This study provides both analysis and a refined, research-ready implementation of Tang and Kucukelbir's Variational Deep Q Network, a novel approach to maximising the efficiency of exploration in complex learning environments using Variational Bayesian Inference. Alongside reference implementations of both Traditional and Double Deep Q Networks, a small novel contribution is presented - the Double Variational Deep Q Network, which incorporates improvements to increase the stability and robustness of inference-based learning. Finally, an evaluation and discussion of the effectiveness of these approaches is discussed in the wider context of Bayesian Deep Learning.
We introduce a novel rule-based approach for handling regression problems. The new methodology carries elements from two frameworks: (i) it provides information about the uncertainty of the parameters of interest using Bayesian inference, and (ii) it allows the incorporation of expert knowledge through rule-based systems. The blending of those two different frameworks can be particularly beneficial for various domains (e.g. engineering), where, even though the significance of uncertainty quantification motivates a Bayesian approach, there is no simple way to incorporate researcher intuition into the model. We validate our models by applying them to synthetic applications: a simple linear regression problem and two more complex structures based on partial differential equations. Finally, we review the advantages of our methodology, which include the simplicity of the implementation, the uncertainty reduction due to the added information and, in some occasions, the derivation of better point predictions, and we address limitations, mainly from the computational complexity perspective, such as the difficulty in choosing an appropriate algorithm and the added computational burden.
Finding the model that best describes a high-dimensional dataset is a daunting task, even more so if one aims to consider all possible high-order patterns of the data, going beyond pairwise models. For binary data, we show that this task becomes feasible when restricting the search to a family of simple models, that we call Minimally Complex Models (MCMs). MCMs are maximum entropy models that have interactions of arbitrarily high order grouped into independent components of minimal complexity. They are simple in information-theoretic terms, which means they can only fit well certain types of data patterns and are therefore easy to falsify. We show that Bayesian model selection restricted to these models is computationally feasible and has many advantages. First, the model evidence, which balances goodness-of-fit against complexity, can be computed efficiently without any parameter fitting, enabling very fast explorations of the space of MCMs. Second, the family of MCMs is invariant under gauge transformations, which can be used to develop a representation-independent approach to statistical modeling. For small systems (up to 15 variables), combining these two results allows us to select the best MCM among all, even though the number of models is already extremely large. For larger systems, we propose simple heuristics to find optimal MCMs in reasonable times. Besides, inference and sampling can be performed without any computational effort. Finally, because MCMs have interactions of any order, they can reveal the presence of important high-order dependencies in the data, providing a new approach to explore high-order dependencies in complex systems. We apply our method to synthetic data and real-world examples, illustrating how MCMs portray the structure of dependencies among variables in a simple manner, extracting falsifiable predictions on symmetries and invariance from the data.
A central capability of a long-lived reinforcement learning (RL) agent is to incrementally adapt its behavior as its environment changes, and to incrementally build upon previous experiences to facilitate future learning in real-world scenarios. In this paper, we propose LifeLong Incremental Reinforcement Learning (LLIRL), a new incremental algorithm for efficient lifelong adaptation to dynamic environments. We develop and maintain a library that contains an infinite mixture of parameterized environment models, which is equivalent to clustering environment parameters in a latent space. The prior distribution over the mixture is formulated as a Chinese restaurant process (CRP), which incrementally instantiates new environment models without any external information to signal environmental changes in advance. During lifelong learning, we employ the expectation maximization (EM) algorithm with online Bayesian inference to update the mixture in a fully incremental manner. In EM, the E-step involves estimating the posterior expectation of environment-to-cluster assignments, while the M-step updates the environment parameters for future learning. This method allows for all environment models to be adapted as necessary, with new models instantiated for environmental changes and old models retrieved when previously seen environments are encountered again. Experiments demonstrate that LLIRL outperforms relevant existing methods, and enables effective incremental adaptation to various dynamic environments for lifelong learning.
