We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.
Recent advances in stochastic gradient variational inference have made it possible to perform variational Bayesian inference with posterior approximations containing auxiliary random variables. This enables us to explore a new synthesis of variational inference and Monte Carlo methods where we incorporate one or more steps of MCMC into our variational approximation. By doing so we obtain a rich class of inference algorithms bridging the gap between variational methods and MCMC, and offering the best of both worlds: fast posterior approximation through the maximization of an explicit objective, with the option of trading off additional computation for additional accuracy. We describe the theoretical foundations that make this possible and show some promising first results.
Bayesian predictive inference analyzes a dataset to make predictions about new observations. When a model does not match the data, predictive accuracy suffers. We develop population empirical Bayes (POP-EB), a hierarchical framework that explicitly models the empirical population distribution as part of Bayesian analysis. We introduce a new concept, the latent dataset, as a hierarchical variable and set the empirical population as its prior. This leads to a new predictive density that mitigates model mismatch. We efficiently apply this method to complex models by proposing a stochastic variational inference algorithm, called bumping variational inference (BUMP-VI). We demonstrate improved predictive accuracy over classical Bayesian inference in three models: a linear regression model of health data, a Bayesian mixture model of natural images, and a latent Dirichlet allocation topic model of scientific documents.
BayesPy is an open-source Python software package for performing variational Bayesian inference. It is based on the variational message passing framework and supports conjugate exponential family models. By removing the tedious task of implementing the variational Bayesian update equations, the user can construct models faster and in a less error-prone way. Simple syntax, flexible model construction and efficient inference make BayesPy suitable for both average and expert Bayesian users. It also supports some advanced methods such as stochastic and collapsed variational inference.
Variational inference is a scalable technique for approximate Bayesian inference. Deriving variational inference algorithms requires tedious model-specific calculations; this makes it difficult to automate. We propose an automatic variational inference algorithm, automatic differentiation variational inference (ADVI). The user only provides a Bayesian model and a dataset; nothing else. We make no conjugacy assumptions and support a broad class of models. The algorithm automatically determines an appropriate variational family and optimizes the variational objective. We implement ADVI in Stan (code available now), a probabilistic programming framework. We compare ADVI to MCMC sampling across hierarchical generalized linear models, nonconjugate matrix factorization, and a mixture model. We train the mixture model on a quarter million images. With ADVI we can use variational inference on any model we write in Stan.
Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit, many copula models, and latent Dirichlet allocation (LDA). Bayesian inference involving probability distributions confined to constrained domains could be quite challenging for commonly used sampling algorithms. In this paper, we propose a novel augmentation technique that handles a wide range of constraints by mapping the constrained domain to a sphere in the augmented space. By moving freely on the surface of this sphere, sampling algorithms handle constraints implicitly and generate proposals that remain within boundaries when mapped back to the original space. Our proposed method, called {Spherical Augmentation}, provides a mathematically natural and computationally efficient framework for sampling from constrained probability distributions. We show the advantages of our method over state-of-the-art sampling algorithms, such as exact Hamiltonian Monte Carlo, using several examples including truncated Gaussian distributions, Bayesian Lasso, Bayesian bridge regression, reconstruction of quantized stationary Gaussian process, and LDA for topic modeling.
Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cases, we expect some form of dependency between the draws: the nucleotide at one position in the DNA strand may depend on the preceding nucleotides, children's names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are not naturally captured by the typical Dirichlet-multinomial formulation. Here, we leverage a logistic stick-breaking representation and recent innovations in P\'olya-gamma augmentation to reformulate the multinomial distribution in terms of latent variables with jointly Gaussian likelihoods, enabling us to take advantage of a host of Bayesian inference techniques for Gaussian models with minimal overhead.
We present Montblanc, a GPU implementation of the Radio interferometer measurement equation (RIME) in support of the Bayesian inference for radio observations (BIRO) technique. BIRO uses Bayesian inference to select sky models that best match the visibilities observed by a radio interferometer. To accomplish this, BIRO evaluates the RIME multiple times, varying sky model parameters to produce multiple model visibilities. Chi-squared values computed from the model and observed visibilities are used as likelihood values to drive the Bayesian sampling process and select the best sky model. As most of the elements of the RIME and chi-squared calculation are independent of one another, they are highly amenable to parallel computation. Additionally, Montblanc caters for iterative RIME evaluation to produce multiple chi-squared values. Modified model parameters are transferred to the GPU between each iteration. We implemented Montblanc as a Python package based upon NVIDIA's CUDA architecture. As such, it is easy to extend and implement different pipelines. At present, Montblanc supports point and Gaussian morphologies, but is designed for easy addition of new source profiles. Montblanc's RIME implementation is performant: On an NVIDIA K40, it is approximately 250 times faster than MeqTrees on a dual hexacore Intel E5-2620v2 CPU. Compared to the OSKAR simulator's GPU-implemented RIME components it is 7.7 and 12 times faster on the same K40 for single and double-precision floating point respectively. However, OSKAR's RIME implementation is more general than Montblanc's BIRO-tailored RIME. Theoretical analysis of Montblanc's dominant CUDA kernel suggests that it is memory bound. In practice, profiling shows that is balanced between compute and memory, as much of the data required by the problem is retained in L1 and L2 cache.
