Missing values exist in nearly all clinical studies because data for a variable or question are not collected or not available. Inadequate handling of missing values can lead to biased results and loss of statistical power in analysis. Existing models usually do not consider privacy concerns or do not utilise the inherent correlations across multiple features to impute the missing values. In healthcare applications, we are usually confronted with high dimensional and sometimes small sample size datasets that need more effective augmentation or imputation techniques. Besides, imputation and augmentation processes are traditionally conducted individually. However, imputing missing values and augmenting data can significantly improve generalisation and avoid bias in machine learning models. A Bayesian approach to impute missing values and creating augmented samples in high dimensional healthcare data is proposed in this work. We propose folded Hamiltonian Monte Carlo (F-HMC) with Bayesian inference as a more practical approach to process the cross-dimensional relations by applying a random walk and Hamiltonian dynamics to adapt posterior distribution and generate large-scale samples. The proposed method is applied to a cancer symptom assessment dataset and confirmed to enrich the quality of data in precision, accuracy, recall, F1 score, and propensity metric.
Bayesian inference provides a uniquely rigorous approach to obtain principled justification for uncertainty in predictions, yet it is difficult to articulate suitably general prior belief in the machine learning context, where computational architectures are pure abstractions subject to frequent modifications by practitioners attempting to improve results. Parsimonious inference is an information-theoretic formulation of inference over arbitrary architectures that formalizes Occam's Razor; we prefer simple and sufficient explanations. Our universal hyperprior assigns plausibility to prior descriptions, encoded as sequences of symbols, by expanding on the core relationships between program length, Kolmogorov complexity, and Solomonoff's algorithmic probability. We then cast learning as information minimization over our composite change in belief when an architecture is specified, training data are observed, and model parameters are inferred. By distinguishing model complexity from prediction information, our framework also quantifies the phenomenon of memorization. Although our theory is general, it is most critical when datasets are limited, e.g. small or skewed. We develop novel algorithms for polynomial regression and random forests that are suitable for such data, as demonstrated by our experiments. Our approaches combine efficient encodings with prudent sampling strategies to construct predictive ensembles without cross-validation, thus addressing a fundamental challenge in how to efficiently obtain predictions from data.
This tutorial gives a quick introduction to Variational Bayes (VB), also called Variational Inference or Variational Approximation, from a practical point of view. The paper covers a range of commonly used VB methods and an attempt is made to keep the materials accessible to the wide community of data analysis practitioners. The aim is that the reader can quickly derive and implement their first VB algorithm for Bayesian inference with their data analysis problem. An end-user software package in Matlab together with the documentation can be found at https://vbayeslab.github.io/VBLabDocs/
Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends MFEnKF to work with non-linear couplings between the models. Optimal nonlinear projection and interpolation operators are obtained by appropriately trained physics-informed autoencoders, and this approach allows to construct reduced order surrogate models with less error than conventional linear methods. Numerical experiments with the canonical Lorenz '96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity filtering.
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $\Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.
This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [65]. It also includes a family of priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for p unknown covariates the method can be implemented exactly online with a cost of $O(p^3)$ in computation and $O(p^2)$ in memory per iteration -- in other words, the cost per iteration is independent of n, and in principle infinite data can be considered. For large $p$ an approximation is able to achieve promising results for a cost of $O(p)$ per iteration, in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles $n = 65536$, $p = 131073$ on a laptop in less than 30 minutes, and $n = 10^5$, $p = 2.1 \times 10^6$ overnight.
