We introduce a flexible, scalable Bayesian inference framework for nonlinear dynamical systems characterised by distinct and hierarchical variability at the individual, group, and population levels. Our model class is a generalisation of nonlinear mixed-effects (NLME) dynamical systems, the statistical workhorse for many experimental sciences. We cast parameter inference as stochastic optimisation of an end-to-end differentiable, block-conditional variational autoencoder. We specify the dynamics of the data-generating process as an ordinary differential equation (ODE) such that both the ODE and its solver are fully differentiable. This model class is highly flexible: the ODE right-hand sides can be a mixture of user-prescribed or "white-box" sub-components and neural network or "black-box" sub-components. Using stochastic optimisation, our amortised inference algorithm could seamlessly scale up to massive data collection pipelines (common in labs with robotic automation). Finally, our framework supports interpretability with respect to the underlying dynamics, as well as predictive generalization to unseen combinations of group components (also called "zero-shot" learning). We empirically validate our method by predicting the dynamic behaviour of bacteria that were genetically engineered to function as biosensors. Our implementation of the framework, the dataset, and all code to reproduce the experimental results is available at https://www.github.com/Microsoft/vi-hds .
Modern deep neural networks are well known to be brittle in the face of unknown data instances and recognition of the latter remains a challenge. Although it is inevitable for continual-learning systems to encounter such unseen concepts, the corresponding literature appears to nonetheless focus primarily on alleviating catastrophic interference with learned representations. In this work, we introduce a probabilistic approach that connects these perspectives based on variational inference in a single deep autoencoder model. Specifically, we propose to bound the approximate posterior by fitting regions of high density on the basis of correctly classified data points. These bounds are shown to serve a dual purpose: unseen unknown out-of-distribution data can be distinguished from already trained known tasks towards robust application. Simultaneously, to retain already acquired knowledge, a generative replay process can be narrowed to strictly in-distribution samples, in order to significantly alleviate catastrophic interference.
In rank aggregation (RA), a collection of preferences from different users are summarized into a total order under the assumption of homogeneity of users. Model misspecification in RA arises since the homogeneity assumption fails to be satisfied in the complex real-world situation. Existing robust RAs usually resort to an augmentation of the ranking model to account for additional noises, where the collected preferences can be treated as a noisy perturbation of idealized preferences. Since the majority of robust RAs rely on certain perturbation assumptions, they cannot generalize well to agnostic noise-corrupted preferences in the real world. In this paper, we propose CoarsenRank, which possesses robustness against model misspecification. Specifically, the properties of our CoarsenRank are summarized as follows: (1) CoarsenRank is designed for mild model misspecification, which assumes there exist the ideal preferences (consistent with model assumption) that locates in a neighborhood of the actual preferences. (2) CoarsenRank then performs regular RAs over a neighborhood of the preferences instead of the original dataset directly. Therefore, CoarsenRank enjoys robustness against model misspecification within a neighborhood. (3) The neighborhood of the dataset is defined via their empirical data distributions. Further, we put an exponential prior on the unknown size of the neighborhood, and derive a much-simplified posterior formula for CoarsenRank under particular divergence measures. (4) CoarsenRank is further instantiated to Coarsened Thurstone, Coarsened Bradly-Terry, and Coarsened Plackett-Luce with three popular probability ranking models. Meanwhile, tractable optimization strategies are introduced with regards to each instantiation respectively. In the end, we apply CoarsenRank on four real-world datasets.
Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical architecture, HINT allows sampling from a joint distribution p(y,x) and the corresponding posterior p(x|y) using a single invertible network. We evaluate our method on some standard data sets and benchmark its full power for density estimation and Bayesian inference on a novel data set of 2D shapes in Fourier parameterization, which enables consistent visualization of samples for different dimensionalities.
We propose a new scalable multi-class Gaussian process classification approach building on a novel modified softmax likelihood function. The new likelihood has two benefits: it leads to well-calibrated uncertainty estimates and allows for an efficient latent variable augmentation. The augmented model has the advantage that it is conditionally conjugate leading to a fast variational inference method via block coordinate ascent updates. Previous approaches suffered from a trade-off between uncertainty calibration and speed. Our experiments show that our method leads to well-calibrated uncertainty estimates and competitive predictive performance while being up to two orders faster than the state of the art.
Reinforcement learning agents are faced with two types of uncertainty. Epistemic uncertainty stems from limited data and is useful for exploration, whereas aleatoric uncertainty arises from stochastic environments and must be accounted for in risk-sensitive applications. We highlight the challenges involved in simultaneously estimating both of them, and propose a framework for disentangling and estimating these uncertainties on learned Q-values. We derive unbiased estimators of these uncertainties and introduce an uncertainty-aware DQN algorithm, which we show exhibits safe learning behavior and outperforms other DQN variants on the MinAtar testbed.
