The concept of mathematical modeling is widespread across almost all of the fields of contemporary science and engineering. Because of the existing necessity of predictions the behavior of natural phenomena, the researchers develop more and more complex models. However, despite their ability to better forecasting, the problem of an appropriate fitting ground truth data to those, high-dimensional and nonlinear models seems to be inevitable. In order to deal with this demanding problem the entire discipline of data assimilation has been developed. Basing on the Human and Nature Dynamics (HANDY) model, we have presented a detailed and comprehensive comparison of Approximate Bayesian Computation (classic data assimilation method) and a novelty approach of Supermodeling. Furthermore, with the usage of Sensitivity Analysis, we have proposed the methodology to reduce the number of coupling coefficients between submodels and as a consequence to increase the speed of the Supermodel converging. In addition, we have demonstrated that usage of Approximate Bayesian Computation method with the knowledge about parameters' sensitivities could result with satisfactory estimation of the initial parameters. However, we have also presented the mentioned methodology as unable to achieve similar predictions to Approximate Bayesian Computation. Finally, we have proved that Supermodeling with synchronization via the most sensitive variable could effect with the better forecasting for chaotic as well as more stable systems than the Approximate Bayesian Computation. What is more, we have proposed the adequate methodologies.

We recapitulate the Bayesian formulation of neural network based classifiers and show that, while sampling from the posterior does indeed lead to better generalisation than is obtained by standard optimisation of the cost function, even better performance can in general be achieved by sampling finite temperature ($T$) distributions derived from the posterior. Taking the example of two different deep (3 hidden layers) classifiers for MNIST data, we find quite different $T$ values to be appropriate in each case. In particular, for a typical neural network classifier a clear minimum of the test error is observed at $T>0$. This suggests an early stopping criterion for full batch simulated annealing: cool until the average validation error starts to increase, then revert to the parameters with the lowest validation error. As $T$ is increased classifiers transition from accurate classifiers to classifiers that have higher training error than assigning equal probability to each class. Efficient studies of these temperature-induced effects are enabled using a replica-exchange Hamiltonian Monte Carlo simulation technique. Finally, we show how thermodynamic integration can be used to perform model selection for deep neural networks. Similar to the Laplace approximation, this approach assumes that the posterior is dominated by a single mode. Crucially, however, no assumption is made about the shape of that mode and it is not required to precisely compute and invert the Hessian.

We introduce Bayesian least-squares policy iteration (BLSPI), an off-policy, model-free, policy iteration algorithm that uses the Bayesian least-squares temporal-difference (BLSTD) learning algorithm to evaluate policies. An online variant of BLSPI has been also proposed, called randomised BLSPI (RBLSPI), that improves its policy based on an incomplete policy evaluation step. In online setting, the exploration-exploitation dilemma should be addressed as we try to discover the optimal policy by using samples collected by ourselves. RBLSPI exploits the advantage of BLSTD to quantify our uncertainty about the value function. Inspired by Thompson sampling, RBLSPI first samples a value function from a posterior distribution over value functions, and then selects actions based on the sampled value function. The effectiveness and the exploration abilities of RBLSPI are demonstrated experimentally in several environments.

Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard tools for estimating graphical models under sparsity conditions. Extensions of these methods to more complex settings where data are contaminated with additive or multiplicative noise have been developed in recent years. In these settings, however, the relative performance of different methods is not well understood and algorithmic gaps still exist. In particular, in high-dimensional settings these methods require using non-positive semidefinite matrices as inputs, presenting novel optimization challenges. We develop an alternating direction method of multipliers (ADMM) algorithm for these problems, providing a feasible algorithm to estimate precision matrices with indefinite input and potentially nonconvex penalties. We compare this method with existing alternative solutions and empirically characterize the tradeoffs between them. Finally, we use this method to explore the networks among US senators estimated from voting records data.

Stochastic block models (SBMs) have been playing an important role in modeling clusters or community structures of network data. But, it is incapable of handling several complex features ubiquitously exhibited in real-world networks, one of which is the power-law degree characteristic. To this end, we propose a new variant of SBM, termed power-law degree SBM (PLD-SBM), by introducing degree decay variables to explicitly encode the varying degree distribution over all nodes. With an exponential prior, it is proved that PLD-SBM approximately preserves the scale-free feature in real networks. In addition, from the inference of variational E-Step, PLD-SBM is indeed to correct the bias inherited in SBM with the introduced degree decay factors. Furthermore, experiments conducted on both synthetic networks and two real-world datasets including Adolescent Health Data and the political blogs network verify the effectiveness of the proposed model in terms of cluster prediction accuracies.

