The brain interprets ambiguous sensory information faster and more reliably than modern computers, using neurons that are slower and less reliable than logic gates. But Bayesian inference, which underpins many computational models of perception and cognition, appears computationally challenging even given modern transistor speeds and energy budgets. The computational principles and structures needed to narrow this gap are unknown. Here we show how to build fast Bayesian computing machines using intentionally stochastic, digital parts, narrowing this efficiency gap by multiple orders of magnitude. We find that by connecting stochastic digital components according to simple mathematical rules, one can build massively parallel, low precision circuits that solve Bayesian inference problems and are compatible with the Poisson firing statistics of cortical neurons. We evaluate circuits for depth and motion perception, perceptual learning and causal reasoning, each performing inference over 10,000+ latent variables in real time - a 1,000x speed advantage over commodity microprocessors. These results suggest a new role for randomness in the engineering and reverse-engineering of intelligent computation.
Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic properties of atomic nuclei to discover the structure, reaction state and chemical environment of molecules. We propose a probabilistic generative model and inference procedures for NMR spectroscopy. Specifically, we use a weighted sum of trigonometric functions undergoing exponential decay to model free induction decay (FID) signals. We discuss the challenges in estimating the components of this general model -- amplitudes, phase shifts, frequencies, decay rates, and noise variances -- and offer practical solutions. We compare with conventional Fourier transform spectroscopy for estimating the relative concentrations of chemicals in a mixture, using synthetic and experimentally acquired FID signals. We find the proposed model is particularly robust to low signal to noise ratios (SNR), and overlapping peaks in the Fourier transform of the FID, enabling accurate predictions (e.g., 1% sensitivity at low SNR) which are not possible with conventional spectroscopy (5% sensitivity).
Bio-image analysis is challenging due to inhomogeneous intensity distributions and high levels of noise in the images. Bayesian inference provides a principled way for regularizing the problem using prior knowledge. A fundamental choice is how one measures "distances" between shapes in an image. It has been shown that the straightforward geometric L2 distance is degenerate and leads to pathological situations. This is avoided when using Sobolev gradients, rendering the segmentation problem less ill-posed. The high computational cost and implementation overhead of Sobolev gradients, however, have hampered practical applications. We show how particle methods as applied to image segmentation allow for a simple and computationally efficient implementation of Sobolev gradients. We show that the evaluation of Sobolev gradients amounts to particle-particle interactions along the contour in an image. We extend an existing particle-based segmentation algorithm to using Sobolev gradients. Using synthetic and real-world images, we benchmark the results for both 2D and 3D images using piecewise smooth and piecewise constant region models. The present particle approximation of Sobolev gradients is 2.8 to 10 times faster than the previous reference implementation, but retains the known favorable properties of Sobolev gradients. This speedup is achieved by using local particle-particle interactions instead of solving a global Poisson equation at each iteration. The computational time per iteration is higher for Sobolev gradients than for L2 gradients. Since Sobolev gradients precondition the optimization problem, however, a smaller number of overall iterations may be necessary for the algorithm to converge, which can in some cases amortize the higher per-iteration cost.
The goal of machine learning is to develop predictors that generalize well to test data. Ideally, this is achieved by training on an almost infinitely large training data set that captures all variations in the data distribution. In practical learning settings, however, we do not have infinite data and our predictors may overfit. Overfitting may be combatted, for example, by adding a regularizer to the training objective or by defining a prior over the model parameters and performing Bayesian inference. In this paper, we propose a third, alternative approach to combat overfitting: we extend the training set with infinitely many artificial training examples that are obtained by corrupting the original training data. We show that this approach is practical and efficient for a range of predictors and corruption models. Our approach, called marginalized corrupted features (MCF), trains robust predictors by minimizing the expected value of the loss function under the corruption model. We show empirically on a variety of data sets that MCF classifiers can be trained efficiently, may generalize substantially better to test data, and are also more robust to feature deletion at test time.
We present a general framework for accelerating a large class of widely used Markov chain Monte Carlo (MCMC) algorithms. Our approach exploits fast, iterative approximations to the target density to speculatively evaluate many potential future steps of the chain in parallel. The approach can accelerate computation of the target distribution of a Bayesian inference problem, without compromising exactness, by exploiting subsets of data. It takes advantage of whatever parallel resources are available, but produces results exactly equivalent to standard serial execution. In the initial burn-in phase of chain evaluation, it achieves speedup over serial evaluation that is close to linear in the number of available cores.