We introduce the Variational Holder (VH) bound as an alternative to Variational Bayes (VB) for approximate Bayesian inference. Unlike VB which typically involves maximization of a non-convex lower bound with respect to the variational parameters, the VH bound involves minimization of a convex upper bound to the intractable integral with respect to the variational parameters. Minimization of the VH bound is a convex optimization problem; hence the VH method can be applied using off-the-shelf convex optimization algorithms and the approximation error of the VH bound can also be analyzed using tools from convex optimization literature. We present experiments on the task of integrating a truncated multivariate Gaussian distribution and compare our method to VB, EP and a state-of-the-art numerical integration method for this problem.
Factorial hidden Markov models (FHMMs) are powerful tools of modeling sequential data. Learning FHMMs yields a challenging simultaneous model selection issue, i.e., selecting the number of multiple Markov chains and the dimensionality of each chain. Our main contribution is to address this model selection issue by extending Factorized Asymptotic Bayesian (FAB) inference to FHMMs. First, we offer a better approximation of marginal log-likelihood than the previous FAB inference. Our key idea is to integrate out transition probabilities, yet still apply the Laplace approximation to emission probabilities. Second, we prove that if there are two very similar hidden states in an FHMM, i.e. one is redundant, then FAB will almost surely shrink and eliminate one of them, making the model parsimonious. Experimental results show that FAB for FHMMs significantly outperforms state-of-the-art nonparametric Bayesian iFHMM and Variational FHMM in model selection accuracy, with competitive held-out perplexity.
Networks capture our intuition about relationships in the world. They describe the friendships between Facebook users, interactions in financial markets, and synapses connecting neurons in the brain. These networks are richly structured with cliques of friends, sectors of stocks, and a smorgasbord of cell types that govern how neurons connect. Some networks, like social network friendships, can be directly observed, but in many cases we only have an indirect view of the network through the actions of its constituents and an understanding of how the network mediates that activity. In this work, we focus on the problem of latent network discovery in the case where the observable activity takes the form of a mutually-excitatory point process known as a Hawkes process. We build on previous work that has taken a Bayesian approach to this problem, specifying prior distributions over the latent network structure and a likelihood of observed activity given this network. We extend this work by proposing a discrete-time formulation and developing a computationally efficient stochastic variational inference (SVI) algorithm that allows us to scale the approach to long sequences of observations. We demonstrate our algorithm on the calcium imaging data used in the Chalearn neural connectomics challenge.
Many modern data analysis problems involve inferences from streaming data. However, streaming data is not easily amenable to the standard probabilistic modeling approaches, which assume that we condition on finite data. We develop population variational Bayes, a new approach for using Bayesian modeling to analyze streams of data. It approximates a new type of distribution, the population posterior, which combines the notion of a population distribution of the data with Bayesian inference in a probabilistic model. We study our method with latent Dirichlet allocation and Dirichlet process mixtures on several large-scale data sets.
We present the first fully variational Bayesian inference scheme for continuous Gaussian-process-modulated Poisson processes. Such point processes are used in a variety of domains, including neuroscience, geo-statistics and astronomy, but their use is hindered by the computational cost of existing inference schemes. Our scheme: requires no discretisation of the domain; scales linearly in the number of observed events; and is many orders of magnitude faster than previous sampling based approaches. The resulting algorithm is shown to outperform standard methods on synthetic examples, coal mining disaster data and in the prediction of Malaria incidences in Kenya.
We present a scalable Bayesian model for low-rank factorization of massive tensors with binary observations. The proposed model has the following key properties: (1) in contrast to the models based on the logistic or probit likelihood, using a zero-truncated Poisson likelihood for binary data allows our model to scale up in the number of \emph{ones} in the tensor, which is especially appealing for massive but sparse binary tensors; (2) side-information in form of binary pairwise relationships (e.g., an adjacency network) between objects in any tensor mode can also be leveraged, which can be especially useful in "cold-start" settings; and (3) the model admits simple Bayesian inference via batch, as well as \emph{online} MCMC; the latter allows scaling up even for \emph{dense} binary data (i.e., when the number of ones in the tensor/network is also massive). In addition, non-negative factor matrices in our model provide easy interpretability, and the tensor rank can be inferred from the data. We evaluate our model on several large-scale real-world binary tensors, achieving excellent computational scalability, and also demonstrate its usefulness in leveraging side-information provided in form of mode-network(s).
