The Viterbi process is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. The existence of such a process suggests that approximate estimation using optimization algorithms which process data segments in parallel may be accurate. For models on state-space $\mathbb{R}^{d}$ satisfying a new "decay-convexity" condition, we approach the existence of the Viterbi process through fixed points of ordinary differential equations in a certain infinite dimensional Hilbert space. Quantitative bounds on the distance to the Viterbi process show that approximate estimation via parallelization can indeed be accurate and scaleable to high-dimensional problems because the rate of convergence to the Viterbi process does not necessarily depend on $d$.

Learning robot objective functions from human input has become increasingly important, but state-of-the-art techniques assume that the human's desired objective lies within the robot's hypothesis space. When this is not true, even methods that keep track of uncertainty over the objective fail because they reason about which hypothesis might be correct, and not whether any of the hypotheses are correct. We focus specifically on learning from physical human corrections during the robot's task execution, where not having a rich enough hypothesis space leads to the robot updating its objective in ways that the person did not actually intend. We observe that such corrections appear irrelevant to the robot, because they are not the best way of achieving any of the candidate objectives. Instead of naively trusting and learning from every human interaction, we propose robots learn conservatively by reasoning in real time about how relevant the human's correction is for the robot's hypothesis space. We test our inference method in an experiment with human interaction data, and demonstrate that this alleviates unintended learning in an in-person user study with a 7DoF robot manipulator.

Compared to the classic latent factor models [14], latent feature models have two main benefits: (1) interpretablity: the binary codes directly reveal whether certain features exist in the data, thus provide more interpretable latent profiles [25]; (2) scalability: compared with real-valued codes, binary codes require fewer bits to store, thereby cutting down the memory footprint, making it easier to deploy into memory constrained environments such as mobile devices. Tensor latent feature models generalize traditional matrix latent feature models to represent high-order correlation structures in the data. For example, in spatiotemporal recommender systems, the observations are user activities over different locations and time. We want to learn the latent features and codes that correspond to user, space and time simultaneously without assuming conditional independence of these three dimensions. In this case, we can first represent such data as a high-order tensor and assign binary codes encoding presence or absence of rows for individual modes of the tensor. These codes can then help us answer the "when" and "where" questions regarding the learned user preferences. Besides the non-convex formulation of the maximum likelihood estimation (MLE) objective, learning latent feature models is further complicated by the combinatorial nature of the codes.

Bayesian neural networks (BNNs) hold great promise as a flexible and principled solution to deal with uncertainty when learning from finite data. Among approaches to realize probabilistic inference in deep neural networks, variational Bayes (VB) is theoretically grounded, generally applicable, and computationally efficient. With wide recognition of potential advantages, why is it that variational Bayes has seen very limited practical use for BNNs in real applications? We argue that variational inference in neural networks is fragile: successful implementations require careful initialization and tuning of prior variances, as well as controlling the variance of Monte Carlo gradient estimates. We fix VB and turn it into a robust inference tool for Bayesian neural networks. We achieve this with two innovations: first, we introduce a novel deterministic method to approximate moments in neural networks, eliminating gradient variance; second, we introduce a hierarchical prior for parameters and a novel empirical Bayes procedure for automatically selecting prior variances. Combining these two innovations, the resulting method is highly efficient and robust. On the application of heteroscedastic regression we demonstrate strong predictive performance over alternative approaches.

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.

We formulate stochastic gradient descent (SGD) as a Bayesian filtering problem. Inference in the Bayesian setting naturally gives rise to BRMSprop and BAdam: Bayesian variants of RMSprop and Adam. Remarkably, the Bayesian approach recovers many features of state-of-the-art adaptive SGD methods, including amoungst others root-mean-square normalization, Nesterov acceleration and AdamW. As such, the Bayesian approach provides one explanation for the empirical effectiveness of state-of-the-art adaptive SGD algorithms. Empirically comparing BRMSprop and BAdam with naive RMSprop and Adam on MNIST, we find that Bayesian methods have the potential to considerably reduce test loss and classification error.

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consists of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a Bayesian state estimation problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers, which were formulated in terms of generating synthetic measurements of the vector field, come out as specific approximations. We derive novel solvers, both Gaussian and non-Gaussian, from the Bayesian state estimation problem posed in this paper and compare them with other probabilistic solvers in illustrative experiments.

Many scientifically well-motivated statistical models in natural, engineering and environmental sciences are specified through a generative process, but in some cases it may not be possible to write down a likelihood for these models analytically. Approximate Bayesian computation (ABC) methods, which allow Bayesian inference in these situations, are typically computationally intensive. Recently, computationally attractive empirical likelihood based ABC methods have been suggested in the literature. These methods heavily rely on the availability of a set of suitable analytically tractable estimating equations. We propose an easy-to-use empirical likelihood ABC method, where the only inputs required are a choice of summary statistic, it's observed value, and the ability to simulate summary statistics for any parameter value under the model. It is shown that the posterior obtained using the proposed method is consistent, and its performance is explored using various examples.

The Normalizing Flow (NF) models a general probability density by estimating an invertible transformation applied on samples drawn from a known distribution. We introduce a new type of NF, called Deep Diffeomorphic Normalizing Flow (DDNF). A diffeomorphic flow is an invertible function where both the function and its inverse are smooth. We construct the flow using an ordinary differential equation (ODE) governed by a time-varying smooth vector field. We use a neural network to parametrize the smooth vector field and a recursive neural network (RNN) for approximating the solution of the ODE. Each cell in the RNN is a residual network implementing one Euler integration step. The architecture of our flow enables efficient likelihood evaluation, straightforward flow inversion, and results in highly flexible density estimation. An end-to-end trained DDNF achieves competitive results with state-of-the-art methods on a suite of density estimation and variational inference tasks. Finally, our method brings concepts from Riemannian geometry that, we believe, can open a new research direction for neural density estimation.

We propose deep convolutional Gaussian processes, a deep Gaussian process architecture with convolutional structure. The model is a principled Bayesian framework for detecting hierarchical combinations of local features for image classification. We demonstrate greatly improved image classification performance compared to current Gaussian process approaches on the MNIST and CIFAR-10 datasets. In particular, we improve CIFAR-10 accuracy by over 10 percentage points.