One of the primary goals of systems neuroscience is to relate the structure of neural circuits to their function, yet patterns of connectivity are difficult to establish when recording from large populations in behaving organisms. Many previous approaches have attempted to estimate functional connectivity between neurons using statistical modeling of observational data, but these approaches rely heavily on parametric assumptions and are purely correlational. Recently, however, holographic photostimulation techniques have made it possible to precisely target selected ensembles of neurons, offering the possibility of establishing direct causal links. Here, we propose a method based on noisy group testing that drastically increases the efficiency of this process in sparse networks. By stimulating small ensembles of neurons, we show that it is possible to recover binarized network connectivity with a number of tests that grows only logarithmically with population size under minimal statistical assumptions. Moreover, we prove that our approach, which reduces to an efficiently solvable convex optimization problem, can be related to Variational Bayesian inference on the binary connection weights, and we derive rigorous bounds on the posterior marginals. This allows us to extend our method to the streaming setting, where continuously updated posteriors allow for optional stopping, and we demonstrate the feasibility of inferring connectivity for networks of up to tens of thousands of neurons online. Finally, we show how our work can be theoretically linked to compressed sensing approaches, and compare results for connectivity inference in different settings.
We formulate approximate Bayesian inference in non-conjugate temporal and spatio-temporal Gaussian process models as a simple parameter update rule applied during Kalman smoothing. This viewpoint encompasses most inference schemes, including expectation propagation (EP), the classical (Extended, Unscented, etc.) Kalman smoothers, and variational inference. We provide a unifying perspective on these algorithms, showing how replacing the power EP moment matching step with linearisation recovers the classical smoothers. EP provides some benefits over the traditional methods via introduction of the so-called cavity distribution, and we combine these benefits with the computational efficiency of linearisation, providing extensive empirical analysis demonstrating the efficacy of various algorithms under this unifying framework. We provide a fast implementation of all methods in JAX.
Forecasting the effect of COVID-19 is essential to design policies that may prepare us to handle the pandemic. Many methods have already been proposed, particularly, to forecast reported cases and deaths at country-level and state-level. Many of these methods are based on traditional epidemiological model which rely on simulations or Bayesian inference to simultaneously learn many parameters at a time. This makes them prone to over-fitting and slow execution. We propose an extension to our model SIkJ$\alpha$ to forecast deaths and show that it can consider the effect of many complexities of the epidemic process and yet be simplified to a few parameters that are learned using fast linear regressions. We also present an evaluation of our method against seven approaches currently being used by the CDC, based on their two weeks forecast at various times during the pandemic. We demonstrate that our method achieves better root mean squared error compared to these seven approaches during majority of the evaluation period. Further, on a 2 core desktop machine, our approach takes only 3.18s to tune hyper-parameters, learn parameters and generate 100 days of forecasts of reported cases and deaths for all the states in the US. The total execution time for 184 countries is 11.83s and for all the US counties ($>$ 3000) is 101.03s.
Continuously-indexed flows (CIFs) have recently achieved improvements over baseline normalizing flows on a variety of density estimation tasks. CIFs do not possess a closed-form marginal density, and so, unlike standard flows, cannot be plugged in directly to a variational inference (VI) scheme in order to produce a more expressive family of approximate posteriors. However, we show here how CIFs can be used as part of an auxiliary VI scheme to formulate and train expressive posterior approximations in a natural way. We exploit the conditional independence structure of multi-layer CIFs to build the required auxiliary inference models, which we show empirically yield low-variance estimators of the model evidence. We then demonstrate the advantages of CIFs over baseline flows in VI problems when the posterior distribution of interest possesses a complicated topology, obtaining improved results in both the Bayesian inference and surrogate maximum likelihood settings.
While deep learning methods continue to improve in predictive accuracy on a wide range of application domains, significant issues remain with other aspects of their performance including their ability to quantify uncertainty and their robustness. Recent advances in approximate Bayesian inference hold significant promise for addressing these concerns, but the computational scalability of these methods can be problematic when applied to large-scale models. In this paper, we describe initial work on the development ofURSABench(the Uncertainty, Robustness, Scalability, and Accu-racy Benchmark), an open-source suite of bench-marking tools for comprehensive assessment of approximate Bayesian inference methods with a focus on deep learning-based classification tasks
Bayesian coresets have emerged as a promising approach for implementing scalable Bayesian inference. The Bayesian coreset problem involves selecting a (weighted) subset of the data samples, such that the posterior inference using the selected subset closely approximates the posterior inference using the full dataset. This manuscript revisits Bayesian coresets through the lens of sparsity constrained optimization. Leveraging recent advances in accelerated optimization methods, we propose and analyze a novel algorithm for coreset selection. We provide explicit convergence rate guarantees and present an empirical evaluation on a variety of benchmark datasets to highlight our proposed algorithm's superior performance compared to state-of-the-art on speed and accuracy.
