Multi-task learning requires accurate identification of the correlations between tasks. In real-world time-series, tasks are rarely perfectly temporally aligned; traditional multi-task models do not account for this and subsequent errors in correlation estimation will result in poor predictive performance and uncertainty quantification. We introduce a method that automatically accounts for temporal misalignment in a unified generative model that improves predictive performance. Our method uses Gaussian processes (GPs) to model the correlations both within and between the tasks. Building on the previous work by Kazlauskaiteet al. [2019], we include a separate monotonic warp of the input data to model temporal misalignment. In contrast to previous work, we formulate a lower bound that accounts for uncertainty in both the estimates of the warping process and the underlying functions. Also, our new take on a monotonic stochastic process, with efficient path-wise sampling for the warp functions, allows us to perform full Bayesian inference in the model rather than MAP estimates. Missing data experiments, on synthetic and real time-series, demonstrate the advantages of accounting for misalignments (vs standard unaligned method) as well as modelling the uncertainty in the warping process(vs baseline MAP alignment approach).

From the previous article, we know that Bayesian Neural Network would treat the model weights and outputs as variables. Instead of finding a set of optimal estimates, we are fitting the probability distributions for them. But the problem is "How can we know what their distributions look like?" To answer this, you have to learn what prior, posterior, and Bayes' theorem are. In the following, we will use an example for illustration.

We present a mathematical framework and computational methods to optimally design a finite number of sequential experiments. We formulate this sequential optimal experimental design (sOED) problem as a finite-horizon partially observable Markov decision process (POMDP) in a Bayesian setting and with information-theoretic utilities. It is built to accommodate continuous random variables, general non-Gaussian posteriors, and expensive nonlinear forward models. sOED then seeks an optimal design policy that incorporates elements of both feedback and lookahead, generalizing the suboptimal batch and greedy designs. We solve for the sOED policy numerically via policy gradient (PG) methods from reinforcement learning, and derive and prove the PG expression for sOED. Adopting an actor-critic approach, we parameterize the policy and value functions using deep neural networks and improve them using gradient estimates produced from simulated episodes of designs and observations. The overall PG-sOED method is validated on a linear-Gaussian benchmark, and its advantages over batch and greedy designs are demonstrated through a contaminant source inversion problem in a convection-diffusion field.

Dynamic Linear Models (DLMs) are commonly employed for time series analysis due to their versatile structure, simple recursive updating, and probabilistic forecasting. However, the options for count time series are limited: Gaussian DLMs require continuous data, while Poisson-based alternatives often lack sufficient modeling flexibility. We introduce a novel methodology for count time series by warping a Gaussian DLM. The warping function has two components: a transformation operator that provides distributional flexibility and a rounding operator that ensures the correct support for the discrete data-generating process. Importantly, we develop conjugate inference for the warped DLM, which enables analytic and recursive updates for the state space filtering and smoothing distributions. We leverage these results to produce customized and efficient computing strategies for inference and forecasting, including Monte Carlo simulation for offline analysis and an optimal particle filter for online inference. This framework unifies and extends a variety of discrete time series models and is valid for natural counts, rounded values, and multivariate observations. Simulation studies illustrate the excellent forecasting capabilities of the warped DLM. The proposed approach is applied to a multivariate time series of daily overdose counts and demonstrates both modeling and computational successes.

We study acquisition functions for active learning (AL) for text classification. The Expected Loss Reduction (ELR) method focuses on a Bayesian estimate of the reduction in classification error, recently updated with Mean Objective Cost of Uncertainty (MOCU). We convert the ELR framework to estimate the increase in (strictly proper) scores like log probability or negative mean square error, which we call Bayesian Estimate of Mean Proper Scores (BEMPS). We also prove convergence results borrowing techniques used with MOCU. In order to allow better experimentation with the new acquisition functions, we develop a complementary batch AL algorithm, which encourages diversity in the vector of expected changes in scores for unlabelled data. To allow high performance text classifiers, we combine ensembling and dynamic validation set construction on pretrained language models. Extensive experimental evaluation then explores how these different acquisition functions perform. The results show that the use of mean square error and log probability with BEMPS yields robust acquisition functions, which consistently outperform the others tested.