Many real-world mission-critical applications require continual online learning from noisy data and real-time decision making with a defined confidence level. Probabilistic models and stochastic neural networks can explicitly handle uncertainty in data and allow adaptive learning-on-the-fly, but their implementation in a low-power substrate remains a challenge. Here, we introduce a novel hardware fabric that implements a new class of stochastic NN called Neural-Sampling-Machine that exploits stochasticity in synaptic connections for approximate Bayesian inference. Harnessing the inherent non-linearities and stochasticity occurring at the atomic level in emerging materials and devices allows us to capture the synaptic stochasticity occurring at the molecular level in biological synapses. We experimentally demonstrate in-silico hybrid stochastic synapse by pairing a ferroelectric field-effect transistor -based analog weight cell with a two-terminal stochastic selector element. Such a stochastic synapse can be integrated within the well-established crossbar array architecture for compute-in-memory. We experimentally show that the inherent stochastic switching of the selector element between the insulator and metallic state introduces a multiplicative stochastic noise within the synapses of NSM that samples the conductance states of the FeFET, both during learning and inference. We perform network-level simulations to highlight the salient automatic weight normalization feature introduced by the stochastic synapses of the NSM that paves the way for continual online learning without any offline Batch Normalization. We also showcase the Bayesian inferencing capability introduced by the stochastic synapse during inference mode, thus accounting for uncertainty in data. We report 98.25%accuracy on standard image classification task as well as estimation of data uncertainty in rotated samples.
We propose injective generative models called Trumpets that generalize invertible normalizing flows. The proposed generators progressively increase dimension from a low-dimensional latent space. We demonstrate that Trumpets can be trained orders of magnitudes faster than standard flows while yielding samples of comparable or better quality. They retain many of the advantages of the standard flows such as training based on maximum likelihood and a fast, exact inverse of the generator. Since Trumpets are injective and have fast inverses, they can be effectively used for downstream Bayesian inference. To wit, we use Trumpet priors for maximum a posteriori estimation in the context of image reconstruction from compressive measurements, outperforming competitive baselines in terms of reconstruction quality and speed. We then propose an efficient method for posterior characterization and uncertainty quantification with Trumpets by taking advantage of the low-dimensional latent space.
We develop a variational Bayesian (VB) approach for estimating large-scale dynamic network models in the network autoregression framework. The VB approach allows for the automatic identification of the dynamic structure of such a model and obtains a direct approximation of the posterior density. Compared to Markov Chain Monte Carlo (MCMC) based sampling approaches, the VB approach achieves enhanced computational efficiency without sacrificing estimation accuracy. In the simulation study conducted here, the proposed VB approach detects various types of proper active structures for dynamic network models. Compared to the alternative approach, the proposed method achieves similar or better accuracy, and its computational time is halved. In a real data analysis scenario of day-ahead natural gas flow prediction in the German gas transmission network with 51 nodes between October 2013 and September 2015, the VB approach delivers promising forecasting accuracy along with clearly detected structures in terms of dynamic dependence.
Approximate inference in Bayesian deep networks exhibits a dilemma of how to yield high fidelity posterior approximations while maintaining computational efficiency and scalability. We tackle this challenge by introducing a novel variational structured approximation inspired by the Bayesian interpretation of Dropout regularization. Concretely, we focus on the inflexibility of the factorized structure in Dropout posterior and then propose an improved method called Variational Structured Dropout (VSD). VSD employs an orthogonal transformation to learn a structured representation on the variational Gaussian noise with plausible complexity, and consequently induces statistical dependencies in the approximate posterior. Theoretically, VSD successfully addresses the pathologies of previous Variational Dropout methods and thus offers a standard Bayesian justification. We further show that VSD induces an adaptive regularization term with several desirable properties which contribute to better generalization. Finally, we conduct extensive experiments on standard benchmarks to demonstrate the effectiveness of VSD over state-of-the-art variational methods on predictive accuracy, uncertainty estimation, and out-of-distribution detection.
A simulation is useful when the phenomenon of interest is either expensive to regenerate or irreproducible with the same context. Recently, Bayesian inference on the distribution of the simulation input parameter has been implemented sequentially to minimize the required simulation budget for the task of simulation validation to the real-world. However, the Bayesian inference is still challenging when the ground-truth posterior is multi-modal with a high-dimensional simulation output. This paper introduces a regularization technique, namely Neural Posterior Regularization (NPR), which enforces the model to explore the input parameter space effectively. Afterward, we provide the closed-form solution of the regularized optimization that enables analyzing the effect of the regularization. We empirically validate that NPR attains the statistically significant gain on benchmark performances for diverse simulation tasks.