Markov chain Monte Carlo (MCMC) is a popular and successful general-purpose tool for Bayesian inference. However, MCMC cannot be practically applied to large data sets because of the prohibitive cost of evaluating every likelihood term at every iteration. Here we present Firefly Monte Carlo (FlyMC) an auxiliary variable MCMC algorithm that only queries the likelihoods of a potentially small subset of the data at each iteration yet simulates from the exact posterior distribution, in contrast to recent proposals that are approximate even in the asymptotic limit. FlyMC is compatible with a wide variety of modern MCMC algorithms, and only requires a lower bound on the per-datum likelihood factors. In experiments, we find that FlyMC generates samples from the posterior more than an order of magnitude faster than regular MCMC, opening up MCMC methods to larger datasets than were previously considered feasible.
The main challenges that arise when adopting Gaussian Process priors in probabilistic modeling are how to carry out exact Bayesian inference and how to account for uncertainty on model parameters when making model-based predictions on out-of-sample data. Using probit regression as an illustrative working example, this paper presents a general and effective methodology based on the pseudo-marginal approach to Markov chain Monte Carlo that efficiently addresses both of these issues. The results presented in this paper show improvements over existing sampling methods to simulate from the posterior distribution over the parameters defining the covariance function of the Gaussian Process prior. This is particularly important as it offers a powerful tool to carry out full Bayesian inference of Gaussian Process based hierarchic statistical models in general. The results also demonstrate that Monte Carlo based integration of all model parameters is actually feasible in this class of models providing a superior quantification of uncertainty in predictions. Extensive comparisons with respect to state-of-the-art probabilistic classifiers confirm this assertion.
We marry ideas from deep neural networks and approximate Bayesian inference to derive a generalised class of deep, directed generative models, endowed with a new algorithm for scalable inference and learning. Our algorithm introduces a recognition model to represent approximate posterior distributions, and that acts as a stochastic encoder of the data. We develop stochastic back-propagation -- rules for back-propagation through stochastic variables -- and use this to develop an algorithm that allows for joint optimisation of the parameters of both the generative and recognition model. We demonstrate on several real-world data sets that the model generates realistic samples, provides accurate imputations of missing data and is a useful tool for high-dimensional data visualisation.
Kernel methods have revolutionized the fields of pattern recognition and machine learning. Their success, however, critically depends on the choice of kernel parameters. Using Gaussian process (GP) classification as a working example, this paper focuses on Bayesian inference of covariance (kernel) parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is that, compared to standard optimization of kernel parameters, they have been systematically demonstrated to be superior in quantifying uncertainty in predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed as a practical inference tool for GP models. In particular, it amounts in replacing the analytically intractable marginal likelihood by an unbiased estimate obtainable by approximate methods and importance sampling. After discussing the potential drawbacks in employing importance sampling, this paper proposes the application of annealed importance sampling. The results empirically demonstrate that compared to importance sampling, annealed importance sampling can reduce the variance of the estimate of the marginal likelihood exponentially in the number of data at a computational cost that scales only polynomially. The results on real data demonstrate that employing annealed importance sampling in the Pseudo-Marginal MCMC approach represents a step forward in the development of fully automated exact inference engines for GP models.
We evaluate the following Machine Learning techniques for Green Energy (Wind, Solar) Prediction: Bayesian Inference, Neural Networks, Support Vector Machines, Clustering techniques (PCA). Our objective is to predict green energy using weather forecasts, predict deviations from forecast green energy, find correlation amongst different weather parameters and green energy availability, recover lost or missing energy (/ weather) data. We use historical weather data and weather forecasts for the same.
This paper presents an approximate method for performing Bayesian inference in models with conditional independence over a decentralized network of learning agents. The method first employs variational inference on each individual learning agent to generate a local approximate posterior, the agents transmit their local posteriors to other agents in the network, and finally each agent combines its set of received local posteriors. The key insight in this work is that, for many Bayesian models, approximate inference schemes destroy symmetry and dependencies in the model that are crucial to the correct application of Bayes' rule when combining the local posteriors. The proposed method addresses this issue by including an additional optimization step in the combination procedure that accounts for these broken dependencies. Experiments on synthetic and real data demonstrate that the decentralized method provides advantages in computational performance and predictive test likelihood over previous batch and distributed methods.