Objective: To compare different deep learning architectures for predicting the risk of readmission within 30 days of discharge from the intensive care unit (ICU). The interpretability of attention-based models is leveraged to describe patients-at-risk. Methods: Several deep learning architectures making use of attention mechanisms, recurrent layers, neural ordinary differential equations (ODEs), and medical concept embeddings with time-aware attention were trained using publicly available electronic medical record data (MIMIC-III) associated with 45,298 ICU stays for 33,150 patients. Bayesian inference was used to compute the posterior over weights of an attention-based model. Odds ratios associated with an increased risk of readmission were computed for static variables. Diagnoses, procedures, medications, and vital signs were ranked according to the associated risk of readmission. Results: A recurrent neural network, with time dynamics of code embeddings computed by neural ODEs, achieved the highest average precision of 0.331 (AUROC: 0.739, F1-Score: 0.372). Predictive accuracy was comparable across neural network architectures. Groups of patients at risk included those suffering from infectious complications, with chronic or progressive conditions, and for whom standard medical care was not suitable. Conclusions: Attention-based networks may be preferable to recurrent networks if an interpretable model is required, at only marginal cost in predictive accuracy.
How can one perform Bayesian inference on stochastic simulators with intractable likelihoods? A recent approach is to learn the posterior from adaptively proposed simulations using neural network-based conditional density estimators. However, existing methods are limited to a narrow range of proposal distributions or require importance weighting that can limit performance in practice. Here we present automatic posterior transformation (APT), a new sequential neural posterior estimation method for simulation-based inference. APT can modify the posterior estimate using arbitrary, dynamically updated proposals, and is compatible with powerful flow-based density estimators. It is more flexible, scalable and efficient than previous simulation-based inference techniques. APT can operate directly on high-dimensional time series and image data, opening up new applications for likelihood-free inference.
We use ensemble machine learning algorithms to study the evolution of magnetic fields in magnetohydrodynamic (MHD) turbulence that is helically forced. We perform direct numerical simulations of helically forced turbulence using mean field formalism, with electromotive force (EMF) modeled both as a linear and non-linear function of the mean magnetic field and current density. The form of the EMF is determined using regularized linear regression and random forests. We also compare various analytical models to the data using Bayesian inference with Markov Chain Monte Carlo (MCMC) sampling. Our results demonstrate that linear regression is largely successful at predicting the EMF and the use of more sophisticated algorithms (random forests, MCMC) do not lead to significant improvement in the fits. We conclude that the data we are looking at is effectively low dimensional and essentially linear. Finally, to encourage further exploration by the community, we provide all of our simulation data and analysis scripts as open source IPython notebooks.
An important task in machine learning and statistics is the approximation of a probability measure by an empirical measure supported on a discrete point set. Stein Points are a class of algorithms for this task, which proceed by sequentially minimising a Stein discrepancy between the empirical measure and the target and, hence, require the solution of a non-convex optimisation problem to obtain each new point. This paper removes the need to solve this optimisation problem by, instead, selecting each new point based on a Markov chain sample path. This significantly reduces the computational cost of Stein Points and leads to a suite of algorithms that are straightforward to implement. The new algorithms are illustrated on a set of challenging Bayesian inference problems, and rigorous theoretical guarantees of consistency are established.
We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation locations. Motivated by recent progress in the related problem of batch Bayesian optimisation, we develop various batch-sequential design strategies which allow to run some of the potentially costly simulations in parallel. We analyse the properties of the resulting method theoretically and empirically. Experiments with several toy problems and simulation models suggest that our method is robust, highly parallelisable, and sample-efficient.
We recapitulate the Bayesian formulation of neural network based classifiers and show that, while sampling from the posterior does indeed lead to better generalisation than is obtained by standard optimisation of the cost function, even better performance can in general be achieved by sampling finite temperature ($T$) distributions derived from the posterior. Taking the example of two different deep (3 hidden layers) classifiers for MNIST data, we find quite different $T$ values to be appropriate in each case. In particular, for a typical neural network classifier a clear minimum of the test error is observed at $T>0$. This suggests an early stopping criterion for full batch simulated annealing: cool until the average validation error starts to increase, then revert to the parameters with the lowest validation error. As $T$ is increased classifiers transition from accurate classifiers to classifiers that have higher training error than assigning equal probability to each class. Efficient studies of these temperature-induced effects are enabled using a replica-exchange Hamiltonian Monte Carlo simulation technique. Finally, we show how thermodynamic integration can be used to perform model selection for deep neural networks. Similar to the Laplace approximation, this approach assumes that the posterior is dominated by a single mode. Crucially, however, no assumption is made about the shape of that mode and it is not required to precisely compute and invert the Hessian.
We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes' rule as infinite-dimensional optimization problem (Csiszar, 1975; Donsker and Varadhan; 1975, Zellner; 1988). First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.
