Traditionally, the field of computational Bayesian statistics has been divided into two main subfields: variational methods and Markov chain Monte Carlo (MCMC). In recent years, however, several methods have been proposed based on combining variational Bayesian inference and MCMC simulation in order to improve their overall accuracy and computational efficiency. This marriage of fast evaluation and flexible approximation provides a promising means of designing scalable Bayesian inference methods. In this paper, we explore the possibility of incorporating variational approximation into a state-of-the-art MCMC method, Hamiltonian Monte Carlo (HMC), to reduce the required gradient computation in the simulation of Hamiltonian flow, which is the bottleneck for many applications of HMC in big data problems. To this end, we use a {\it free-form} approximation induced by a fast and flexible surrogate function based on single-hidden layer feedforward neural networks. The surrogate provides sufficiently accurate approximation while allowing for fast exploration of parameter space, resulting in an efficient approximate inference algorithm. We demonstrate the advantages of our method on both synthetic and real data problems.
For big data analysis, high computational cost for Bayesian methods often limits their applications in practice. In recent years, there have been many attempts to improve computational efficiency of Bayesian inference. Here we propose an efficient and scalable computational technique for a state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian Monte Carlo (HMC). The key idea is to explore and exploit the structure and regularity in parameter space for the underlying probabilistic model to construct an effective approximation of its geometric properties. To this end, we build a surrogate function to approximate the target distribution using properly chosen random bases and an efficient optimization process. The resulting method provides a flexible, scalable, and efficient sampling algorithm, which converges to the correct target distribution. We show that by choosing the basis functions and optimization process differently, our method can be related to other approaches for the construction of surrogate functions such as generalized additive models or Gaussian process models. Experiments based on simulated and real data show that our approach leads to substantially more efficient sampling algorithms compared to existing state-of-the art methods.
We consider the inference of the structure of an undirected graphical model in an exact Bayesian framework. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. This task would be intractable without any restriction on the considered graphs, so we limit our exploration to mixtures of spanning trees. We consider the inference of the structure of an undirected graphical model in a Bayesian framework. To avoid convergence issues and highly demanding Monte Carlo sampling, we focus on exact inference. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. To this aim, we restrict the set of considered graphs to mixtures of spanning trees. We investigate under which conditions on the priors - on both tree structures and parameters - exact Bayesian inference can be achieved. Under these conditions, we derive a fast an exact algorithm to compute the posterior probability for an edge to belong to {the tree model} using an algebraic result called the Matrix-Tree theorem. We show that the assumption we have made does not prevent our approach to perform well on synthetic and flow cytometry data.
We develop the theory and practice of an approach to modelling and probabilistic inference in causal networks that is suitable when application-specific or analysis-specific constraints should inform such inference or when little or no data for the learning of causal network structure or probability values at nodes are available. Constrained Bayesian Networks generalize a Bayesian Network such that probabilities can be symbolic, arithmetic expressions and where the meaning of the network is constrained by finitely many formulas from the theory of the reals. A formal semantics for constrained Bayesian Networks over first-order logic of the reals is given, which enables non-linear and non-convex optimisation algorithms that rely on decision procedures for this logic, and supports the composition of several constrained Bayesian Networks. A non-trivial case study in arms control, where few or no data are available to assess the effectiveness of an arms inspection process, evaluates our approach. An open-access prototype implementation of these foundations and their algorithms uses the SMT solver Z3 as decision procedure, leverages an open-source package for Bayesian inference to symbolic computation, and is evaluated experimentally.