When working with multimodal Bayesian posterior distributions, Markov chain Monte Carlo (MCMC) algorithms have difficulty moving between modes, and default variational or mode-based approximate inferences will understate posterior uncertainty. And, even if the most important modes can be found, it is difficult to evaluate their relative weights in the posterior. Here we propose an approach using parallel runs of MCMC, variational, or mode-based inference to hit as many modes or separated regions as possible and then combine these using Bayesian stacking, a scalable method for constructing a weighted average of distributions. The result from stacking efficiently samples from multimodal posterior distribution, minimizes cross validation prediction error, and represents the posterior uncertainty better than variational inference, but it is not necessarily equivalent, even asymptotically, to fully Bayesian inference. We present theoretical consistency with an example where the stacked inference approximates the true data generating process from the misspecified model and a non-mixing sampler, from which the predictive performance is better than full Bayesian inference, hence the multimodality can be considered a blessing rather than a curse under model misspecification. We demonstrate practical implementation in several model families: latent Dirichlet allocation, Gaussian process regression, hierarchical regression, horseshoe variable selection, and neural networks.
A popular theory of perceptual processing holds that the brain learns both a generative model of the world and a paired recognition model using variational Bayesian inference. Most hypotheses of how the brain might learn these models assume that neurons in a population are conditionally independent given their common inputs. This simplification is likely not compatible with the type of local recurrence observed in the brain. Seeking an alternative that is compatible with complex inter-dependencies yet consistent with known biology, we argue here that the cortex may learn with an adversarial algorithm. Many observable symptoms of this approach would resemble known neural phenomena, including wake/sleep cycles and oscillations that vary in magnitude with surprise, and we describe how further predictions could be tested. We illustrate the idea on recurrent neural networks trained to model image and video datasets. This framework for learning brings variational inference closer to neuroscience and yields multiple testable hypotheses.
Variational Bayesian Monte Carlo (VBMC) is a recently introduced framework that uses Gaussian process surrogates to perform approximate Bayesian inference in models with black-box, non-cheap likelihoods. In this work, we extend VBMC to deal with noisy log-likelihood evaluations, such as those arising from simulation-based models. We introduce new `global' acquisition functions, such as expected information gain (EIG) and variational interquantile range (VIQR), which are robust to noise and can be efficiently evaluated within the VBMC setting. In a novel, challenging, noisy-inference benchmark comprising of a variety of models with real datasets from computational and cognitive neuroscience, VBMC+VIQR achieves state-of-the-art performance in recovering the ground-truth posteriors and model evidence. In particular, our method vastly outperforms `local' acquisition functions and other surrogate-based inference methods while keeping a small algorithmic cost. Our benchmark corroborates VBMC as a general-purpose technique for sample-efficient black-box Bayesian inference also with noisy models.
Latent position network models are a versatile tool in network science; applications include clustering entities, controlling for causal confounders, and defining priors over unobserved graphs. Estimating each node's latent position is typically framed as a Bayesian inference problem, with Metropolis within Gibbs being the most popular tool for approximating the posterior distribution. However, it is well-known that Metropolis within Gibbs is inefficient for large networks; the acceptance ratios are expensive to compute, and the resultant posterior draws are highly correlated. In this article, we propose an alternative Markov chain Monte Carlo strategy -- defined using a combination of split Hamiltonian Monte Carlo and Firefly Monte Carlo -- that leverages the posterior distribution's functional form for more efficient posterior computation. We demonstrate that these strategies outperform Metropolis within Gibbs and other algorithms on synthetic networks, as well as on real information-sharing networks of teachers and staff in a school district.
Small disturbances can trigger functional breakdowns in complex systems. A challenging task is to infer the structural cause of a disturbance in a networked system, soon enough to prevent a catastrophe. We present a graph neural network approach, borrowed from the deep learning paradigm, to infer structural perturbations from functional time series. We show our data-driven approach outperforms typical reconstruction methods while meeting the accuracy of Bayesian inference. We validate the versatility and performance of our approach with epidemic spreading, population dynamics, and neural dynamics, on various network structures: random networks, scale-free networks, 25 real food-web systems, and the C. Elegans connectome. Moreover, we report that our approach is robust to data corruption. This work uncovers a practical avenue to study the resilience of real-world complex systems.