Approximate Bayesian computation (ABC) methods provide an elaborate approach to Bayesian inference on complex models, including model choice. Both theoretical arguments and simulation experiments indicate, however, that model posterior probabilities may be poorly evaluated by standard ABC techniques. We propose a novel approach based on a machine learning tool named random forests to conduct selection among the highly complex models covered by ABC algorithms. We thus modify the way Bayesian model selection is both understood and operated, in that we rephrase the inferential goal as a classification problem, first predicting the model that best fits the data with random forests and postponing the approximation of the posterior probability of the predicted MAP for a second stage also relying on random forests. Compared with earlier implementations of ABC model choice, the ABC random forest approach offers several potential improvements: (i) it often has a larger discriminative power among the competing models, (ii) it is more robust against the number and choice of statistics summarizing the data, (iii) the computing effort is drastically reduced (with a gain in computation efficiency of at least fifty), and (iv) it includes an approximation of the posterior probability of the selected model. The call to random forests will undoubtedly extend the range of size of datasets and complexity of models that ABC can handle. We illustrate the power of this novel methodology by analyzing controlled experiments as well as genuine population genetics datasets. The proposed methodologies are implemented in the R package abcrf available on the CRAN.
We consider a network of agents that aim to learn some unknown state of the world using private observations and exchange of beliefs. At each time, agents observe private signals generated based on the true unknown state. Each agent might not be able to distinguish the true state based only on her private observations. This occurs when some other states are observationally equivalent to the true state from the agent's perspective. To overcome this shortcoming, agents must communicate with each other to benefit from local observations. We propose a model where each agent selects one of her neighbors randomly at each time. Then, she refines her opinion using her private signal and the prior of that particular neighbor. The proposed rule can be thought of as a Bayesian agent who cannot recall the priors based on which other agents make inferences. This learning without recall approach preserves some aspects of the Bayesian inference while being computationally tractable. By establishing a correspondence with a random walk on the network graph, we prove that under the described protocol, agents learn the truth exponentially fast in the almost sure sense. The asymptotic rate is expressed as the sum of the relative entropies between the signal structures of every agent weighted by the stationary distribution of the random walk.
A generic algorithm for the extraction of probabilistic (Bayesian) information about model parameters from data is presented. The algorithm propagates an ensemble of particles in the product space of model parameters and outputs. Each particle update consists of a random jump in parameter space followed by a simulation of a model output and a Metropolis acceptance/rejection step based on a comparison of the simulated output to the data. The distance of a particle to the data is interpreted as an energy and the algorithm is reducing the associated temperature of the ensemble such that entropy production is minimized. If this simulated annealing is not too fast compared to the mixing speed in parameter space, the parameter marginal of the ensemble approaches the Bayesian posterior distribution. Annealing is adaptive and depends on certain extensive thermodynamic quantities that can easily be measured throughout run-time. In the general case, we propose annealing with a constant entropy production rate, which is optimal as long as annealing is not too fast. For the practically relevant special case of no prior knowledge, we derive an optimal fast annealing schedule with a non-constant entropy production rate. The algorithm does not require the calculation of the density of the model likelihood, which makes it interesting for Bayesian parameter inference with stochastic models, whose likelihood functions are typically very high dimensional integrals.
We propose a Conditional Density Filtering (C-DF) algorithm for efficient online Bayesian inference. C-DF adapts MCMC sampling to the online setting, sampling from approximations to conditional posterior distributions obtained by propagating surrogate conditional sufficient statistics (a function of data and parameter estimates) as new data arrive. These quantities eliminate the need to store or process the entire dataset simultaneously and offer a number of desirable features. Often, these include a reduction in memory requirements and runtime and improved mixing, along with state-of-the-art parameter inference and prediction. These improvements are demonstrated through several illustrative examples including an application to high dimensional compressed regression. Finally, we show that C-DF samples converge to the target posterior distribution asymptotically as sampling proceeds and more data arrives.
We consider the problem of transforming samples from one continuous source distribution into samples from another target distribution. We demonstrate with optimal transport theory that when the source distribution can be easily sampled from and the target distribution is log-concave, this can be tractably solved with convex optimization. We show that a special case of this, when the source is the prior and the target is the posterior, is Bayesian inference. Here, we can tractably calculate the normalization constant and draw posterior i.i.d. samples. Remarkably, our Bayesian tractability criterion is simply log concavity of the prior and likelihood: the same criterion for tractable calculation of the maximum a posteriori point estimate. With simulated data, we demonstrate how we can attain the Bayes risk in simulations. With physiologic data, we demonstrate improvements over point estimation in intensive care unit outcome prediction and electroencephalography-based sleep staging.
The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.