People have made remarkable progress in game AIs, especially in domain of perfect information game. However, trick-taking poker game, as a popular form of imperfect information game, has been regarded as a challenge for a long time. Since trick-taking game requires high level of not only reasoning, but also inference to excel, it can be a new milestone for imperfect information game AI. We study Gongzhu, a trick-taking game analogous to, but slightly simpler than contract bridge. Nonetheless, the strategies of Gongzhu are complex enough for both human and computer players. We train a strong Gongzhu AI ScrofaZero from \textit{tabula rasa} by deep reinforcement learning, while few previous efforts on solving trick-taking poker game utilize the representation power of neural networks. Also, we introduce new techniques for imperfect information game including stratified sampling, importance weighting, integral over equivalent class, Bayesian inference, etc. Our AI can achieve human expert level performance. The methodologies in building our program can be easily transferred into a wide range of trick-taking games.
Bayesian inference over the reward presents an ideal solution to the ill-posed nature of the inverse reinforcement learning problem. Unfortunately current methods generally do not scale well beyond the small tabular setting due to the need for an inner-loop MDP solver, and even non-Bayesian methods that do themselves scale often require extensive interaction with the environment to perform well, being inappropriate for high stakes or costly applications such as healthcare. In this paper we introduce our method, Approximate Variational Reward Imitation Learning (AVRIL), that addresses both of these issues by jointly learning an approximate posterior distribution over the reward that scales to arbitrarily complicated state spaces alongside an appropriate policy in a completely offline manner through a variational approach to said latent reward. Applying our method to real medical data alongside classic control simulations, we demonstrate Bayesian reward inference in environments beyond the scope of current methods, as well as task performance competitive with focused offline imitation learning algorithms.
We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces the long-standing curse of dimensionality. We overcome this challenge by exploiting the intrinsic low-rank structure in the difference between the posterior and prior distributions. The parameters are projected into a low-dimensional subspace to alleviate the approximation error of KDE in high dimensions. We formulate a projected Wasserstein gradient flow and analyze its convergence property under mild assumptions. Several numerical experiments illustrate the accuracy, convergence, and complexity scalability of pWGD with respect to parameter dimension, sample size, and processor cores.
We consider adversarial training of deep neural networks through the lens of Bayesian learning, and present a principled framework for adversarial training of Bayesian Neural Networks (BNNs) with certifiable guarantees. We rely on techniques from constraint relaxation of non-convex optimisation problems and modify the standard cross-entropy error model to enforce posterior robustness to worst-case perturbations in $\epsilon$-balls around input points. We illustrate how the resulting framework can be combined with methods commonly employed for approximate inference of BNNs. In an empirical investigation, we demonstrate that the presented approach enables training of certifiably robust models on MNIST, FashionMNIST and CIFAR-10 and can also be beneficial for uncertainty calibration. Our method is the first to directly train certifiable BNNs, thus facilitating their deployment in safety-critical applications.
Estimation of spatially-varying parameters for computationally expensive forward models governed by partial differential equations is addressed. A novel multiscale Bayesian inference approach is introduced based on deep probabilistic generative models. Such generative models provide a flexible representation by inferring on each scale a low-dimensional latent encoding while allowing hierarchical parameter generation from coarse- to fine-scales. Combining the multiscale generative model with Markov Chain Monte Carlo (MCMC), inference across scales is achieved enabling us to efficiently obtain posterior parameter samples at various scales. The estimation of coarse-scale parameters using a low-dimensional latent embedding captures global and notable parameter features using an inexpensive but inaccurate solver. MCMC sampling of the fine-scale parameters is enabled by utilizing the posterior information in the immediate coarser-scale. In this way, the global features are identified in the coarse-scale with inference of low-dimensional variables and inexpensive forward computation, and the local features are refined and corrected in the fine-scale. The developed method is demonstrated with two types of permeability estimation for flow in heterogeneous media. One is a Gaussian random field (GRF) with uncertain length scales, and the other is channelized permeability with the two regions defined by different GRFs. The obtained results indicate that the method allows high-dimensional parameter estimation while exhibiting stability, efficiency and accuracy.
