Online learning via Bayes' theorem allows new data to be continuously integrated into an agent's current beliefs. However, a naive application of Bayesian methods in non stationary environments leads to slow adaptation and results in state estimates that may converge confidently to the wrong parameter value. A common solution when learning in changing environments is to discard/downweight past data; however, this simple mechanism of "forgetting" fails to account for the fact that many real-world environments involve revisiting similar states. We propose a new framework, Bayes with Adaptive Memory (BAM), that takes advantage of past experience by allowing the agent to choose which past observations to remember and which to forget. We demonstrate that BAM generalizes many popular Bayesian update rules for non-stationary environments. Through a variety of experiments, we demonstrate the ability of BAM to continuously adapt in an ever-changing world.
The Bayes-Adaptive Markov Decision Process (BAMDP) formalism pursues the Bayes-optimal solution to the exploration-exploitation trade-off in reinforcement learning. As the computation of exact solutions to Bayesian reinforcement-learning problems is intractable, much of the literature has focused on developing suitable approximation algorithms. In this work, before diving into algorithm design, we first define, under mild structural assumptions, a complexity measure for BAMDP planning. As efficient exploration in BAMDPs hinges upon the judicious acquisition of information, our complexity measu…
School of Mathematical and Data Science（数学与数据科学学院）；West Virginia University（西弗吉尼亚大学）；Department of Chemical and Biomedical Engineering（化学与生物医学工程系）；The Medical College of Wisconsin（威斯康星医学院）；Department of Biostatistics and Bioinformatics（生物统计学与生物信息学系）
We present the first PAC optimal algorithm for Bayes-Adaptive Markov Decision Processes (BAMDPs) in continuous state and action spaces, to the best of our knowledge. The BAMDP framework elegantly addresses model uncertainty by incorporating Bayesian belief updates into long-term expected return. However, computing an exact optimal Bayesian policy is intractable. Our key insight is to compute a near-optimal value function by covering the continuous state-belief-action space with a finite set of representative samples and exploiting the Lipschitz continuity of the value function. We prove the ne…
The popularity of Bayesian statistical methods has increased dramatically in recent years across many research areas and industrial applications. This is the result of a variety of methodological advances with faster and cheaper hardware as well as the development of new software tools. Here we introduce an open source Python package named Bambi (BAyesian Model Building Interface) that is built on top of the PyMC probabilistic programming framework and the ArviZ package for exploratory analysis of Bayesian models. Bambi makes it easy to specify complex generalized linear hierarchical models us…
In this paper we present an algorithm for rapid Bayesian analysis that combines the benefits of nested sampling and artificial neural networks. The blind accelerated multimodal Bayesian inference (BAMBI) algorithm implements the MultiNest package for nested sampling as well as the training of an artificial neural network (NN) to learn the likelihood function. In the case of computationally expensive likelihoods, this allows the substitution of a much more rapid approximation in order to increase significantly the speed of the analysis. We begin by demonstrating, with a few toy examples, the ab…
Statistical analyses of directional or angular data have applications in a variety of fields, such as geology, meteorology and bioinformatics. There is substantial literature on descriptive and inferential techniques for univariate angular data, with the bivariate (or more generally, multivariate) cases receiving more attention in recent years. More specifically, the bivariate wrapped normal, von Mises sine and von Mises cosine distributions, and mixtures thereof, have been proposed for practical use. However, there is a lack of software implementing these distributions and the associated infe…
Association testing aims to discover the underlying relationship between genotypes (usually Single Nucleotide Polymorphisms, or SNPs) and phenotypes (attributes, or traits). The typically large data sets used in association testing often contain missing values. Standard statistical methods either impute the missing values using relatively simple assumptions, or delete them, or both, which can generate biased results. Here we describe the Bayesian hierarchical model BAMD (Bayesian Association with Missing Data). BAMD is a Gibbs sampler, in which missing values are multiply imputed based upon al…
Aalto University（阿尔托大学）；Helsinki Institute for Information Technology（赫尔辛基信息技术研究所）；Institute for Molecular Medicine Finland（芬兰分子医学研究所）
Metabolic flux balance analyses are a standard tool in analysing metabolic reaction rates compatible with measurements, steady-state and the metabolic reaction network stoichiometry. Flux analysis methods commonly place unrealistic assumptions on fluxes due to the convenience of formulating the problem as a linear programming model, and most methods ignore the notable uncertainty in flux estimates. We introduce a novel paradigm of Bayesian metabolic flux analysis that models the reactions of the whole genome-scale cellular system in probabilistic terms, and can infer the full flux vector distr…
