A Robust Markov Decision Process (RMDP) is a sequential decision making model that accounts for uncertainty in the parameters of dynamic systems. This uncertainty introduces difficulties in learning an optimal policy, especially for environments with large state spaces. We propose two algorithms, RTD-DQN and Deep-RoK, for solving large-scale RMDPs using nonlinear approximation schemes such as deep neural networks. The RTD-DQN algorithm incorporates the robust Bellman temporal difference error into a robust loss function, yielding robust policies for the agent. The Deep-RoK algorithm is a robust Bayesian method, based on the Extended Kalman Filter (EKF), that accounts for both the uncertainty in the weights of the approximated value function and the uncertainty in the transition probabilities, improving the robustness of the agent. We provide theoretical results for our approach and test the proposed algorithms on a continuous state domain.

We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the coefficients (e.g. material parameters) exhibit random, fine scale variability. The direct solution to such problems requires grids that are small enough to resolve this fine scale variability which unavoidably requires the repeated solution of very large systems of algebraic equations. We establish a physically inspired, data-driven coarse-grained model which learns a low- dimensional set of microstructural features that are predictive of the fine-grained model (FG) response. Once learned, those features provide a sharp distribution over the coarse scale effec- tive coefficients of the PDE that are most suitable for prediction of the fine scale model output. This ultimately allows to replace the computationally expensive FG by a generative proba- bilistic model based on evaluating the much cheaper CG several times. Sparsity enforcing pri- ors further increase predictive efficiency and reveal microstructural features that are important in predicting the FG response. Moreover, the model yields probabilistic rather than single-point predictions, which enables the quantification of the unavoidable epistemic uncertainty that is present due to the information loss that occurs during the coarse-graining process.

To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.

The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propagated by the chain rule through the graph are low variance unbiased estimators of the gradients of the expected loss. While many continuous random variables have such reparameterizations, discrete random variables lack useful reparameterizations due to the discontinuous nature of discrete states. In this work we introduce Concrete random variables---continuous relaxations of discrete random variables. The Concrete distribution is a new family of distributions with closed form densities and a simple reparameterization. Whenever a discrete stochastic node of a computation graph can be refactored into a one-hot bit representation that is treated continuously, Concrete stochastic nodes can be used with automatic differentiation to produce low-variance biased gradients of objectives (including objectives that depend on the log-probability of latent stochastic nodes) on the corresponding discrete graph. We demonstrate the effectiveness of Concrete relaxations on density estimation and structured prediction tasks using neural networks.

Yield curve forecasting is an important problem in finance. In this work we explore the use of Gaussian Processes in conjunction with a dynamic modeling strategy, much like the Kalman Filter, to model the yield curve. Gaussian Processes have been successfully applied to model functional data in a variety of applications. A Gaussian Process is used to model the yield curve. The hyper-parameters of the Gaussian Process model are updated as the algorithm receives yield curve data. Yield curve data is typically available as a time series with a frequency of one day. We compare existing methods to forecast the yield curve with the proposed method. The results of this study showed that while a competing method (a multivariate time series method) performed well in forecasting the yields at the short term structure region of the yield curve, Gaussian Processes perform well in the medium and long term structure regions of the yield curve. Accuracy in the long term structure region of the yield curve has important practical implications. The Gaussian Process framework yields uncertainty and probability estimates directly in contrast to other competing methods. Analysts are frequently interested in this information. In this study the proposed method has been applied to yield curve forecasting, however it can be applied to model high frequency time series data or data streams in other domains.

This paper studies a new Bayesian algorithm for the joint reconstruction and classification of reflectance confocal microscopy (RCM) images, with application to the identification of human skin lentigo. The proposed Bayesian approach takes advantage of the distribution of the multiplicative speckle noise affecting the true reflectivity of these images and of appropriate priors for the unknown model parameters. A Markov chain Monte Carlo (MCMC) algorithm is proposed to jointly estimate the model parameters and the image of true reflectivity while classifying images according to the distribution of their reflectivity. Precisely, a Metropolis-within-Gibbs sampler is investigated to sample the posterior distribution of the Bayesian model associated with RCM images and to build estimators of its parameters, including labels indicating the class of each RCM image. The resulting algorithm is applied to synthetic data and to real images from a clinical study containing healthy and lentigo patients. The lentigo is a hyperplasia that affects the skin.

Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis of data streams and to the one-pass analysis of massive data sets. Applications include network traffic and latency analysis, real time fraud detection and high frequency trading. We introduce new techniques for online quantile estimation based on Hermite series estimators in the settings of static quantile estimation and dynamic quantile estimation. In the static quantile estimation setting we apply the existing Gauss-Hermite expansion in a novel manner. In particular, we exploit the fact that Gauss-Hermite coefficients can be updated in a sequential manner. To treat dynamic quantile estimation we introduce a novel expansion with an exponentially weighted estimator for the Gauss-Hermite coefficients which we term the Exponentially Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing sequential quantile estimation algorithms in that they allow arbitrary quantiles (as opposed to pre-specified quantiles) to be estimated at any point in time. In doing so we provide a solution to online distribution function and online quantile function estimation on data streams. In particular we derive an analytical expression for the CDF and prove consistency results for the CDF under certain conditions. In addition we analyse the associated quantile estimator. Simulation studies and tests on real data reveal the Gauss-Hermite based algorithms to be competitive with a leading existing algorithm.

We harness the power of Bayesian emulation techniques, designed to aid the analysis of complex computer models, to examine the structure of complex Bayesian analyses themselves. These techniques facilitate robust Bayesian analyses and/or sensitivity analyses of complex problems, and hence allow global exploration of the impacts of choices made in both the likelihood and prior specification. We show how previously intractable problems in robustness studies can be overcome using emulation techniques, and how these methods allow other scientists to quickly extract approximations to posterior results corresponding to their own particular subjective specification. The utility and flexibility of our method is demonstrated on a reanalysis of a real application where Bayesian methods were employed to capture beliefs about river flow. We discuss the obvious extensions and directions of future research that such an approach opens up.

Increasingly complex generative models are being used across disciplines as they allow for realistic characterization of data, but a common difficulty with them is the prohibitively large computational cost to evaluate the likelihood function and thus to perform likelihood-based statistical inference. A likelihood-free inference framework has emerged where the parameters are identified by finding values that yield simulated data resembling the observed data. While widely applicable, a major difficulty in this framework is how to measure the discrepancy between the simulated and observed data. Transforming the original problem into a problem of classifying the data into simulated versus observed, we find that classification accuracy can be used to assess the discrepancy. The complete arsenal of classification methods becomes thereby available for inference of intractable generative models. We validate our approach using theory and simulations for both point estimation and Bayesian inference, and demonstrate its use on real data by inferring an individual-based epidemiological model for bacterial infections in child care centers.