Fingerprinting based WLAN indoor positioning system (FWIPS) provides a promising indoor positioning solution to meet the growing interests for indoor location-based services (e.g., indoor way finding or geo-fencing). FWIPS is preferred because it requires no additional infrastructure for deploying an FWIPS and achieving the position estimation by reusing the available WLAN and mobile devices, and capable of providing absolute position estimation. For fingerprinting based positioning (FbP), a model is created to provide reference values of observable features (e.g., signal strength from access point (AP)) as a function of location during offline stage. One widely applied method to build a complete and an accurate reference database (i.e. radio map (RM)) for FWIPS is carrying out a site survey throughout the region of interest (RoI). Along the site survey, the readings of received signal strength (RSS) from all visible APs at each reference point (RP) are collected. This site survey, however, is time-consuming and labor-intensive, especially in the case that the RoI is large (e.g., an airport or a big mall). This bottleneck hinders the wide commercial applications of FWIPS (e.g., proximity promotions in a shopping center). To diminish the cost of site survey, we propose a probabilistic model, which combines fingerprinting based positioning (FbP) and RM generation based on stochastic variational Bayesian inference (SVBI). This SVBI based position and RSS estimation has three properties: i) being able to predict the distribution of the estimated position and RSS, ii) treating each observation of RSS at each RP as an example to learn for FbP and RM generation instead of using the whole RM as an example, and iii) requiring only one time training of the SVBI model for both localization and RSS estimation. These benefits make it outperforms the previous proposed approaches.
User opinions expressed in the form of ratings can influence an individual's view of an item. However, the true quality of an item is often obfuscated by user biases, and it is not obvious from the observed ratings the importance different users place on different aspects of an item. We propose a probabilistic modeling of the observed aspect ratings to infer (i) each user's aspect bias and (ii) latent intrinsic quality of an item. We model multi-aspect ratings as ordered discrete data and encode the dependency between different aspects by using a latent Gaussian structure. We handle the Gaussian-Categorical non-conjugacy using a stick-breaking formulation coupled with P\'{o}lya-Gamma auxiliary variable augmentation for a simple, fully Bayesian inference. On two real world datasets, we demonstrate the predictive ability of our model and its effectiveness in learning explainable user biases to provide insights towards a more reliable product quality estimation.
On a daily investment decision in a security market, the price earnings (PE) ratio is one of the most widely applied methods being used as a firm valuation tool by investment experts. Unfortunately, recent academic developments in financial econometrics and machine learning rarely look at this tool. In practice, fundamental PE ratios are often estimated only by subjective expert opinions. The purpose of this research is to formalize a process of fundamental PE estimation by employing advanced dynamic Bayesian network (DBN) methodology. The estimated PE ratio from our model can be used either as a information support for an expert to make investment decisions, or as an automatic trading system illustrated in experiments. Forward-backward inference and EM parameter estimation algorithms are derived with respect to the proposed DBN structure. Unlike existing works in literatures, the economic interpretation of our DBN model is well-justified by behavioral finance evidences of volatility. A simple but practical trading strategy is invented based on the result of Bayesian inference. Extensive experiments show that our trading strategy equipped with the inferenced PE ratios consistently outperforms standard investment benchmarks.
Higher-level cognition depends on the ability to learn models of the world. We can characterize this at the computational level as a structure-learning problem with the goal of best identifying the prevailing causal relationships among a set of relata. However, the computational cost of performing exact Bayesian inference over causal models grows rapidly as the number of relata increases. This implies that the cognitive processes underlying causal learning must be substantially approximate. A powerful class of approximations that focuses on the sequential absorption of successive inputs is captured by the Neurath's ship metaphor in philosophy of science, where theory change is cast as a stochastic and gradual process shaped as much by people's limited willingness to abandon their current theory when considering alternatives as by the ground truth they hope to approach. Inspired by this metaphor and by algorithms for approximating Bayesian inference in machine learning, we propose an algorithmic-level model of causal structure learning under which learners represent only a single global hypothesis that they update locally as they gather evidence. We propose a related scheme for understanding how, under these limitations, learners choose informative interventions that manipulate the causal system to help elucidate its workings. We find support for our approach in the analysis of four experiments.
Predicated on the increasing abundance of electronic health records, we investi- gate the problem of inferring individualized treatment effects using observational data. Stemming from the potential outcomes model, we propose a novel multi- task learning framework in which factual and counterfactual outcomes are mod- eled as the outputs of a function in a vector-valued reproducing kernel Hilbert space (vvRKHS). We develop a nonparametric Bayesian method for learning the treatment effects using a multi-task Gaussian process (GP) with a linear coregion- alization kernel as a prior over the vvRKHS. The Bayesian approach allows us to compute individualized measures of confidence in our estimates via pointwise credible intervals, which are crucial for realizing the full potential of precision medicine. The impact of selection bias is alleviated via a risk-based empirical Bayes method for adapting the multi-task GP prior, which jointly minimizes the empirical error in factual outcomes and the uncertainty in (unobserved) counter- factual outcomes. We conduct experiments on observational datasets for an inter- ventional social program applied to premature infants, and a left ventricular assist device applied to cardiac patients wait-listed for a heart transplant. In both experi- ments, we show that our method significantly outperforms the state-of-the-art.