An ideal cognitively-inspired memory system would compress and organize incoming items. The Kanerva Machine (Wu et al, 2018) is a Bayesian model that naturally implements online memory compression. However, the organization of the Kanerva Machine is limited by its use of a single Gaussian random matrix for storage. Here we introduce the Product Kanerva Machine, which dynamically combines many smaller Kanerva Machines. Its hierarchical structure provides a principled way to abstract invariant features and gives scaling and capacity advantages over single Kanerva Machines. We show that it can ex…
Cyprus University of Technology（塞浦路斯理工大学）；University of Patras（帕特雷大学）；Rutgers University（罗格斯大学）
Memory-efficient continuous Sign Language Translation is a significant challenge for the development of assisted technologies with real-time applicability for the deaf. In this work, we introduce a paradigm of designing recurrent deep networks whereby the output of the recurrent layer is derived from appropriate arguments from nonparametric statistics. A novel variational Bayesian sequence-to-sequence network architecture is proposed that consists of a) a full Gaussian posterior distribution for data-driven memory compression and b) a nonparametric Indian Buffet Process prior for regularizatio…
Memory by Design: Bayesian Layers for Sequence Modeling
Champalimaud Research, Champalimaud Foundation, Portugal（恰帕拉马德研究、恰帕拉马德基金会、葡萄牙）；Center for Neural Science, New York University, USA（神经科学中心、纽约大学、美国）；Center for Data Science, New York University, USA（数据科学中心、纽约大学、美国）；RyvivyR Inc., NY, USA（RyvivyR公司、纽约、美国）
The joint modeling of longitudinal and time-to-event data is an important tool of growing popularity to gain insights into the association between a biomarker and an event process. We develop a general framework of flexible additive joint models that allows the specification of a variety of effects, such as smooth nonlinear, time-varying and random effects, in the longitudinal and survival parts of the models. Our extensions are motivated by the investigation of the relationship between fluctuating disease-specific markers, in this case autoantibodies, and the progression to the autoimmune dis…
Inference in the presence of outliers is an important field of research as outliers are ubiquitous and may arise across a variety of problems and domains. Bayesian optimization is method that heavily relies on probabilistic inference. This allows outstanding sample efficiency because the probabilistic machinery provides a memory of the whole optimization process. However, that virtue becomes a disadvantage when the memory is populated with outliers, inducing bias in the estimation. In this paper, we present an empirical evaluation of Bayesian optimization methods in the presence of outliers. T…
Value-at-Risk (VaR) and Expected Shortfall (ES) are widely used in the financial sector to measure the market risk and manage the extreme market movement. The recent link between the quantile score function and the Asymmetric Laplace density has led to a flexible likelihood-based framework for joint modelling of VaR and ES. It is of high interest in financial applications to be able to capture the underlying joint dynamics of these two quantities. We address this problem by developing a hybrid model that is based on the Asymmetric Laplace quasi-likelihood and employs the Long Short-Term Memory…
Joint models of longitudinal and survival data have become an important tool for modeling associations between longitudinal biomarkers and event processes. The association between marker and log-hazard is assumed to be linear in existing shared random effects models, with this assumption usually remaining unchecked. We present an extended framework of flexible additive joint models that allows the estimation of nonlinear, covariate specific associations by making use of Bayesian P-splines. Our joint models are estimated in a Bayesian framework using structured additive predictors for all model…
University of Toronto（多伦多大学）；Stanford Unviersity（斯坦福大学）
We perform scalable approximate inference in continuous-depth Bayesian neural networks. In this model class, uncertainty about separate weights in each layer gives hidden units that follow a stochastic differential equation. We demonstrate gradient-based stochastic variational inference in this infinite-parameter setting, producing arbitrarily-flexible approximate posteriors. We also derive a novel gradient estimator that approaches zero variance as the approximate posterior over weights approaches the true posterior. This approach brings continuous-depth Bayesian neural nets to a competitive …
In the Bayesian approach to sequential decision making, exact calculation of the (subjective) utility is intractable. This extends to most special cases of interest, such as reinforcement learning problems. While utility bounds are known to exist for this problem, so far none of them were particularly tight. In this paper, we show how to efficiently calculate a lower bound, which corresponds to the utility of a near-optimal memoryless policy for the decision problem, which is generally different from both the Bayes-optimal policy and the policy which is optimal for the expected MDP under the c…
In a typical online learning scenario, a learner is required to process a large data stream using a small memory buffer. Such a requirement is usually in conflict with a learner's primary pursuit of prediction accuracy. To address this dilemma, we introduce a novel Bayesian online classi cation algorithm, called the Virtual Vector Machine. The virtual vector machine allows you to smoothly trade-off prediction accuracy with memory size. The virtual vector machine summarizes the information contained in the preceding data stream by a Gaussian distribution over the classi cation weights plus a co…
Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this paper, we study Bayesian optimization for constrained problems in the general case that noise may be present in the constraint functions, and the objective and constraints may be evaluated independently. We provide motivating practical examples, and present a general framework to solve such problems. We demonstrate the effectiveness of our approach on optimi…
University of Chicago（芝加哥大学）；Arizona State University（亚利桑那州立大学）
Bayesian additive regression trees (BART) (Chipman et. al., 2010) is a powerful predictive model that often outperforms alternative models at out-of-sample prediction. BART is especially well-suited to settings with unstructured predictor variables and substantial sources of unmeasured variation as is typical in the social, behavioral and health sciences. This paper develops a modified version of BART that is amenable to fast posterior estimation. We present a stochastic hill climbing algorithm that matches the remarkable predictive accuracy of previous BART implementations, but is many times …
Fert Beijing Research Institute（费尔特北京研究院）；BDBC（BDBC（北京大数据研究院））；Qingdao Research Institute, Beihang University（北京航空航天大学青岛研究院）；Beihang University（北京航空航天大学）；Delft University of Technology（代尔夫特理工大学）
Bayesian method is capable of capturing real world uncertainties/incompleteness and properly addressing the over-fitting issue faced by deep neural networks. In recent years, Bayesian Neural Networks (BNNs) have drawn tremendous attentions of AI researchers and proved to be successful in many applications. However, the required high computation complexity makes BNNs difficult to be deployed in computing systems with limited power budget. In this paper, an efficient BNN inference flow is proposed to reduce the computation cost then is evaluated by means of both software and hardware implementat…
In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easil…
Bayesian Attention Mechanism: A Probabilistic Framework for Positional Encoding and Context Length Extrapolation
MALTA – Machine Learning Theory and Applications Lab, PUCRS – Porto Alegre, Brazil（MALTA机器学习理论与应用实验室，PUCRS-波士顿实验室，巴西）；Kunumi Institute, Brazil（昆米研究所，巴西）
For a Gaussian time series with long-memory behavior, we use the FEXP-model for semi-parametric estimation of the long-memory parameter $d$. The true spectral density $f_o$ is assumed to have long-memory parameter $d_o$ and a FEXP-expansion of Sobolev-regularity $\be > 1$. We prove that when $k$ follows a Poisson or geometric prior, or a sieve prior increasing at rate $n^{\frac{1}{1+2\be}}$, $d$ converges to $d_o$ at a suboptimal rate. When the sieve prior increases at rate $n^{\frac{1}{2\be}}$ however, the minimax rate is almost obtained. Our results can be seen as a Bayesian equivalent of th…
Microsoft Research（微软研究院）；University of Sydney（悉尼大学）
We provide a method for approximating Bayesian inference using rejection sampling. We not only make the process efficient, but also dramatically reduce the memory required relative to conventional methods by combining rejection sampling with particle filtering. We also provide an approximate form of rejection sampling that makes rejection filtering tractable in cases where exact rejection sampling is not efficient. Finally, we present several numerical examples of rejection filtering that show its ability to track time dependent parameters in online settings and also benchmark its performance …
In forecasting problems it is important to know whether or not recent events represent a regime change (low long-term predictive potential), or rather a local manifestation of longer term effects (potentially higher predictive potential). Mathematically, a key question is about whether the underlying stochastic process exhibits "memory", and if so whether the memory is "long" in a precise sense. Being able to detect or rule out such effects can have a profound impact on speculative investment (e.g., in financial markets) and inform public policy (e.g., characterising the size and timescales of…
In relational event networks, the tendency for actors to interact with each other depends greatly on the past interactions between the actors in a social network. Both the quantity of past interactions and the time that elapsed since the past interactions occurred affect the actors' decision-making to interact with other actors in the network. Recently occurred events generally have a stronger influence on current interaction behavior than past events that occurred a long time ago--a phenomenon known as "memory decay". Previous studies either predefined a short-run and long-run memory or fixed…
University of Oxford（牛津大学）；University of Amsterdam（阿姆斯特丹大学）
We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In particular, partially stochastic networks with only $n$ stochastic biases are universal probabilistic predictors for $n$-dimensional predictive problems. In empirical investigations, we find no systematic benefit of full stochasticity across four different inference modalities an…
We introduce a new class of semiparametric latent variable models for long memory discretized event data. The proposed methodology is motivated by a study of bird vocalizations in the Amazon rain forest; the timings of vocalizations exhibit self-similarity and long range dependence ruling out models based on Poisson processes. The proposed class of FRActional Probit (FRAP) models is based on thresholding of a latent process consisting of an additive expansion of a smooth Gaussian process with a fractional Brownian motion. We develop a Bayesian approach to inference using Markov chain Monte Car…
