Rethinking the Role of Bayesianism
in the Age of Modern AI

In the era of large-scale AI, foundation models, and deep learning, the Bayesian paradigm is evolving beyond traditional posteriors. In this short talk, I introduced three distinct yet interconnected approaches to Bayesian inference: Generalized Bayes, Variational Inference, and the Laplace approximation. These methodologies offer a rich continuum of ways to be Bayesian in modern machine learning, balancing theoretical rigor and computational scalability. Through this lens, we reflect on the role of Bayesianism in contemporary AI, addressing both its theoretical developments and its place within the broader statistical community.

This talk revisited the questions posed by Yee Whye Teh in his 2017 NeurIPS Breiman lecture, substituting foundation models for deep learning. Two empirical observations were presented, demonstrating the value of probabilistic thinking in understanding foundation models. Firstly, the marginal likelihood was shown to provide insight into the phenomenon of “grokking” or delayed generalization in neural networks. Secondly, a simple approximation scheme for Bernoulli and softmax likelihoods was introduced, which can outperform more principled approaches in certain cases. The talk also explored the potential benefits of Bayesian foundation models, highlighting their adaptability to various tasks that require uncertainty quantification, such as active learning. The discussion concluded with a call to action for the community to define tasks that would benefit from Bayesian foundation models, and to investigate the role of uncertainty in these applications.

I started by discussing some obstacles for Bayesian inference in DL: namely, the intractability of posterior distributions over the weights, as well as the existence of multiple minima. A recent conjecture by Entezari et al. suggests that, modulo permutation symmetries, there may be only one minimum when training with SGD. What are the implications for Bayesian inference in deep learning if the conjecture were true? Would the challenges we currently face when working in the weight space geometry be resolved?

I then discussed alternative geometries. I brought up the example of sped up training when imposing the “right” structure on the feature space (Malinar et al., “Emergence in non-neural models”). This example demonstrates the potential of working in the feature/function space geometry, though realizing this potential is tricky, and right now figuring out the structure of the features at best can be done post-training.

Finally, I discussed the role of the “right” initialization, that, historically, enabled deep neural network training (He initialization) and scaling ($\mu$-P initialization). I raised a question of what would be an analogue of signal propagation for Bayesian analysis. Perhaps what we are missing is the “right” initialization for priors that would enable scaling.

In the literature of Bayesian neural networks, there has been considerable interest in studying uncertainty. There have been numerous attempts to define uncertainties, and a converging point has been to use information theoretic measures to do so. There have been proposals to decompose the total predictive uncertainty, measured through the entropy of the predictive distribution obtained by marginalizing parameters, into (mainly) aleatoric and epistemic uncertainties. While these measures make intuitive sense, in my presentation I tried to make two points on the usefulness of these measures. First, these measures of uncertainty have been criticized by some recent works (e.g., Wimmer et al., UAI 2023 and Schweighofer et al., arXiv:2410.10786), which point out some pathological behaviors in some specific cases. Second, I presented a simple polynomial regression example with a dataset generated from a fixed order polynomial; in this example, I reported total, epistemic and aleatoric uncertainties with respect to modeling choices with increasing model order and with respect to increasing number of observations. These results show some sensible behavior, but suggest some difficulties in using them in an actionable way to compare, select, or improve models. This motivates studying alternative ways to study uncertainties with the aim of benchmarking models and inference methods, and I welcomed discussions on this front.

Teaching, not research, is the principal way for academics to create value. While industry and academia compete with each other in research, teaching is unique to academia. However, we rarely talk about teaching this way, and most academics are primarily evaluated based on their research contribution, rather their teaching impact. When we teach, we actively decide what to teach and what not. By deciding what not to teach, we decide what to let die. Academics who want to ensure their research work outlasts them should ultimately actively aim to get their work immortalized in the teaching canon. Writing papers is not the best way to achieve this. Teaching, then, is an active act of forming the body of knowledge for entire generations of students, of formalizing the field into a canon. Because no course is ever complete, it would be an overall loss of knowledge if every student only learned from one and the same course. If each university teaches ML differently, this allows for some diversity in teaching ML, leading to more knowledge retained in distribution. There are many valuable concepts (e.g. Bayesian Nonparametrics, Sum Product, and more generally message passing in graphical models, Support and Relevance Vector Machines) that have now dropped from the widely taught curriculum, and are thus slated to vanish from memory. In my talk, I used my own “Probabilistic ML” course in Tübingen as a trigger for discussion of the above points.

The biggest challenge with modern AI is that it requires a huge amount of resources which favors monopoly and harmful for the society in the long run. We need to fix this and develop better systems that are more sustainable, transparent, and trustworthy. To do so, we must retrain less and reuse more (the previously trained models). I argue that lifelong learning is the ultimate goal and designing incremental, continual, federated, active learning is extremely important. These are the problems where Bayesian principles shine. Our group has focused on such problems for the last many years which I will briefly present.