Many applications of machine learning, for example in health care, would benefit from methods that can guarantee privacy of data subjects. Differential privacy (DP) has become established as a standard for protecting learning results. The standard DP algorithms require a single trusted party to have access to the entire data, which is a clear weakness. We consider DP Bayesian learning in a distributed setting, where each party only holds a single sample or a few samples of the data. We propose a learning strategy based on a secure multi-party sum function for aggregating summaries from data holders and the Gaussian mechanism for DP. Our method builds on an asymptotically optimal and practically efficient DP Bayesian inference with rapidly diminishing extra cost.
In this work, we investigate the value of uncertainty modeling in 3D super-resolution with convolutional neural networks (CNNs). Deep learning has shown success in a plethora of medical image transformation problems, such as super-resolution (SR) and image synthesis. However, the highly ill-posed nature of such problems results in inevitable ambiguity in the learning of networks. We propose to account for intrinsic uncertainty through a per-patch heteroscedastic noise model and for parameter uncertainty through approximate Bayesian inference in the form of variational dropout. We show that the combined benefits of both lead to the state-of-the-art performance SR of diffusion MR brain images in terms of errors compared to ground truth. We further show that the reduced error scores produce tangible benefits in downstream tractography. In addition, the probabilistic nature of the methods naturally confers a mechanism to quantify uncertainty over the super-resolved output. We demonstrate through experiments on both healthy and pathological brains the potential utility of such an uncertainty measure in the risk assessment of the super-resolved images for subsequent clinical use.
In statistical learning for real-world large-scale data problems, one must often resort to "streaming" algorithms which operate sequentially on small batches of data. In this work, we present an analysis of the information-theoretic limits of mini-batch inference in the context of generalized linear models and low-rank matrix factorization. In a controlled Bayes-optimal setting, we characterize the optimal performance and phase transitions as a function of mini-batch size. We base part of our results on a detailed analysis of a mini-batch version of the approximate message-passing algorithm (Mini-AMP), which we introduce. Additionally, we show that this theoretical optimality carries over into real-data problems by illustrating that Mini-AMP is competitive with standard streaming algorithms for clustering.
Stochastic gradient MCMC (SG-MCMC) algorithms have proven useful in scaling Bayesian inference to large datasets under an assumption of i.i.d data. We instead develop an SG-MCMC algorithm to learn the parameters of hidden Markov models (HMMs) for time-dependent data. There are two challenges to applying SG-MCMC in this setting: The latent discrete states, and needing to break dependencies when considering minibatches. We consider a marginal likelihood representation of the HMM and propose an algorithm that harnesses the inherent memory decay of the process. We demonstrate the effectiveness of our algorithm on synthetic experiments and an ion channel recording data, with runtimes significantly outperforming batch MCMC.
Undirected graphical models are applied in genomics, protein structure prediction, and neuroscience to identify sparse interactions that underlie discrete data. Although Bayesian methods for inference would be favorable in these contexts, they are rarely used because they require doubly intractable Monte Carlo sampling. Here, we develop a framework for scalable Bayesian inference of discrete undirected models based on two new methods. The first is Persistent VI, an algorithm for variational inference of discrete undirected models that avoids doubly intractable MCMC and approximations of the partition function. The second is Fadeout, a reparameterization approach for variational inference under sparsity-inducing priors that captures a posteriori correlations between parameters and hyperparameters with noncentered parameterizations. We find that, together, these methods for variational inference substantially improve learning of sparse undirected graphical models in simulated and real problems from physics and biology.
We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the construction of power law models in several applications. Our construction readily extends to capture the structure of latent factors, similarly to stochastic blockmodels, while maintaining its power law degree distribution. The BFRY random variables are well approximated by gamma random variables in a variational Bayesian inference routine, which we apply to several network datasets for which power law degree distributions are a natural assumption. By learning the parameters of the BFRY distribution via probabilistic inference, we are able to automatically select the appropriate power law behavior from the data. In order to further scale our inference procedure, we adopt stochastic gradient ascent routines where the gradients are computed on minibatches (i.e., subsets) of the edges in the graph.
