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Empirical Bayes (EB) improves the accuracy of simultaneous inference "by learning from the experience of others" (Efron, 2012). Classical EB theory focuses on latent variables that are iid draws from a fitted prior (Efron, 2019). Modern applications, however, feature complex structure, like arrays, spatial processes, or covariates. How can we apply EB ideas to these settings? We propose a generalized approach to empirical Bayes based on the notion of probabilistic symmetry. Our method pairs a simultaneous inference problem-with an unknown prior-to a symmetry assumption on the joint distribution of the latent variables. Each symmetry implies an ergodic decomposition, which we use to derive a corresponding empirical Bayes method. We call this methodBayesian empirical Bayes (BEB). We show how BEB recovers the classical methods of empirical Bayes, which implicitly assume exchangeability. We then use it to extend EB to other probabilistic symmetries: (i) EB matrix recovery for arrays and graphs; (ii) covariate-assisted EB for conditional data; (iii) EB spatial regression under shift invariance. We develop scalable algorithms based on variational inference and neural networks. In simulations, BEB outperforms existing approaches to denoising arrays and spatial data. On real data, we demonstrate BEB by denoising a cancer gene-expression matrix and analyzing spatial air-quality data from New York City.

Bayesian optimization usually assumes that a Bayesian prior is given.

The task of quantifying the inherent uncertainty associated with neural network predictions is a key challenge in artificial intelligence. Bayesian neural networks (BNNs) and deep ensembles are among the most prominent approaches to tackle this task. Both approaches produce predictions by computing an expectation of neural network outputs over some distribution on the corresponding weights; this distribution is given by the posterior in the case of BNNs, and by a mixture of point masses for ensembles. Inspired by recent work showing that the distribution used by ensembles can be understood as a posterior corresponding to a learned data-dependent prior, we propose last layer empirical Bayes (LLEB). LLEB instantiates a learnable prior as a normalizing flow, which is then trained to maximize the evidence lower bound; to retain tractability we use the flow only on the last layer. We show why LLEB is well motivated, and how it interpolates between standard BNNs and ensembles in terms of the strength of the prior that they use. LLEB performs on par with existing approaches, highlighting that empirical Bayes is a promising direction for future research in uncertainty quantification.

Quantifying uncertainty in neural networks is a highly relevant problem which is essential to many applications. The two predominant paradigms to tackle this task are Bayesian neural networks (BNNs) and deep ensembles. Despite some similarities between these two approaches, they are typically surmised to lack a formal connection and are thus understood as fundamentally different. BNNs are often touted as more principled due to their reliance on the Bayesian paradigm, whereas ensembles are perceived as more ad-hoc; yet, deep ensembles tend to empirically outperform BNNs, with no satisfying explanation as to why this is the case. In this work we bridge this gap by showing that deep ensembles perform exact Bayesian averaging with a posterior obtained with an implicitly learned data-dependent prior. In other words deep ensembles are Bayesian, or more specifically, they implement an empirical Bayes procedure wherein the prior is learned from the data. This perspective offers two main benefits: (i) it theoretically justifies deep ensembles and thus provides an explanation for their strong empirical performance; and (ii) inspection of the learned prior reveals it is given by a mixture of point masses -- the use of such a strong prior helps elucidate observed phenomena about ensembles. Overall, our work delivers a newfound understanding of deep ensembles which is not only of interest in it of itself, but which is also likely to generate future insights that drive empirical improvements for these models.

The Poisson compound decision problem is a classical problem in statistics, for which parametric and nonparametric empirical Bayes methodologies are available to estimate the Poisson's means in static or batch domains. In this paper, we consider the Poisson compound decision problem in a streaming or online domain. By relying on a quasi-Bayesian approach, often referred to as Newton's algorithm, we obtain sequential Poisson's mean estimates that are of easy evaluation, computationally efficient and with a constant computational cost as data increase, which is desirable for streaming data. Large sample asymptotic properties of the proposed estimates are investigated, also providing frequentist guarantees in terms of a regret analysis. We validate empirically our methodology, both on synthetic and real data, comparing against the most popular alternatives.

