Stochastic Variational Bayesian Inference for a Nonlinear Forward Model (2007.01675v1)

Abstract: 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.

• The paper proposes a stochastic VB method that avoids Taylor approximations to infer nonlinear model parameters from complex, noisy data.
• It demonstrates competitive parameter recovery and efficiency by leveraging sampling and batch processing on simulated and arterial spin labelling MRI data.
• Tuning key parameters like learning rate and sample size proved essential for balancing exploration and convergence stability in challenging noise conditions.

Overview of Stochastic Variational Bayesian Inference for Nonlinear Forward Models

The paper presents a novel approach to Bayesian inference for nonlinear model parameters using a stochastic version of Variational Bayes (VB). Traditionally, Bayesian methods offer principled approaches for parameter inference in nonlinear models, particularly valuable in noisy data contexts often encountered in fields like medical imaging. However, existing analytical variational formulations, such as those employing Taylor series expansions, impose constraints that can limit their applicability and flexibility, particularly when dealing with complex non-linearities and noise models.

In addressing these limitations, the authors propose a stochastic VB method that mitigates the need for approximations inherent to analytical solutions. The new stochastic formulation harnesses the flexibility of choosing less constrained, non-conjugate prior distributions and embarks on a sampling-driven inference process distinct from deterministic updates characteristic of analytical VB algorithms. This approach permits broader applicability, supporting various noise and prior models that the analytical variant may not accommodate efficiently.

The paper evaluates the effectiveness of this stochastic VB method on both simulated data and actual perfusion data from Arterial Spin Labelling (ASL) MRI, a technique prominently utilized in neuroimaging research. It further investigates the implications of various algorithmic parameters, such as learning rate, sample size, and batch size, on inference accuracy and computational efficiency. The stochastic VB approach is validated against the established analytical VB method, showcasing that competitive parameter recovery can be achieved with comparable computational overhead, even in data conditions traditionally deemed more challenging due to noise levels or model complexity.

Numerical and Algorithmic Insights

A deep exploration of the stochastic VB algorithm delineates several numerical and operational benefits. The authors demonstrate that, unlike deterministic VB, the stochastic method can more broadly explore the parameter space, inherently reducing susceptibility to becoming trapped in local minima. This outcome is particularly advantageous in nonlinear contexts where the free energy landscape may feature numerous such minima. However, this flexibility introduces complexities in convergence behavior, necessitating careful tuning of algorithm parameters. Notably, the paper finds that moderate learning rates (around 0.05) and small, yet adequate, posterior sample sizes tend to offer a balance between exploration and convergence stability.

Moreover, by leveraging batch processing, the stochastic VB method can significantly reduce computation time, a critical consideration in practical applications like brain imaging where thousands of serial datasets require simultaneous processing. The authors acknowledge that while a single epoch—complete over the dataset—traditionally dictates efficiency, additional computational parallelism could further enhance throughput, suggesting future avenues for GPU implementations.

Practical and Theoretical Implications and Future Prospects

From a practical standpoint, this work suggests that stochastic VB could supplant analytical methods in a host of nonlinear applications, particularly where hierarchical or complex noise models challenge current Bayesian paradigms. Its inherent flexibility to address non-linearity without excessive computational burden has ramifications for time-sensitive and large-scale data fields such as neuroimaging, climate modeling, and beyond.

Theoretically, the approach underscores the potential evolvement of Bayesian inference methods beyond traditional reliance on tractability constraints. By accommodating a wider variety of prior and noise configurations, the stochastic VB framework broadens the horizon for applications that were previously deemed computationally prohibitive under strict conjugate or linearizing assumptions.

Future work might explore enhancement potentials in adaptive learning schedules and hybridizations with deep learning frameworks, perhaps integrating further into neural network architectures for end-to-end learning tasks across expansive datasets. Further scalability considerations could be examined via distributed computing paradigms as data dimensions and complexity continue to rise. As such, the paper not only advances non-linear Bayesian inference methodologies but also sets the stage for evolving research avenues in statistical learning.