Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution data and makes the unrealistic assumption that out-of-distribution (OOD) data is known at training time. In this work we propose the Posterior Network (PostNet), which uses Normalizing Flows to predict an individual closed-form posterior distribution over predicted probabilites for any input sample. The posterior distributions learned by PostNet accurately reflect uncertainty for in- and out-of-distribution data -- without requiring access to OOD data at training time. PostNet achieves state-of-the art results in OOD detection and in uncertainty calibration under dataset shifts.

Approximate Bayesian computation (ABC) is a likelihood-free inference method that has been employed in various applications. However, ABC is sensitive to outliers, which is caused by an inappropriate choice of the data discrepancy measure. In this paper, we propose to use a nearest-neighbor-based $\gamma$-divergence estimator as a data discrepancy measure. We show that our estimator possesses a suitable robustness property called the redescending property. In addition, our estimator enjoys various desirable properties such as high flexibility, asymptotic unbiasedness, almost sure convergence, and linear time complexity. Through experiments, we demonstrate that our method achieves significantly higher robustness than existing discrepancy measures.

Offline reinforcement learning seeks to utilize offline (observational) data to guide the learning of (causal) sequential decision making strategies. The hope is that offline reinforcement learning coupled with function approximation methods (to deal with the curse of dimensionality) can provide a means to help alleviate the excessive sample complexity burden in modern sequential decision making problems. However, the extent to which this broader approach can be effective is not well understood, where the literature largely consists of sufficient conditions. This work focuses on the basic question of what are necessary representational and distributional conditions that permit provable sample-efficient offline reinforcement learning. Perhaps surprisingly, our main result shows that even if: i) we have realizability in that the true value function of \emph{every} policy is linear in a given set of features and 2) our off-policy data has good coverage over all features (under a strong spectral condition), then any algorithm still (information-theoretically) requires a number of offline samples that is exponential in the problem horizon in order to non-trivially estimate the value of \emph{any} given policy. Our results highlight that sample-efficient offline policy evaluation is simply not possible unless significantly stronger conditions hold; such conditions include either having low distribution shift (where the offline data distribution is close to the distribution of the policy to be evaluated) or significantly stronger representational conditions (beyond realizability).

Discrete latent spaces in variational autoencoders have been shown to effectively capture the data distribution for many real-world problems such as natural language understanding, human intent prediction, and visual scene representation. However, discrete latent spaces need to be sufficiently large to capture the complexities of real-world data, rendering downstream tasks computationally challenging. For instance, performing motion planning in a high-dimensional latent representation of the environment could be intractable. We consider the problem of sparsifying the discrete latent space of a trained conditional variational autoencoder, while preserving its learned multimodality. As a post hoc latent space reduction technique, we use evidential theory to identify the latent classes that receive direct evidence from a particular input condition and filter out those that do not. Experiments on diverse tasks, such as image generation and human behavior prediction, demonstrate the effectiveness of our proposed technique at reducing the discrete latent sample space size of a model while maintaining its learned multimodality.

Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model's predictions. For this reason, we propose predictive complexity priors: a functional prior that is defined by comparing the model's predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to high-dimensional regression, reasoning over neural network depth, and sharing of statistical strength for few-shot learning.

A decision tree is commonly restricted to use a single hyperplane to split the covariate space at each of its internal nodes. It often requires a large number of nodes to achieve high accuracy, hurting its interpretability. In this paper, we propose convex polytope trees (CPT) to expand the family of decision trees by an interpretable generalization of their decision boundary. The splitting function at each node of CPT is based on the logical disjunction of a community of differently weighted probabilistic linear decision-makers, which also geometrically corresponds to a convex polytope in the covariate space. We use a nonparametric Bayesian prior at each node to infer the community's size, encouraging simpler decision boundaries by shrinking the number of polytope facets. We develop a greedy method to efficiently construct CPT and scalable end-to-end training algorithms for the tree parameters when the tree structure is given. We empirically demonstrate the efficiency of CPT over existing state-of-the-art decision trees in several real-world classification and regression tasks from diverse domains.

Domains where supervised models are deployed often come with task-specific constraints, such as prior expert knowledge on the ground-truth function, or desiderata like safety and fairness. We introduce a novel probabilistic framework for reasoning with such constraints and formulate a prior that enables us to effectively incorporate them into Bayesian neural networks (BNNs), including a variant that can be amortized over tasks. The resulting Output-Constrained BNN (OC-BNN) is fully consistent with the Bayesian framework for uncertainty quantification and is amenable to black-box inference. Unlike typical BNN inference in uninterpretable parameter space, OC-BNNs widen the range of functional knowledge that can be incorporated, especially for model users without expertise in machine learning. We demonstrate the efficacy of OC-BNNs on real-world datasets, spanning multiple domains such as healthcare, criminal justice, and credit scoring.

Compared to the conditional mean as a simple point estimator, the conditional density function is more informative to describe the distributions with multi-modality, asymmetry or heteroskedasticity. In this paper, we propose a novel parametric conditional density estimation method by showing the connection between the general density and the likelihood function of inhomogeneous Poisson process models. The maximum likelihood estimates can be obtained via weighted logistic regressions, and the computation can be significantly relaxed by combining a block-wise alternating maximization scheme and local case-control sampling. We also provide simulation studies for illustration.

A common problem in neuroscience is to elucidate the collective neural representations of behaviorally important variables such as head direction, spatial location, upcoming movements, or mental spatial transformations. Often, these latent variables are internal constructs not directly accessible to the experimenter. Here, we propose a new probabilistic latent variable model to simultaneously identify the latent state and the way each neuron contributes to its representation in an unsupervised way. In contrast to previous models which assume Euclidean latent spaces, we embrace the fact that latent states often belong to symmetric manifolds such as spheres, tori, or rotation groups of various dimensions. We therefore propose the manifold Gaussian process latent variable model (mGPLVM), where neural responses arise from (i) a shared latent variable living on a specific manifold, and (ii) a set of non-parametric tuning curves determining how each neuron contributes to the representation. Cross-validated comparisons of models with different topologies can be used to distinguish between candidate manifolds, and variational inference enables quantification of uncertainty. We demonstrate the validity of the approach on several synthetic datasets, as well as on calcium recordings from the ellipsoid body of Drosophila melanogaster and extracellular recordings from the mouse anterodorsal thalamic nucleus. These circuits are both known to encode head direction, and mGPLVM correctly recovers the ring topology expected from neural populations representing a single angular variable.

Among existing uncertainty estimation approaches, Dirichlet Prior Network (DPN) distinctly models different predictive uncertainty types. However, for in-domain examples with high data uncertainties among multiple classes, even a DPN model often produces indistinguishable representations from the out-of-distribution (OOD) examples, compromising their OOD detection performance. We address this shortcoming by proposing a novel loss function for DPN to maximize the \textit{representation gap} between in-domain and OOD examples. Experimental results demonstrate that our proposed approach consistently improves OOD detection performance.