models. We use this divergence to study the feature importance explainability tradeoffs with essential notions in modern machine learning, such as privacy and fairness. Furthermore, BIF can work on two levels: global explanation (feature importance across all data instances) and local explanation (individual feature importance for each data instance). We show the effectiveness of our method on a variety of synthetic and real datasets, taking into account both tabular and image datasets. The code is available at https://github.com/kamadforge/featimp_dp.
Existing medical image super-resolution methods rely on pairs of low- and high- resolution images to learn a mapping in a fully supervised manner. However, such image pairs are often not available in clinical practice. In this paper, we address super-resolution problem in a real-world scenario using unpaired data and synthesize linearly \textbf{eight times} higher resolved Micro-CT images of temporal bone structure, which is embedded in the inner ear. We explore cycle-consistency generative adversarial networks for super-resolution task and equip the translation approach with Bayesian inference. We further introduce \emph{Hu Moment distance} the evaluation metric to quantify the shape of the temporal bone. We evaluate our method on a public inner ear CT dataset and have seen both visual and quantitative improvement over state-of-the-art deep-learning-based methods. In addition, we perform a multi-rater visual evaluation experiment and find that trained experts consistently rate the proposed method the highest quality scores among all methods. Furthermore, we are able to quantify uncertainty in the unpaired translation task and the uncertainty map can provide structural information of the temporal bone.
Variational Bayes (VB) is a popular scalable alternative to Markov chain Monte Carlo for Bayesian inference. We study a mean-field spike and slab VB approximation of widely used Bayesian model selection priors in sparse high-dimensional logistic regression. We provide non-asymptotic theoretical guarantees for the VB posterior in both $\ell_2$ and prediction loss for a sparse truth, giving optimal (minimax) convergence rates. Since the VB algorithm does not depend on the unknown truth to achieve optimality, our results shed light on effective prior choices. We confirm the improved performance of our VB algorithm over common sparse VB approaches in a numerical study.
We investigate the problem of reliably assessing group fairness when labeled examples are few but unlabeled examples are plentiful. We propose a general Bayesian framework that can augment labeled data with unlabeled data to produce more accurate and lower-variance estimates compared to methods based on labeled data alone. Our approach estimates calibrated scores for unlabeled examples in each group using a hierarchical latent variable model conditioned on labeled examples. This in turn allows for inference of posterior distributions with associated notions of uncertainty for a variety of group fairness metrics. We demonstrate that our approach leads to significant and consistent reductions in estimation error across multiple well-known fairness datasets, sensitive attributes, and predictive models. The results show the benefits of using both unlabeled data and Bayesian inference in terms of assessing whether a prediction model is fair or not.
In machine learning and computer vision, optimal transport has had significant success in learning generative models and defining metric distances between structured and stochastic data objects, that can be cast as probability measures. The key element of optimal transport is the so called lifting of an \emph{exact} cost (distance) function, defined on the sample space, to a cost (distance) between probability measures over the sample space. However, in many real life applications the cost is \emph{stochastic}: e.g., the unpredictable traffic flow affects the cost of transportation between a factory and an outlet. To take this stochasticity into account, we introduce a Bayesian framework for inferring the optimal transport plan distribution induced by the stochastic cost, allowing for a principled way to include prior information and to model the induced stochasticity on the transport plans. Additionally, we tailor an HMC method to sample from the resulting transport plan posterior distribution.
Underlying many Bayesian inference techniques that seek to approximate the posterior as a Gaussian distribution is a fundamental linear algebra problem that must be solved for both the mean and key entries of the covariance. Even when the true posterior is not Gaussian (e.g., in the case of nonlinear measurement functions) we can use variational schemes that repeatedly solve this linear algebra problem at each iteration. In most cases, the question is not whether a solution to this problem exists, but rather how we can exploit problem-specific structure to find it efficiently. Our contribution is to clearly state the Fundamental Linear Algebra Problem of Gaussian Inference (FLAPOGI) and to provide a novel presentation (using Kronecker algebra) of the not-so-well-known result of Takahashi et al. (1973) that makes it possible to solve for key entries of the covariance matrix. We first provide a global solution and then a local version that can be implemented using local message passing amongst a collection of agents calculating in parallel. Contrary to belief propagation, our local scheme is guaranteed to converge in both the mean and desired covariance quantities to the global solution even when the underlying factor graph is loopy; in the case of synchronous updates, we provide a bound on the number of iterations required for convergence. Compared to belief propagation, this guaranteed convergence comes at the cost of additional storage, calculations, and communication links in the case of loops; however, we show how these can be automatically constructed on the fly using only local information.
Bayesian inference without the likelihood evaluation, or likelihood-free inference, has been a key research topic in simulation studies for gaining quantitatively validated simulation models on real-world datasets. As the likelihood evaluation is inaccessible, previous papers train the amortized neural network to estimate the ground-truth posterior for the simulation of interest. Training the network and accumulating the dataset alternatively in a sequential manner could save the total simulation budget by orders of magnitude. In the data accumulation phase, the new simulation inputs are chosen within a portion of the total simulation budget to accumulate upon the collected dataset. This newly accumulated data degenerates because the set of simulation inputs is hardly mixed, and this degenerated data collection process ruins the posterior inference. This paper introduces a new sampling approach, called Neural Proposal (NP), of the simulation input that resolves the biased data collection as it guarantees the i.i.d. sampling. The experiments show the improved performance of our sampler, especially for the simulations with multi-modal posteriors.
Animals are equipped with a rich innate repertoire of sensory, behavioral and motor skills, which allows them to interact with the world immediately after birth. At the same time, many behaviors are highly adaptive and can be tailored to specific environments by means of learning. In this work, we use mathematical analysis and the framework of meta-learning (or 'learning to learn') to answer when it is beneficial to learn such an adaptive strategy and when to hard-code a heuristic behavior. We find that the interplay of ecological uncertainty, task complexity and the agents' lifetime has crucial effects on the meta-learned amortized Bayesian inference performed by an agent. There exist two regimes: One in which meta-learning yields a learning algorithm that implements task-dependent information-integration and a second regime in which meta-learning imprints a heuristic or 'hard-coded' behavior. Further analysis reveals that non-adaptive behaviors are not only optimal for aspects of the environment that are stable across individuals, but also in situations where an adaptation to the environment would in fact be highly beneficial, but could not be done quickly enough to be exploited within the remaining lifetime. Hard-coded behaviors should hence not only be those that always work, but also those that are too complex to be learned within a reasonable time frame.