This talk examined the assumptions underlying traditional Bayesianism, specifically: (A1) the model is well-specified, (A2) beliefs are captured in the prior, and (A3) computation is tractable. The limitations of these assumptions in the context of modern AI were discussed, and several alternative approaches were presented to address model mispecification, including likelihood tempering, robust loss functions yielding a pseudo-posterior, and martingale posteriors. These methods represent a departure from traditional Bayesianism and can be seen as part of a broader class of “post-Bayesian” approaches, which aim to adapt Bayesian foundations to the complexities of modern AI applications.

A paradigm shift is on-going in the deep learning field, where researchers are racing in training multimodal foundation models bigger than ever. Currently non-probabilistic approaches dominate the playground of big foundation models, where most Bayesian neural network approaches are prohibitively expensive to compete. But I claim the practitioner’s impression that “Bayesian neural networks with approximate inference are always more expensive than simple deterministic neural networks” will no longer be true in the near future. To partly prove my point, I discuss an example from my previous and on-going work which has made Bayesian neural networks with time and memory complexity lower than deterministic neural networks. The final piece of the puzzle is to translate the time complexity gains to real-time savings, and here the key is to leverage GPU compute more efficiently. I argue that the aspect of algorithm-hardware co-design has been largely ignored by the Bayesian deep learning community, and I hope my encouraging initial result will inspire future Bayesian ML/DL developments by better leveraging contemporary computing hardware.

We propose a novel approach to sequential Bayesian inference based on variational Bayes (VB). The key insight is that, in the online setting, we do not need to add the KL term to regularize to the prior (which comes from the posterior at the previous timestep); instead we can optimize just the expected log-likelihood, performing a single step of natural gradient descent starting at the prior predictive. We prove this method recovers exact Bayesian inference if the model is conjugate. We also show how to compute an efficient deterministic approximation to the VB objective, as well as our simplified objective, when the variational distribution is Gaussian or a sub-family, including the case of a diagonal plus low-rank precision matrix. We show empirically that our method outperforms other online VB methods in the non-conjugate setting, such as online learning for neural networks, especially when controlling for computational costs.

Bayesian approaches have the potential to mitigate problems with neural networks (NNs) such as overconfidence and lack of robustness. However, computation is a major obstacle to performing high-fidelity posterior inference. In this talk, I will first present our research on scalable variational approximations based on subnetworks. Only a subset of the NN is given a Bayesian treatment, and we find this is enough to perform competitive uncertainty estimation. I will then go on to further justify subnetwork inference, not simply for its computational benefits, but from the theoretical insight that these NNs have as rich a posterior predictive distribution as fully-stochastic NNs. Moreover, across various inference schemes, we observe no empirical benefit to using fully stochastic NNs. I will close by questioning whether a fully-Bayesian treatment of NNs can ever have a benefit.

The ideas of aleatoric and epistemic uncertainty are widely used to reason about the probabilistic predictions of machine-learning models. This talk highlights that the common understanding of these ideas is not self-consistent, with the terms overloaded to refer to some quite distinct precise mathematical quantities. This has caused a number of issues with the narrative in the literature, including a number of spurious associations, such as between the data-generating process and a model’s predictive uncertainty, between parameter stochasticity and the reducibility of predictive uncertainty, and between subjective uncertainty estimates and objective measures of predictive performance. We suggested a new ontology for how to talk about uncertainty decomposition, and discussed what quantities we actually wish to estimate in practice, and how to go about this.

I discussed the challenges in modeling complex data, where one necessarily fails to meet the high bar set by theory motivating subjective Bayesianism. I briefly posited that perhaps trying out models and sticking with those that work is a deep part of the success of Bayesianism. I then proposed another approach, rooted in regret theory, where one aims to control the difference between the losses incurred by their own actions and the losses that would have been incurred by any in a set of (comparator) policies. I introduced exponential weights, a classical algorithm, that delivers nontrivial regret guarantees when competing against a finite comparator set, and showed that exponential weights is, de facto, updating a (generalized Bayesian) posterior over the comparator set, which it uses to guide actions. Unlike Bayesian guarantees for posterior updating and prediction, which rest on the validity of the model and prior, regret guarantees for these very same algorithms hold without assumption, offering a new perspective which much wider purview.

This talk explored the role of Bayesian methods in selecting neural network architectures, with the goal of improving generalization performance. It was argued that machine learning excels at optimizing function spaces, but regularization is inherently tied to the chosen function space. It was then proposed to use Bayesian model selection to choose between architectures, rather than just parameters, by evaluating the marginal likelihood of the data given the architecture. This approach allows for the selection of models that generalize better outside the training data regime. The talk highlighted the challenges of computing the marginal likelihood, which is often approximated using point estimates. It also discussed the potential benefits of using Bayesian model selection to understand causality and causal direction, where the prior defines a notion of simplicity that informs model choice. By parametrizing equivariances, Bayesian model selection can also help choose appropriate transformations, leading to more robust and generalizable models.