Principal Component Analysis (PCA) minimizes the reconstruction error given a class of linear models of fixed component dimensionality. Probabilistic PCA adds a probabilistic structure by learning the probability distribution of the PCA latent space weights, thus creating a generative model. Autoencoders (AE) minimize the reconstruction error in a class of nonlinear models of fixed latent space dimensionality and outperform PCA at fixed dimensionality. Here, we introduce the Probabilistic Autoencoder (PAE) that learns the probability distribution of the AE latent space weights using a normalizing flow (NF). The PAE is fast and easy to train and achieves small reconstruction errors, high sample quality, and good performance in downstream tasks. We compare the PAE to Variational AE (VAE), showing that the PAE trains faster, reaches a lower reconstruction error, and produces good sample quality without requiring special tuning parameters or training procedures. We further demonstrate that the PAE is a powerful model for performing the downstream tasks of probabilistic image reconstruction in the context of Bayesian inference of inverse problems for inpainting and denoising applications. Finally, we identify latent space density from NF as a promising outlier detection metric.
Through sequential construction of posteriors on observing data online, Bayes' theorem provides a natural framework for continual learning. We develop Variational Auto-Regressive Gaussian Processes (VAR-GPs), a principled posterior updating mechanism to solve sequential tasks in continual learning. By relying on sparse inducing point approximations for scalable posteriors, we propose a novel auto-regressive variational distribution which reveals two fruitful connections to existing results in Bayesian inference, expectation propagation and orthogonal inducing points. Mean predictive entropy estimates show VAR-GPs prevent catastrophic forgetting, which is empirically supported by strong performance on modern continual learning benchmarks against competitive baselines. A thorough ablation study demonstrates the efficacy of our modeling choices.
High dimensional integration is essential to many areas of science, ranging from particle physics to Bayesian inference. Approximating these integrals is hard, due in part to the difficulty of locating and sampling from regions of the integration domain that make significant contributions to the overall integral. Here, we present a new algorithm called Tree Quadrature (TQ) that separates this sampling problem from the problem of using those samples to produce an approximation of the integral. TQ places no qualifications on how the samples provided to it are obtained, allowing it to use state-of-the-art sampling algorithms that are largely ignored by existing integration algorithms. Given a set of samples, TQ constructs a surrogate model of the integrand in the form of a regression tree, with a structure optimised to maximise integral precision. The tree divides the integration domain into smaller containers, which are individually integrated and aggregated to estimate the overall integral. Any method can be used to integrate each individual container, so existing integration methods, like Bayesian Monte Carlo, can be combined with TQ to boost their performance. On a set of benchmark problems, we show that TQ provides accurate approximations to integrals in up to 15 dimensions; and in dimensions 4 and above, it outperforms simple Monte Carlo and the popular Vegas method.
The COVID-19 pandemic has highlighted the importance of in-silico epidemiological modelling in predicting the dynamics of infectious diseases to inform health policy and decision makers about suitable prevention and containment strategies. Work in this setting involves solving challenging inference and control problems in individual-based models of ever increasing complexity. Here we discuss recent breakthroughs in machine learning, specifically in simulation-based inference, and explore its potential as a novel venue for model calibration to support the design and evaluation of public health interventions. To further stimulate research, we are developing software interfaces that turn two cornerstone COVID-19 and malaria epidemiology models COVID-sim, (https://github.com/mrc-ide/covid-sim/) and OpenMalaria (https://github.com/SwissTPH/openmalaria) into probabilistic programs, enabling efficient interpretable Bayesian inference within those simulators.
Group testing is a method of identifying infected patients by performing tests on a pool of specimens collected from patients. For the case in which the test returns a false result with finite probability, we propose Bayesian inference and a corresponding belief propagation (BP) algorithm to identify the infected patients from the results of tests performed on the pool. We show that the true-positive rate is improved by taking into account the credible interval of a point estimate of each patient. Further, the prevalence and the error probability in the test are estimated by combining an expectation-maximization method with the BP algorithm. As another approach, we introduce a hierarchical Bayes model to identify the infected patients and estimate the prevalence. By comparing these methods, we formulate a guide for practical usage.