Recent developments in data science and big data research have produced an abundance of large data sets that are too big to be analyzed in their entirety, due to limits on either computer memory or storage capacity. Here, we introduce our R package 'BayesSummaryStatLM' for Bayesian linear regression models with Markov chain Monte Carlo implementation that overcomes these limitations. Our Bayesian models use only summary statistics of data as input; these summary statistics can be calculated from subsets of big data and combined over subsets. Thus, complete data sets do not need to be read into…
Working memory (WM) was one of the first cognitive processes studied with functional magnetic resonance imaging. With now over 20 years of studies on WM, each study with tiny sample sizes, there is a need for meta-analysis to identify the brain regions that are consistently activated by WM tasks, and to understand the interstudy variation in those activations. However, current methods in the field cannot fully account for the spatial nature of neuroimaging meta-analysis data or the heterogeneity observed among WM studies. In this work, we propose a fully Bayesian random-effects metaregression …
A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density $f(λ)$ can be written as $f(λ)=|λ|^{-2d}g(|λ|)$, where $0<d<1/2$ (resp., $-1/2<d<0$), and $g$ is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both $d$ and $g$, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our…
We present an end-to-end trained memory system that quickly adapts to new data and generates samples like them. Inspired by Kanerva's sparse distributed memory, it has a robust distributed reading and writing mechanism. The memory is analytically tractable, which enables optimal on-line compression via a Bayesian update-rule. We formulate it as a hierarchical conditional generative model, where memory provides a rich data-dependent prior distribution. Consequently, the top-down memory and bottom-up perception are combined to produce the code representing an observation. Empirically, we demonst…
Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=μ$ and covariance function $γ(τ)=E(X_t-μ)(X_{t+τ}-μ)$. Let $f(λ)$ be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density $f(λ)$ can be written as the product of a slowly varying function $\tilde{f}(λ)$ and the quantity $λ^{-2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, ou…
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 l…
We introduce a dynamic generative model, Bayesian allocation model (BAM), which establishes explicit connections between nonnegative tensor factorization (NTF), graphical models of discrete probability distributions and their Bayesian extensions, and the topic models such as the latent Dirichlet allocation. BAM is based on a Poisson process, whose events are marked by using a Bayesian network, where the conditional probability tables of this network are then integrated out analytically. We show that the resulting marginal process turns out to be a Polya urn, an integer valued self-reinforcing …
Attack graphs are a powerful tool for security risk assessment by analysing network vulnerabilities and the paths attackers can use to compromise network resources. The uncertainty about the attacker's behaviour makes Bayesian networks suitable to model attack graphs to perform static and dynamic analysis. Previous approaches have focused on the formalization of attack graphs into a Bayesian model rather than proposing mechanisms for their analysis. In this paper we propose to use efficient algorithms to make exact inference in Bayesian attack graphs, enabling the static and dynamic network ri…
Studies of memory trajectories using longitudinal data often result in highly non-representative samples due to selective study enrollment and attrition. An additional bias comes from practice effects that result in improved or maintained performance due to familiarity with test content or context. These challenges may bias study findings and severely distort the ability to generalize to the target population. In this study we propose an approach for estimating the finite population mean of a longitudinal outcome conditioning on being alive at a specific time point. We develop a flexible Bayes…
We consider alternate formulations of recently proposed hierarchical Nearest Neighbor Gaussian Process (NNGP) models (Datta et al., 2016a) for improved convergence, faster computing time, and more robust and reproducible Bayesian inference. Algorithms are defined that improve CPU memory management and exploit existing high-performance numerical linear algebra libraries. Computational and inferential benefits are assessed for alternate NNGP specifications using simulated datasets and remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory U…
The present article is the reply to the discussion of our earlier "Not only defended but also applied" (arXiv:1006.5366, to appear in The American Statistician) that arose from our memory of a particularly intemperate anti-Bayesian statement in Feller's beautiful and classic book on probability theory. We felt that it was worth exploring the very extremeness of Feller's words, along with similar anti-Bayesian remarks by others, in order to better understand the background underlying controversies that still exist regarding the foundations of statistics. We thank the four discussants of our art…