In images collected by astronomical surveys, stars and galaxies often overlap visually. Deblending is the task of distinguishing and characterizing individual light sources in survey images. We propose StarNet, a Bayesian method to deblend sources in astronomical images of crowded star fields. StarNet leverages recent advances in variational inference, including amortized variational distributions and an optimization objective targeting an expectation of the forward KL divergence. In our experiments with SDSS images of the M2 globular cluster, StarNet is substantially more accurate than two competing methods: Probabilistic Cataloging (PCAT), a method that uses MCMC for inference, and DAOPHOT, a software pipeline employed by SDSS for deblending. In addition, the amortized approach to inference gives StarNet the scaling characteristics necessary to perform Bayesian inference on modern astronomical surveys.
The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered equation to noise and the complexities of model selection. In this work, we propose an advanced Bayesian sparse learning algorithm for PDE discovery with variable coefficients, predominantly when the coefficients are spatially or temporally dependent. Specifically, we apply threshold Bayesian group Lasso regression with a spike-and-slab prior (tBGL-SS) and leverage a Gibbs sampler for Bayesian posterior estimation of PDE coefficients. This approach not only enhances the robustness of point estimation with valid uncertainty quantification but also relaxes the computational burden from Bayesian inference through the integration of coefficient thresholds as an approximate MCMC method. Moreover, from the quantified uncertainties, we propose a Bayesian total error bar criteria for model selection, which outperforms classic metrics including the root mean square and the Akaike information criterion. The capability of this method is illustrated by the discovery of several classical benchmark PDEs with spatially or temporally varying coefficients from solution data obtained from the reference simulations. In the experiments, we show that the tBGL-SS method is more robust than the baseline methods under noisy environments and provides better model selection criteria along the regularization path.
Recent works have presented promising results from the application of machine learning (ML) to the modeling of flow rates in oil and gas wells. Encouraging results and advantageous properties of ML models, such as computationally cheap evaluation and ease of calibration to new data, have sparked optimism for the development of data-driven virtual flow meters (VFMs). Data-driven VFMs are developed in the small data regime, where it is important to question the uncertainty and robustness of models. The modeling of uncertainty may help to build trust in models, which is a prerequisite for industrial applications. The contribution of this paper is the introduction of a probabilistic VFM based on Bayesian neural networks. Uncertainty in the model and measurements is described, and the paper shows how to perform approximate Bayesian inference using variational inference. The method is studied by modeling on a large and heterogeneous dataset, consisting of 60 wells across five different oil and gas assets. The predictive performance is analyzed on historical and future test data, where an average error of 4-6% and 8-13% is achieved for the 50% best performing models, respectively. Variational inference appears to provide more robust predictions than the reference approach on future data. Prediction performance and uncertainty calibration is explored in detail and discussed in light of four data challenges. The findings motivate the development of alternative strategies to improve the robustness of data-driven VFMs.
We establish exact asymptotic expressions for the normalized mutual information and minimum mean-square-error (MMSE) of sparse linear regression in the sub-linear sparsity regime. Our result is achieved by a generalization of the adaptive interpolation method in Bayesian inference for linear regimes to sub-linear ones. A modification of the well-known approximate message passing algorithm to approach the MMSE fundamental limit is also proposed, and its state evolution is rigorously analyzed. Our results show that the traditional linear assumption between the signal dimension and number of observations in the replica and adaptive interpolation methods is not necessary for sparse signals. They also show how to modify the existing well-known AMP algorithms for linear regimes to sub-linear ones.