Traditional deep Gaussian processes model the data evolution using a discrete hierarchy, whereas differential Gaussian processes (DIFFGPs) represent the evolution as an infinitely deep Gaussian process. However, prior DIFFGP methods often overlook the uncertainty of kernel hyperparameters and assume them to be fixed and time-invariant, failing to leverage the unique synergy between continuous-time models and approximate inference. In this work, we propose a fully Bayesian approach that treats the kernel hyperparameters as random variables and constructs coupled stochastic differential equations (SDEs) to learn their posterior distribution and that of inducing points. By incorporating estimation uncertainty on hyperparameters, our method enhances the model's flexibility and adaptability to complex dynamics. Additionally, our approach provides a time-varying, comprehensive, and realistic posterior approximation through coupling variables using SDE methods. Experimental results demonstrate the advantages of our method over traditional approaches, showcasing its superior performance in terms of flexibility, accuracy, and other metrics. Our work opens up exciting research avenues for advancing Bayesian inference and offers a powerful modeling tool for continuous-time Gaussian processes.

We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multi-spiked tensor model in the high-dimensional regime. This multi-index model arises from the tensor principal component analysis (PCA) problem, which aims to infer $r$ unknown, orthogonal signal vectors within the $N$-dimensional unit sphere through maximum likelihood estimation from noisy observations of an order-$p$ tensor. We determine the number of samples and the conditions on the signal-to-noise ratios (SNRs) required to efficiently recover the unknown spikes from natural initializations. Specifically, we distinguish between three types of recovery: exact recovery of each spike, recovery of a permutation of all spikes, and recovery of the correct subspace spanned by the signal vectors. We show that with online SGD, it is possible to recover all spikes provided a number of sample scaling as $N^{p-2}$, aligning with the computational threshold identified in the rank-one tensor PCA problem [Ben Arous, Gheissari, Jagannath 2020, 2021]. For gradient flow, we show that the algorithmic threshold to efficiently recover the first spike is also of order $N^{p-2}$. However, recovering the subsequent directions requires the number of samples to scale as $N^{p-1}$. Our results are obtained through a detailed analysis of a low-dimensional system that describes the evolution of the correlations between the estimators and the spikes. In particular, the hidden vectors are recovered one by one according to a sequential elimination phenomenon: as one correlation exceeds a critical threshold, all correlations sharing a row or column index decrease and become negligible, allowing the subsequent correlation to grow and become macroscopic. The sequence in which correlations become macroscopic depends on their initial values and on the associated SNRs.

Expected free energy (EFE) is a central quantity in active inference which has recently gained popularity due to its intuitive decomposition of the expected value of control into a pragmatic and an epistemic component. While numerous conjectures have been made to justify EFE as a decision making objective function, the most widely accepted is still its intuitiveness and resemblance to variational free energy in approximate Bayesian inference. In this work, we take a bottom up approach and ask: taking EFE as given, what's the resulting agent's optimality gap compared with a reward-driven reinforcement learning (RL) agent, which is well understood? By casting EFE under a particular class of belief MDP and using analysis tools from RL theory, we show that EFE approximates the Bayes optimal RL policy via information value. We discuss the implications for objective specification of active inference agents.

Schr\"{o}dinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr\"{o}dinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schr\"{o}dinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schr\"{o}dinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

The Variational Bayesian method (VB) is used to solve the probability distributions of latent variables with the minimum free energy criterion. This criterion is not easy to understand, and the computation is complex. For these reasons, this paper proposes the Semantic Variational Bayes' method (SVB). The Semantic Information Theory the author previously proposed extends the rate-distortion function R(D) to the rate-fidelity function R(G), where R is the minimum mutual information for given semantic mutual information G. SVB came from the parameter solution of R(G), where the variational and iterative methods originated from Shannon et al.'s research on the rate-distortion function. The constraint functions SVB uses include likelihood, truth, membership, similarity, and distortion functions. SVB uses the maximum information efficiency (G/R) criterion, including the maximum semantic information criterion for optimizing model parameters and the minimum mutual information criterion for optimizing the Shannon channel. For the same tasks, SVB is computationally simpler than VB. The computational experiments in the paper include 1) using a mixture model as an example to show that the mixture model converges as G/R increases; 2) demonstrating the application of SVB in data compression with a group of error ranges as the constraint; 3) illustrating how the semantic information measure and SVB can be used for maximum entropy control and reinforcement learning in control tasks with given range constraints, providing numerical evidence for balancing control's purposiveness and efficiency. Further research is needed to apply SVB to neural networks and deep learning.