In this paper we develop a new Bayesian inference method for low rank matrix reconstruction. We call the new method the Relevance Singular Vector Machine (RSVM) where appropriate priors are defined on the singular vectors of the underlying matrix to promote low rank. To accelerate computations, a numerically efficient approximation is developed. The proposed algorithms are applied to matrix completion and matrix reconstruction problems and their performance is studied numerically.
We consider the problem of learning by demonstration from agents acting in unknown stochastic Markov environments or games. Our aim is to estimate agent preferences in order to construct improved policies for the same task that the agents are trying to solve. To do so, we extend previous probabilistic approaches for inverse reinforcement learning in known MDPs to the case of unknown dynamics or opponents. We do this by deriving two simplified probabilistic models of the demonstrator's policy and utility. For tractability, we use maximum a posteriori estimation rather than full Bayesian inference. Under a flat prior, this results in a convex optimisation problem. We find that the resulting algorithms are highly competitive against a variety of other methods for inverse reinforcement learning that do have knowledge of the dynamics.
In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are independent, the off-diagonal loadings are normally distributed, and the diagonal loadings follow a truncated normal distribution. This prior specification, however, depends in an important way on how the variables and associated rows of the loading matrix are ordered. We show how a minor modification of the approach allows one to compute with the identifiable lower triangular loading matrix but maintain invariance properties under reordering of the variables.
This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.
This paper presents a computational model of concept learning using Bayesian inference for a grammatically structured hypothesis space, and test the model on multisensory (visual and haptics) recognition of 3D objects. The study is performed on a set of artificially generated 3D objects known as fribbles, which are complex, multipart objects with categorical structures. The goal of this work is to develop a working multisensory representational model that integrates major themes on concepts and concepts learning from the cognitive science literature. The model combines the representational power of a probabilistic generative grammar with the inferential power of Bayesian induction.
Gibbs sampling is a workhorse for Bayesian inference but has several limitations when used for parameter estimation, and is often much slower than non-sampling inference methods. SAME (State Augmentation for Marginal Estimation) \cite{Doucet99,Doucet02} is an approach to MAP parameter estimation which gives improved parameter estimates over direct Gibbs sampling. SAME can be viewed as cooling the posterior parameter distribution and allows annealed search for the MAP parameters, often yielding very high quality (lower loss) estimates. But it does so at the expense of additional samples per iteration and generally slower performance. On the other hand, SAME dramatically increases the parallelism in the sampling schedule, and is an excellent match for modern (SIMD) hardware. In this paper we explore the application of SAME to graphical model inference on modern hardware. We show that combining SAME with factored sample representation (or approximation) gives throughput competitive with the fastest symbolic methods, but with potentially better quality. We describe experiments on Latent Dirichlet Allocation, achieving speeds similar to the fastest reported methods (online Variational Bayes) and lower cross-validated loss than other LDA implementations. The method is simple to implement and should be applicable to many other models.
In this paper, we discuss the method of Bayesian regression and its efficacy for predicting price variation of Bitcoin, a recently popularized virtual, cryptographic currency. Bayesian regression refers to utilizing empirical data as proxy to perform Bayesian inference. We utilize Bayesian regression for the so-called "latent source model". The Bayesian regression for "latent source model" was introduced and discussed by Chen, Nikolov and Shah (2013) and Bresler, Chen and Shah (2014) for the purpose of binary classification. They established theoretical as well as empirical efficacy of the method for the setting of binary classification. In this paper, instead we utilize it for predicting real-valued quantity, the price of Bitcoin. Based on this price prediction method, we devise a simple strategy for trading Bitcoin. The strategy is able to nearly double the investment in less than 60 day period when run against real data trace.
Probabilistic approaches for tensor factorization aim to extract meaningful structure from incomplete data by postulating low rank constraints. Recently, variational Bayesian (VB) inference techniques have successfully been applied to large scale models. This paper presents full Bayesian inference via VB on both single and coupled tensor factorization models. Our method can be run even for very large models and is easily implemented. It exhibits better prediction performance than existing approaches based on maximum likelihood on several real-world datasets for missing link prediction problem.
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank. In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.