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Register a new userStochastic Variational Bayesian Inference for a Nonlinear Forward Model
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Variational Bayes (VB) has been used to facilitate the calculation of the posterior distribution in the context of Bayesian inference of the parameters of nonlinear models from data. Previously an analytical formulation of VB has been derived for nonlinear model inference on data with additive gaussian noise as an alternative to nonlinear least squares. Here a stochastic solution is derived that avoids some of the approximations required of the analytical formulation, offering a solution that can be more flexibly deployed for nonlinear model inference problems. The stochastic VB solution was used for inference on a biexponential toy case and the algorithmic parameter space explored, before being deployed on real data from a magnetic resonance imaging study of perfusion. The new method was found to achieve comparable parameter recovery to the analytic solution and be competitive in terms of computational speed despite being reliant on sampling.
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Neural processes (NPs) constitute a family of variational approximate models for stochastic processes with promising properties in computational efficiency and uncertainty quantification. These processes use neural networks with latent variable inputs to induce predictive distributions. However, the expressiveness of vanilla NPs is limited as they only use a global latent variable, while target specific local variation may be crucial sometimes. To address this challenge, we investigate NPs systematically and present a new variant of NP model that we call Doubly Stochastic Variational Neural Process (DSVNP). This model combines the global latent variable and local latent variables for prediction. We evaluate this model in several experiments, and our results demonstrate competitive prediction performance in multi-output regression and uncertainty estimation in classification.
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.
Stochastic variational inference allows for fast posterior inference in complex Bayesian models. However, the algorithm is prone to local optima which can make the quality of the posterior approximation sensitive to the choice of hyperparameters and initialization. We address this problem by replacing the natural gradient step of stochastic varitional inference with a trust-region update. We show that this leads to generally better results and reduced sensitivity to hyperparameters. We also describe a new strategy for variational inference on streaming data and show that here our trust-region method is crucial for getting good performance.
Bayesian methods have proved powerful in many applications for the inference of model parameters from data. These methods are based on Bayes theorem, which itself is deceptively simple. However, in practice the computations required are intractable even for simple cases. Hence methods for Bayesian inference have historically either been significantly approximate, e.g., the Laplace approximation, or achieve samples from the exact solution at significant computational expense, e.g., Markov Chain Monte Carlo methods. Since around the year 2000 so-called Variational approaches to Bayesian inference have been increasingly deployed. In its most general form Variational Bayes (VB) involves approximating the true posterior probability distribution via another more manageable distribution, the aim being to achieve as good an approximation as possible. In the original FMRIB Variational Bayes tutorial we documented an approach to VB based that took a mean field approach to forming the approximate posterior, required the conjugacy of prior and likelihood, and exploited the Calculus of Variations, to derive an iterative series of update equations, akin to Expectation Maximisation. In this tutorial we revisit VB, but now take a stochastic approach to the problem that potentially circumvents some of the limitations imposed by the earlier methodology. This new approach bears a lot of similarity to, and has benefited from, computational methods applied to machine learning algorithms. Although, what we document here is still recognisably Bayesian inference in the classic sense, and not an attempt to use machine learning as a black-box to solve the inference problem.
A high level of physical detail in a molecular model improves its ability to perform high accuracy simulations, but can also significantly affect its complexity and computational cost. In some situations, it is worthwhile to add additional complexity to a model to capture properties of interest; in others, additional complexity is unnecessary and can make simulations computationally infeasible. In this work we demonstrate the use of Bayes factors for molecular model selection, using Monte Carlo sampling techniques to evaluate the evidence for different levels of complexity in the two-centered Lennard-Jones + quadrupole (2CLJQ) fluid model. Examining three levels of nested model complexity, we demonstrate that the use of variable quadrupole and bond length parameters in this model framework is justified only sometimes. We also explore the effect of the Bayesian prior distribution on the Bayes factors, as well as ways to propose meaningful prior distributions. This Bayesian Markov Chain Monte Carlo (MCMC) process is enabled by the use of analytical surrogate models that accurately approximate the physical properties of interest. This work paves the way for further atomistic model selection work via Bayesian inference and surrogate modeling
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