Dynamic Bayesian Networks (DBNs) are a class of Probabilistic Graphical Models that enable the modeling of a Markovian dynamic process through defining the kernel transition by the DAG structure of the graph found to fit a dataset. There are a number of structure learners than enable one to find the structure of a DBN to fit data, each of which with its own set of particular advantages and disadvantages. The structure of a DBN itself presents transparent criteria in order to identify causal discovery between variables. However, without the presence of large quantities of data, identifying a ground truth causal structure becomes unrealistic in practice. However, one can consider a procedure by which a set of graphs identifying structure are computed as approximate noisy solutions, and subsequently amortized in a broader statistical procedure fitting a mixture of DBNs. Each component of the mixture presents an alternative hypothesis on the causal structure. From the mixture weights, one can also compute the Bayes Factors comparing the preponderance of evidence between different models. This presents a natural opportunity for the development of Empirical Bayesian methods.

We formally map the problem of sampling from an unknown distribution with density $p_X$ in $\mathbb{R}^d$ to the problem of learning and sampling $p_\mathbf{Y}$ in $\mathbb{R}^{Md}$ obtained by convolving $p_X$ with a fixed factorial kernel: $p_\mathbf{Y}$ is referred to as M-density and the factorial kernel as multimeasurement noise model (MNM). The M-density is smoother than $p_X$, easier to learn and sample from, yet for large $M$ the two problems are mathematically equivalent since $X$ can be estimated exactly given $\mathbf{Y}=\mathbf{y}$ using the Bayes estimator $\widehat{x}(\mathbf{y})=\mathbb{E}[X\vert\mathbf{Y}=\mathbf{y}]$. To formulate the problem, we derive $\widehat{x}(\mathbf{y})$ for Poisson and Gaussian MNMs expressed in closed form in terms of unnormalized $p_\mathbf{Y}$. This leads to a simple least-squares objective for learning parametric energy and score functions. We present various parametrization schemes of interest, including one in which studying Gaussian M-densities directly leads to multidenoising autoencoders--this is the first theoretical connection made between denoising autoencoders and empirical Bayes in the literature. Samples from $p_X$ are obtained by walk-jump sampling (Saremi & Hyvarinen, 2019) via underdamped Langevin MCMC (walk) to sample from $p_\mathbf{Y}$ and the multimeasurement Bayes estimation of $X$ (jump). We study permutation invariant Gaussian M-densities on MNIST, CIFAR-10, and FFHQ-256 datasets, and demonstrate the effectiveness of this framework for realizing fast-mixing stable Markov chains in high dimensions.

We revisit empirical Bayes in the absence of a tractable likelihood function, as is typical in scientific domains relying on computer simulations. We investigate how the empirical Bayesian can make use of neural density estimators first to use all noise-corrupted observations to estimate a prior or source distribution over uncorrupted samples, and then to perform single-observation posterior inference using the fitted source distribution. We propose an approach based on the direct maximization of the log-marginal likelihood of the observations, examining both biased and de-biased estimators, and comparing to variational approaches. We find that, up to symmetries, a neural empirical Bayes approach recovers ground truth source distributions. With the learned source distribution in hand, we show the applicability to likelihood-free inference and examine the quality of the resulting posterior estimates. Finally, we demonstrate the applicability of Neural Empirical Bayes on an inverse problem from collider physics.

We unify empirical Bayes and variational Bayes for approximating unnormalized densities. This framework, named unnormalized variational Bayes (UVB), is based on formulating a latent variable model for the random variable $Y=X+N(0,\sigma^2 I_d)$ and using the evidence lower bound (ELBO), computed by a variational autoencoder, as a parametrization of the energy function of $Y$ which is then used to estimate $X$ with the empirical Bayes least-squares estimator. In this intriguing setup, the $\textit{gradient}$ of the ELBO with respect to noisy inputs plays the central role in learning the energy function. Empirically, we demonstrate that UVB has a higher capacity to approximate energy functions than the parametrization with MLPs as done in neural empirical Bayes (DEEN). We especially showcase $\sigma=1$, where the differences between UVB and DEEN become visible and qualitative in the denoising experiments. For this high level of noise, the distribution of $Y$ is very smoothed and we demonstrate that one can traverse in a single run $-$ without a restart $-$ all MNIST classes in a variety of styles via walk-jump sampling with a fast-mixing Langevin MCMC sampler. We finish by probing the encoder/decoder of the trained models and confirm UVB $\neq$ VAE.