One's ability to learn a generative model of the world without supervision depends on the extent to which one can construct abstract knowledge representations that generalize across experiences. To this end, capturing an accurate statistical structure from observational data provides useful inductive biases that can be transferred to novel environments. Here, we tackle the problem of learning to control dynamical systems by applying Bayesian nonparametric methods, which is applied to solve visual servoing tasks. This is accomplished by first learning a state space representation, then inferring environmental dynamics and improving the policies through imagined future trajectories. Bayesian nonparametric models provide automatic model adaptation, which not only combats underfitting and overfitting, but also allows the model's unbounded dimension to be both flexible and computationally tractable. By employing Gaussian processes to discover latent world dynamics, we mitigate common data efficiency issues observed in reinforcement learning and avoid introducing explicit model bias by describing the system's dynamics. Our algorithm jointly learns a world model and policy by optimizing a variational lower bound of a log-likelihood with respect to the expected free energy minimization objective function. Finally, we compare the performance of our model with the state-of-the-art alternatives for continuous control tasks in simulated environments.

A Bayesian Network (BN) is a probabilistic graphical model consisting of a directed acyclic graph (DAG), where each node is a random variable represented as a function of its parents. We present a novel approach capable of learning the global DAG structure of a BN and modelling linear and non-linear local relationships between variables. We achieve this by a combination of feature selection to reduce the search space for local relationships, and extending the widely used score-and-search approach to support modelling relationships between variables as Multivariate Adaptive Regression Splines (MARS). MARS are polynomial regression models represented as piecewise spline functions - this lets us model non-linear relationships without the risk of overfitting that a single polynomial regression model would bring. The combination allows us to learn relationships in all bnlearn benchmark instances within minutes and enables us to scale to networks of over a thousand nodes

There are proposals that extend the classical generalized additive models (GAMs) to accommodate high-dimensional data ($p>>n$) using group sparse regularization. However, the sparse regularization may induce excess shrinkage when estimating smoothing functions, damaging predictive performance. Moreover, most of these GAMs consider an "all-in-all-out" approach for functional selection, rendering them difficult to answer if nonlinear effects are necessary. While some Bayesian models can address these shortcomings, using Markov chain Monte Carlo algorithms for model fitting creates a new challenge, scalability. Hence, we propose Bayesian hierarchical generalized additive models as a solution: we consider the smoothing penalty for proper shrinkage of curve interpolation and separation of smoothing function linear and nonlinear spaces. A novel spike-and-slab spline prior is proposed to select components of smoothing functions. Two scalable and deterministic algorithms, EM-Coordinate Descent and EM-Iterative Weighted Least Squares, are developed for different utilities. Simulation studies and metabolomics data analyses demonstrate improved predictive or computational performance against state-of-the-art models, mgcv, COSSO and sparse Bayesian GAM. The software implementation of the proposed models is freely available via an R package BHAM.

In recent years, local differential privacy (LDP) has emerged as a technique of choice for privacy-preserving data collection in several scenarios when the aggregator is not trustworthy. LDP provides client-side privacy by adding noise at the user's end. Thus, clients need not rely on the trustworthiness of the aggregator. In this work, we provide a noise-aware probabilistic modeling framework, which allows Bayesian inference to take into account the noise added for privacy under LDP, conditioned on locally perturbed observations. Stronger privacy protection (compared to the central model) provided by LDP protocols comes at a much harsher privacy-utility trade-off. Our framework tackles several computational and statistical challenges posed by LDP for accurate uncertainty quantification under Bayesian settings. We demonstrate the efficacy of our framework in parameter estimation for univariate and multi-variate distributions as well as logistic and linear regression.

Aggregated predictors are obtained by making a set of basic predictors vote according to some weights, that is, to some probability distribution. Randomized predictors are obtained by sampling in a set of basic predictors, according to some prescribed probability distribution. Thus, aggregated and randomized predictors have in common that they are not defined by a minimization problem, but by a probability distribution on the set of predictors. In statistical learning theory, there is a set of tools designed to understand the generalization ability of such procedures: PAC-Bayesian or PAC-Bayes bounds. Since the original PAC-Bayes bounds of D. McAllester, these tools have been considerably improved in many directions (we will for example describe a simplified version of the localization technique of O. Catoni that was missed by the community, and later rediscovered as "mutual information bounds"). Very recently, PAC-Bayes bounds received a considerable attention: for example there was workshop on PAC-Bayes at NIPS 2017, "(Almost) 50 Shades of Bayesian Learning: PAC-Bayesian trends and insights", organized by B. Guedj, F. Bach and P. Germain. One of the reason of this recent success is the successful application of these bounds to neural networks by G. Dziugaite and D. Roy. An elementary introduction to PAC-Bayes theory is still missing. This is an attempt to provide such an introduction.