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order structures or resulting in undesired tensors, respectively. This paper introduces a probabilistic vectorial dimensionality reduction model for tensorial data. The model represents a tensor by employing a linear combination of same order basis tensors, thus it offers a mechanism to directly reduce a tensor to a vector. Under this expression, the projection base of the model is based on the tensor CandeComp/PARAFAC (CP) decomposition and the number of free parameters in the model only grows linearly with the number of modes rather than exponentially. A Bayesian inference has been established via the variational EM approach. A criterion to set the parameters (factor number of CP decomposition and the number of extracted features) is empirically given. The model outperforms several existing PCA-based methods and CP decomposition on several publicly available databases in terms of classification and clustering accuracy.
Decoding human brain activities via functional magnetic resonance imaging (fMRI) has gained increasing attention in recent years. While encouraging results have been reported in brain states classification tasks, reconstructing the details of human visual experience still remains difficult. Two main challenges that hinder the development of effective models are the perplexing fMRI measurement noise and the high dimensionality of limited data instances. Existing methods generally suffer from one or both of these issues and yield dissatisfactory results. In this paper, we tackle this problem by casting the reconstruction of visual stimulus as the Bayesian inference of missing view in a multiview latent variable model. Sharing a common latent representation, our joint generative model of external stimulus and brain response is not only "deep" in extracting nonlinear features from visual images, but also powerful in capturing correlations among voxel activities of fMRI recordings. The nonlinearity and deep structure endow our model with strong representation ability, while the correlations of voxel activities are critical for suppressing noise and improving prediction. We devise an efficient variational Bayesian method to infer the latent variables and the model parameters. To further improve the reconstruction accuracy, the latent representations of testing instances are enforced to be close to that of their neighbours from the training set via posterior regularization. Experiments on three fMRI recording datasets demonstrate that our approach can more accurately reconstruct visual stimuli.
In this paper, we study the trade-offs of different inference approaches for Bayesian matrix factorisation methods, which are commonly used for predicting missing values, and for finding patterns in the data. In particular, we consider Bayesian nonnegative variants of matrix factorisation and tri-factorisation, and compare non-probabilistic inference, Gibbs sampling, variational Bayesian inference, and a maximum-a-posteriori approach. The variational approach is new for the Bayesian nonnegative models. We compare their convergence, and robustness to noise and sparsity of the data, on both synthetic and real-world datasets. Furthermore, we extend the models with the Bayesian automatic relevance determination prior, allowing the models to perform automatic model selection, and demonstrate its efficiency.
We present a simple and effective framework for simultaneous semantic segmentation and instance segmentation with Fully Convolutional Networks (FCNs). The method, called BiSeg, predicts instance segmentation as a posterior in Bayesian inference, where semantic segmentation is used as a prior. We extend the idea of position-sensitive score maps used in recent methods to a fusion of multiple score maps at different scales and partition modes, and adopt it as a robust likelihood for instance segmentation inference. As both Bayesian inference and map fusion are performed per pixel, BiSeg is a fully convolutional end-to-end solution that inherits all the advantages of FCNs. We demonstrate state-of-the-art instance segmentation accuracy on PASCAL VOC.
Markov chain Monte Carlo (MCMC) algorithms have become powerful tools for Bayesian inference. However, they do not scale well to large-data problems. Divide-and-conquer strategies, which split the data into batches and, for each batch, run independent MCMC algorithms targeting the corresponding subposterior, can spread the computational burden across a number of separate workers. The challenge with such strategies is in recombining the subposteriors to approximate the full posterior. By creating a Gaussian-process approximation for each log-subposterior density we create a tractable approximation for the full posterior. This approximation is exploited through three methodologies: firstly a Hamiltonian Monte Carlo algorithm targeting the expectation of the posterior density provides a sample from an approximation to the posterior; secondly, evaluating the true posterior at the sampled points leads to an importance sampler that, asymptotically, targets the true posterior expectations; finally, an alternative importance sampler uses the full Gaussian-process distribution of the approximation to the log-posterior density to re-weight any initial sample and provide both an estimate of the posterior expectation and a measure of the uncertainty in it.