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 function. 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 modern machine learning tasks such as reasoning over neural network depth and sharing of statistical strength for few-shot learning.
Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.
Leveraging the wealth of unlabeled data produced in recent years provides great potential for improving supervised models. When the cost of acquiring labels is high, probabilistic active learning methods can be used to greedily select the most informative data points to be labeled. However, for many large-scale problems standard greedy procedures become computationally infeasible and suffer from negligible model change. In this paper, we introduce a novel Bayesian batch active learning approach that mitigates these issues. Our approach is motivated by approximating the complete data posterior of the model parameters. While naive batch construction methods result in correlated queries, our algorithm produces diverse batches that enable efficient active learning at scale. We derive interpretable closed-form solutions akin to existing active learning procedures for linear models, and generalize to arbitrary models using random projections. We demonstrate the benefits of our approach on several large-scale regression and classification tasks.
Recent work has shown that deep generative models can assign higher likelihood to out-of-distribution data sets than to their training data. We posit that this phenomenon is caused by a mismatch between the model's typical set and its areas of high probability density. In-distribution inputs should reside in the former but not necessarily in the latter, as previous work has presumed. To determine whether or not inputs reside in the typical set, we propose a statistically principled, easy-to-implement test using the empirical distribution of model likelihoods. The test is model agnostic and widely applicable, only requiring that the likelihood can be computed or closely approximated. We report experiments showing that our procedure can successfully detect the out-of-distribution sets in several of the challenging cases reported by Nalisnick et al. (2019).
We propose a neural hybrid model consisting of a linear model defined on a set of features computed by a deep, invertible transformation (i.e. a normalizing flow). An attractive property of our model is that both p(features), the features' density, and p(targets | features), the predictive distribution, can be computed exactly in a single feed-forward pass. We show that our hybrid model, despite the invertibility constraints, achieves similar accuracy to purely predictive models. Yet the generative component remains a good model of the input features despite the hybrid optimization objective. This offers additional capabilities such as detection of out-of-distribution inputs and enabling semi-supervised learning. The availability of the exact joint density p(targets, features) also allows us to compute many quantities readily, making our hybrid model a useful building block for downstream applications of probabilistic deep learning.
A neural network deployed in the wild may be asked to make predictions for inputs that were drawn from a different distribution than that of the training data. A plethora of work has demonstrated that it is easy to find or synthesize inputs for which a neural network is highly confident yet wrong. Generative models are widely viewed to be robust to such mistaken confidence as modeling the density of the input features can be used to detect novel, out-of-distribution inputs. In this paper we challenge this assumption. We find that the model density from flow-based models, VAEs and PixelCNN cannot distinguish images of common objects such as dogs, trucks, and horses (i.e. CIFAR-10) from those of house numbers (i.e. SVHN), assigning a higher likelihood to the latter when the model is trained on the former. We focus our analysis on flow-based generative models in particular since they are trained and evaluated via the exact marginal likelihood. We find such behavior persists even when we restrict the flow models to constant-volume transformations. These transformations admit some theoretical analysis, and we show that the difference in likelihoods can be explained by the location and variances of the data and the model curvature, which shows that such behavior is more general and not just restricted to the pairs of datasets used in our experiments. Our results caution against using the density estimates from deep generative models to identify inputs similar to the training distribution, until their behavior on out-of-distribution inputs is better understood.
Dropout regularization of deep neural networks has been a mysterious yet effective tool to prevent overfitting. Explanations for its success range from the prevention of "co-adapted" weights to it being a form of cheap Bayesian inference. We propose a novel framework for understanding multiplicative noise in neural networks, considering continuous distributions as well as Bernoulli (i.e. dropout). We show that multiplicative noise induces structured shrinkage priors on a network's weights. We derive the equivalence through reparametrization properties of scale mixtures and not via any approximation. Given the equivalence, we then show that dropout's usual Monte Carlo training objective approximates marginal MAP estimation. We analyze this MAP objective under strong shrinkage, showing the expanded parametrization (i.e. likelihood noise) is more stable than a hierarchical representation. Lastly, we derive analogous priors for ResNets, RNNs, and CNNs and reveal their equivalent implementation as noise.
Analysis of flow cytometry data is an essential tool for clinical diagnosis of hematological and immunological conditions. Current clinical workflows rely on a manual process called gating to classify cells into their canonical types. This dependence on human annotation limits the rate, reproducibility, and complexity of flow cytometry analysis. In this paper, we propose using Mondrian processes to perform automated gating by incorporating prior information of the kind used by gating technicians. The method segments cells into types via Bayesian nonparametric trees. Examining the posterior over trees allows for interpretable visualizations and uncertainty quantification - two vital qualities for implementation in clinical practice.
Informative Bayesian priors are often difficult to elicit, and when this is the case, modelers usually turn to noninformative or objective priors. However, objective priors such as the Jeffreys and reference priors are not tractable to derive for many models of interest. We address this issue by proposing techniques for learning reference prior approximations: we select a parametric family and optimize a black-box lower bound on the reference prior objective to find the member of the family that serves as a good approximation. We experimentally demonstrate the method's effectiveness by recovering Jeffreys priors and learning the Variational Autoencoder's reference prior.