Unexpected stimuli induce "error" or "surprise" signals in the brain. The theory of predictive coding promises to explain these observations in terms of Bayesian inference by suggesting that the cortex implements variational inference in a probabilistic graphical model. However, when applied to machine learning tasks, this family of algorithms has yet to perform on par with other variational approaches in high-dimensional, structured inference problems. To address this, we introduce a novel predictive coding algorithm for structured generative models, that we call divide-and-conquer predictive coding (DCPC). DCPC differs from other formulations of predictive coding, as it respects the correlation structure of the generative model and provably performs maximum-likelihood updates of model parameters, all without sacrificing biological plausibility. Empirically, DCPC achieves better numerical performance than competing algorithms and provides accurate inference in a number of problems not previously addressed with predictive coding. We provide an open implementation of DCPC in Pyro on Github.

The ability to detect out-of-distribution (OOD) inputs is critical to guarantee the reliability of classification models deployed in an open environment. A fundamental challenge in OOD detection is that a discriminative classifier is typically trained to estimate the posterior probability p(y|z) for class y given an input z, but lacks the explicit likelihood estimation of p(z) ideally needed for OOD detection. While numerous OOD scoring functions have been proposed for classification models, these estimate scores are often heuristic-driven and cannot be rigorously interpreted as likelihood. To bridge the gap, we propose Intrinsic Likelihood (INK), which offers rigorous likelihood interpretation to modern discriminative-based classifiers. Specifically, our proposed INK score operates on the constrained latent embeddings of a discriminative classifier, which are modeled as a mixture of hyperspherical embeddings with constant norm. We draw a novel connection between the hyperspherical distribution and the intrinsic likelihood, which can be effectively optimized in modern neural networks. Extensive experiments on the OpenOOD benchmark empirically demonstrate that INK establishes a new state-of-the-art in a variety of OOD detection setups, including both far-OOD and near-OOD. Code is available at https://github.com/deeplearning-wisc/ink.

Reconstructing the evolutionary history relating a collection of molecular sequences is the main subject of modern Bayesian phylogenetic inference. However, the commonly used Markov chain Monte Carlo methods can be inefficient due to the complicated space of phylogenetic trees, especially when the number of sequences is large. An alternative approach is variational Bayesian phylogenetic inference (VBPI) which transforms the inference problem into an optimization problem. While effective, the default diagonal lognormal approximation for the branch lengths of the tree used in VBPI is often insufficient to capture the complexity of the exact posterior. In this work, we propose a more flexible family of branch length variational posteriors based on semi-implicit hierarchical distributions using graph neural networks. We show that this semi-implicit construction emits straightforward permutation equivariant distributions, and therefore can handle the non-Euclidean branch length space across different tree topologies with ease. To deal with the intractable marginal probability of semi-implicit variational distributions, we develop several alternative lower bounds for stochastic optimization. We demonstrate the effectiveness of our proposed method over baseline methods on benchmark data examples, in terms of both marginal likelihood estimation and branch length posterior approximation.

We study best-arm identification (BAI) in the fixed-budget setting. Adaptive allocations based on upper confidence bounds (UCBs), such as UCBE, are known to work well in BAI. However, it is well-known that its optimal regret is theoretically dependent on instances, which we show to be an artifact in many fixed-budget BAI problems. In this paper we propose an UCB exploration algorithm that is both theoretically and empirically efficient for the fixed budget BAI problem under a Bayesian setting. The key idea is to learn prior information, which can enhance the performance of UCB-based BAI algorithm as it has done in the cumulative regret minimization problem. We establish bounds on the failure probability and the simple regret for the Bayesian BAI problem, providing upper bounds of order $\tilde{O}(\sqrt{K/n})$, up to logarithmic factors, where $n$ represents the budget and $K$ denotes the number of arms. Furthermore, we demonstrate through empirical results that our approach consistently outperforms state-of-the-art baselines.

Adaptive experimentation can significantly improve statistical power, but standard algorithms overlook important practical issues including batched and delayed feedback, personalization, non-stationarity, multiple objectives, and constraints. To address these issues, the current algorithm design paradigm crafts tailored methods for each problem instance. Since it is infeasible to devise novel algorithms for every real-world instance, practitioners often have to resort to suboptimal approximations that do not address all of their challenges. Moving away from developing bespoke algorithms for each setting, we present a mathematical programming view of adaptive experimentation that can flexibly incorporate a wide range of objectives, constraints, and statistical procedures. By formulating a dynamic program in the batched limit, our modeling framework enables the use of scalable optimization methods (e.g., SGD and auto-differentiation) to solve for treatment allocations. We evaluate our framework on benchmarks modeled after practical challenges such as non-stationarity, personalization, multi-objectives, and constraints. Unlike bespoke algorithms such as modified variants of Thomson sampling, our mathematical programming approach provides remarkably robust performance across instances.