Unstructured data from diverse sources, such as social media and aerial imagery, can provide valuable up-to-date information for intelligent situation assessment. Mining these different information sources could bring major benefits to applications such as situation awareness in disaster zones and mapping the spread of diseases. Such applications depend on classifying the situation across a region of interest, which can be depicted as a spatial "heatmap". Annotating unstructured data using crowdsourcing or automated classifiers produces individual classifications at sparse locations that typically contain many errors. We propose a novel Bayesian approach that models the relevance, error rates and bias of each information source, enabling us to learn a spatial Gaussian Process classifier by aggregating data from multiple sources with varying reliability and relevance. Our method does not require gold-labelled data and can make predictions at any location in an area of interest given only sparse observations. We show empirically that our approach can handle noisy and biased data sources, and that simultaneously inferring reliability and transferring information between neighbouring reports leads to more accurate predictions. We demonstrate our method on two real-world problems from disaster response, showing how our approach reduces the amount of crowdsourced data required and can be used to generate valuable heatmap visualisations from SMS messages and satellite images.

The field of image reconstruction has undergone four waves of methods. The first wave was analytical methods, such as filtered back-projection (FBP) for X-ray computed tomography (CT) and the inverse Fourier transform for magnetic resonance imaging (MRI), based on simple mathematical models for the imaging systems. These methods are typically fast, but have suboptimal properties such as poor resolution-noise trade-off for CT. The second wave was iterative reconstruction methods based on more complete models for the imaging system physics and, where appropriate, models for the sensor statistics. These iterative methods improved image quality by reducing noise and artifacts. The FDA-approved methods among these have been based on relatively simple regularization models. The third wave of methods has been designed to accommodate modified data acquisition methods, such as reduced sampling in MRI and CT to reduce scan time or radiation dose. These methods typically involve mathematical image models involving assumptions such as sparsity or low-rank. The fourth wave of methods replaces mathematically designed models of signals and processes with data-driven or adaptive models inspired by the field of machine learning. This paper reviews the progress in image reconstruction methods with focus on the two most recent trends: methods based on sparsity or low-rank models, and data-driven methods based on machine learning techniques.

This paper introduces a generalized representation of Bayesian inference. It is derived axiomatically, recovering existing Bayesian methods as special cases. We use it to prove that variational inference (VI) based on the Kullback-Leibler Divergence with a variational family Q produces the uniquely optimal Q-constrained approximation to the exact Bayesian inference problem. Surprisingly, this implies that standard VI dominates any other Q-constrained approximation to the exact Bayesian inference problem. This means that alternative Q-constrained approximations such as VI targeted at minimizing other divergences and Expectation Propagation can produce better posteriors than VI only by implicitly targeting more appropriate Bayesian inference problems. Inspired by this, we introduce Generalized Variational Inference (GVI), a modular approach for instead solving such alternative inference problems explicitly. We explore some applications of GVI, including robustness and better marginals. Lastly, we derive black box GVI and apply it to Bayesian Neural Networks as well as Deep Gaussian Processes, where GVI comprehensively outperforms competing methods.

Despite some empirical success at correcting exposure bias in machine translation, scheduled sampling algorithms suffer from a major drawback: they incorrectly assume that words in the reference translations and in sampled sequences are aligned at each time step. Our new differentiable sampling algorithm addresses this issue by optimizing the probability that the reference can be aligned with the sampled output, based on a soft alignment predicted by the model itself. As a result, the output distribution at each time step is evaluated with respect to the whole predicted sequence. Experiments on IWSLT translation tasks show that our approach improves BLEU compared to maximum likelihood and scheduled sampling baselines. In addition, our approach is simpler to train with no need for sampling schedule and yields models that achieve larger improvements with smaller beam sizes.

Despite their unprecedented performance in various domains, utilization of Deep Neural Networks (DNNs) in safety-critical environments is severely limited in the presence of even small adversarial perturbations. The present work develops a randomized approach to detecting such perturbations based on minimum uncertainty metrics that rely on sampling at the hidden layers during the DNN inference stage. The sampling probabilities are designed for effective detection of the adversarially corrupted inputs. Being modular, the novel detector of adversaries can be conveniently employed by any pre-trained DNN at no extra training overhead. Selecting which units to sample per hidden layer entails quantifying the amount of DNN output uncertainty from the viewpoint of Bayesian neural networks, where the overall uncertainty is expressed in terms of its layer-wise components - what also promotes scalability. Sampling probabilities are then sought by minimizing uncertainty measures layer-by-layer, leading to a novel convex optimization problem that admits an exact solver with superlinear convergence rate. By simplifying the objective function, low-complexity approximate solvers are also developed. In addition to valuable insights, these approximations link the novel approach with state-of-the-art randomized adversarial detectors. The effectiveness of the novel detectors in the context of competing alternatives is highlighted through extensive tests for various types of adversarial attacks with variable levels of strength.