Inspired by applications in sports where the skill of players or teams competing against each other varies over time, we propose a probabilistic model of pairwise-comparison outcomes that can capture a wide range of time dynamics. We achieve this by replacing the static parameters of a class of popular pairwise-comparison models by continuous-time Gaussian processes; the covariance function of these processes enables expressive dynamics. We develop an efficient inference algorithm that computes an approximate Bayesian posterior distribution. Despite the flexbility of our model, our inference algorithm requires only a few linear-time iterations over the data and can take advantage of modern multiprocessor computer architectures. We apply our model to several historical databases of sports outcomes and find that our approach outperforms competing approaches in terms of predictive performance, scales to millions of observations, and generates compelling visualizations that help in understanding and interpreting the data.
Dynamic paired comparison models, such as Elo and Glicko, are frequently used for sports prediction and ranking players or teams. We present an alternative dynamic paired comparison model which uses a Gaussian Process (GP) as a prior for the time dynamics rather than the Markovian dynamics usually assumed. In addition, we show that the GP model can easily incorporate covariates. We derive an efficient approximate Bayesian inference procedure based on the Laplace Approximation and sparse linear algebra. We select hyperparameters by maximising their marginal likelihood using Bayesian Optimisation, comparing the results against random search. Finally, we fit and evaluate the model on the 2018 season of ATP tennis matches, where it performs competitively, outperforming Elo and Glicko on log loss, particularly when surface covariates are included.
Robust MDPs (RMDPs) can be used to compute policies with provable worst-case guarantees in reinforcement learning. The quality and robustness of an RMDP solution are determined by the ambiguity set---the set of plausible transition probabilities---which is usually constructed as a multi-dimensional confidence region. Existing methods construct ambiguity sets as confidence regions using concentration inequalities which leads to overly conservative solutions. This paper proposes a new paradigm that can achieve better solutions with the same robustness guarantees without using confidence regions as ambiguity sets. To incorporate prior knowledge, our algorithms optimize the size and position of ambiguity sets using Bayesian inference. Our theoretical analysis shows the safety of the proposed method, and the empirical results demonstrate its practical promise.
Online detection of instantaneous changes in the generative process of a data sequence generally focuses on retrospective inference of such change points without considering their future occurrences. We extend the Bayesian Online Change Point Detection algorithm to also infer the number of time steps until the next change point (i.e., the residual time). This enables to handle observation models which depend on the total segment duration, which is useful to model data sequences with temporal scaling. The resulting inference algorithm for segment detection can be deployed in an online fashion, and we illustrate applications to synthetic and to two medical real-world data sets.
The posteriors over neural network weights are high dimensional and multimodal. Each mode typically characterizes a meaningfully different representation of the data. We develop Cyclical Stochastic Gradient MCMC (SG-MCMC) to automatically explore such distributions. In particular, we propose a cyclical stepsize schedule, where larger steps discover new modes, and smaller steps characterize each mode. We also prove non-asymptotic convergence of our proposed algorithm. Moreover, we provide extensive experimental results, including ImageNet, to demonstrate the scalability and effectiveness of cyclical SG-MCMC in learning complex multimodal distributions, especially for fully Bayesian inference with modern deep neural networks.
Increasingly complex datasets pose a number of challenges for Bayesian inference. Conventional posterior sampling based on Markov chain Monte Carlo can be too computationally intensive, is serial in nature and mixes poorly between posterior modes. Further, all models are misspecified, which brings into question the validity of the conventional Bayesian update. We present a scalable Bayesian nonparametric learning routine that enables posterior sampling through the optimization of suitably randomized objective functions. A Dirichlet process prior on the unknown data distribution accounts for model misspecification, and admits an embarrassingly parallel posterior bootstrap algorithm that generates independent and exact samples from the nonparametric posterior distribution. Our method is particularly adept at sampling from multimodal posterior distributions via a random restart mechanism. We demonstrate our method on Gaussian mixture model and sparse logistic regression examples.
We propose SWA-Gaussian (SWAG), a simple, scalable, and general purpose approach for uncertainty representation and calibration in deep learning. Stochastic Weight Averaging (SWA), which computes the first moment of stochastic gradient descent (SGD) iterates with a modified learning rate schedule, has recently been shown to improve generalization in deep learning. With SWAG, we fit a Gaussian using the SWA solution as the first moment and a low rank plus diagonal covariance also derived from the SGD iterates, forming an approximate posterior distribution over neural network weights; we then sample from this Gaussian distribution to perform Bayesian model averaging. We empirically find that SWAG approximates the shape of the true posterior, in accordance with results describing the stationary distribution of SGD iterates. Moreover, we demonstrate that SWAG performs well on a wide variety of tasks, including out of sample detection, calibration, and transfer learning, in comparison to many popular alternatives including MC dropout, KFAC Laplace, SGLD, and temperature scaling.