Samuel Power

Anish Dhir, Samuel Power, Mark van der Wilk

Much of the causal discovery literature prioritises guaranteeing the identifiability of causal direction in statistical models. For structures within a Markov equivalence class, this requires strong assumptions which may not hold in real-world datasets, ultimately limiting the usability of these methods. Building on previous attempts, we show how to incorporate causal assumptions within the Bayesian framework. Identifying causal direction then becomes a Bayesian model selection problem. This enables us to construct models with realistic assumptions, and consequently allows for the differentiation between Markov equivalent causal structures. We analyse why Bayesian model selection works in situations where methods based on maximum likelihood fail. To demonstrate our approach, we construct a Bayesian non-parametric model that can flexibly model the joint distribution. We then outperform previous methods on a wide range of benchmark datasets with varying data generating assumptions.

Rocco Caprio, Juan Kuntz, Samuel Power et al.

We prove non-asymptotic error bounds for particle gradient descent (PGD, Kuntz et al., 2023), a recently introduced algorithm for maximum likelihood estimation of large latent variable models obtained by discretizing a gradient flow of the free energy. We begin by showing that the flow converges exponentially fast to the free energy's minimizers for models satisfying a condition that generalizes both the log-Sobolev and the Polyak--Łojasiewicz inequalities (LSI and PŁI, respectively). We achieve this by extending a result well-known in the optimal transport literature (that the LSI implies the Talagrand inequality) and its counterpart in the optimization literature (that the PŁI implies the so-called quadratic growth condition), and applying the extension to our new setting. We also generalize the Bakry--Émery Theorem and show that the LSI/PŁI extension holds for models with strongly concave log-likelihoods. For such models, we further control PGD's discretization error and obtain the non-asymptotic error bounds. While we are motivated by the study of PGD, we believe that the inequalities and results we extend may be of independent interest.

Jen Ning Lim, Juan Kuntz, Samuel Power et al.

Maximum likelihood estimation (MLE) of latent variable models is often recast as the minimization of a free energy functional over an extended space of parameters and probability distributions. This perspective was recently combined with insights from optimal transport to obtain novel particle-based algorithms for fitting latent variable models to data. Drawing inspiration from prior works which interpret `momentum-enriched' optimization algorithms as discretizations of ordinary differential equations, we propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. The result is a dynamical system that blends elements of Nesterov's Accelerated Gradient method, the underdamped Langevin diffusion, and particle methods. Under suitable assumptions, we prove that the continuous-time system minimizes the functional. By discretizing the system, we obtain a practical algorithm for MLE in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.

Sebastian W. Ober, Samuel Power, Tom Diethe et al.

We observe that BatchBALD, a popular acquisition function for batch Bayesian active learning for classification, can conflate epistemic and aleatoric uncertainty, leading to suboptimal performance. Motivated by this observation, we propose to focus on the predictive probabilities, which only exhibit epistemic uncertainty. The result is an acquisition function that not only performs better, but is also faster to evaluate, allowing for larger batches than before.