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Operational earthquake forecasting for risk management and communication during seismic sequences depends on our ability to select an optimal forecasting model. To do this, we need to compare the performance of competing models with each other in prospective forecasting mode, and to rank their performance using a fair, reproducible and reliable method. The Collaboratory for the Study of Earthquake Predictability (CSEP) conducts such prospective earthquake forecasting experiments around the globe. One metric that has been proposed to rank competing models is the Parimutuel Gambling score, which has the advantage of allowing alarm-based (categorical) forecasts to be compared with probabilistic ones. Here we examine the suitability of this score for ranking competing earthquake forecasts. First, we prove analytically that this score is in general improper, meaning that, on average, it does not prefer the model that generated the data. Even in the special case where it is proper, we show it can still be used in an improper way. Then, we compare its performance with two commonly-used proper scores (the Brier and logarithmic scores), taking into account the uncertainty around the observed average score. We estimate the confidence intervals for the expected score difference which allows us to define if and when a model can be preferred. Our findings suggest the Parimutuel Gambling score should not be used to distinguishing between multiple competing forecasts. They also enable a more rigorous approach to distinguish between the predictive skills of candidate forecasts in addition to their rankings.

Insights into complex, high-dimensional data can be obtained by discovering features of the data that match or do not match a model of interest. To formalize this task, we introduce the "data selection" problem: finding a lower-dimensional statistic - such as a subset of variables - that is well fit by a given parametric model of interest. A fully Bayesian approach to data selection would be to parametrically model the value of the statistic, nonparametrically model the remaining "background" components of the data, and perform standard Bayesian model selection for the choice of statistic. However, fitting a nonparametric model to high-dimensional data tends to be highly inefficient, statistically and computationally. We propose a novel score for performing both data selection and model selection, the "Stein volume criterion", that takes the form of a generalized marginal likelihood with a kernelized Stein discrepancy in place of the Kullback-Leibler divergence. The Stein volume criterion does not require one to fit or even specify a nonparametric background model, making it straightforward to compute - in many cases it is as simple as fitting the parametric model of interest with an alternative objective function. We prove that the Stein volume criterion is consistent for both data selection and model selection, and we establish consistency and asymptotic normality (Bernstein-von Mises) of the corresponding generalized posterior on parameters. We validate our method in simulation and apply it to the analysis of single-cell RNA sequencing datasets using probabilistic principal components analysis and a spin glass model of gene regulation.

The Intergovernmental Panel on Climate Change proposes different mitigation strategies to achieve the net emissions reductions that would be required to follow a pathway that limits global warming to 1.5{\deg}C with no or limited overshoot. The transition towards a carbon-free society goes through an inevitable increase in the share of renewable generation in the energy mix and a drastic decrease in the total consumption of fossil fuels. Therefore, this thesis studies the integration of renewables in power systems by investigating forecasting and decision-making tools. Indeed, in contrast to conventional power plants, renewable energy is subject to uncertainty. Most of the generation technologies based on renewable sources are non-dispatchable, and their production is stochastic and complex to predict in advance. A high share of renewables is challenging for power systems that have been designed and sized for dispatchable units. In this context, probabilistic forecasts, which aim at modeling the distribution of all possible future realizations, have become a vital tool to equip decision-makers, hopefully leading to better decisions in energy applications. This thesis focuses on two main research questions: (1) How to produce reliable probabilistic renewable generation forecasts, consumption, and electricity prices? (2) How to make decisions with uncertainty using probabilistic forecasts? The thesis perimeter is the energy management of "small" systems such as microgrids at a residential scale on a day-ahead basis. It is divided into two main parts to propose directions to address both research questions (1) a forecasting part; (2) a planning and control part.

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.

Many statistical problems in causal inference involve a probability distribution other than the one from which data are actually observed; as an additional complication, the object of interest is often a marginal quantity of this other probability distribution. This creates many practical complications for statistical inference, even where the problem is non-parametrically identified. Na\"ive attempts to specify a model parametrically can lead to unwanted consequences such as incompatible parametric assumptions or the so-called `g-null paradox'. As a consequence it is difficult to perform likelihood-based inference, or even to simulate from the model in a general way. We introduce the `frugal parameterization', which places the causal effect of interest at its centre, and then build the rest of the model around it. We do this in a way that provides a recipe for constructing a regular, non-redundant parameterization using causal quantities of interest. In the case of discrete variables we use odds ratios to complete the parameterization, while in the continuous case we use copulas. Our methods allow us to construct and simulate from models with parametrically specified causal distributions, and fit them using likelihood-based methods, including fully Bayesian approaches. Models we can fit and simulate from exactly include marginal structural models and structural nested models. Our proposal includes parameterizations for the average causal effect and effect of treatment on the treated, as well as other causal quantities of interest. Our results will allow practitioners to assess their methods against the best possible estimators for correctly specified models, in a way which has previously been impossible.

We initiate the study of incentive-compatible forecasting competitions in which multiple forecasters make predictions about one or more events and compete for a single prize. We have two objectives: (1) to incentivize forecasters to report truthfully and (2) to award the prize to the most accurate forecaster. Proper scoring rules incentivize truthful reporting if all forecasters are paid according to their scores. However, incentives become distorted if only the best-scoring forecaster wins a prize, since forecasters can often increase their probability of having the highest score by reporting more extreme beliefs. In this paper, we introduce two novel forecasting competition mechanisms. Our first mechanism is incentive compatible and guaranteed to select the most accurate forecaster with probability higher than any other forecaster. Moreover, we show that in the standard single-event, two-forecaster setting and under mild technical conditions, no other incentive-compatible mechanism selects the most accurate forecaster with higher probability. Our second mechanism is incentive compatible when forecasters' beliefs are such that information about one event does not lead to belief updates on other events, and it selects the best forecaster with probability approaching 1 as the number of events grows. Our notion of incentive compatibility is more general than previous definitions of dominant strategy incentive compatibility in that it allows for reports to be correlated with the event outcomes. Moreover, our mechanisms are easy to implement and can be generalized to the related problems of outputting a ranking over forecasters and hiring a forecaster with high accuracy on future events.

Spatio-temporal forecasting has numerous applications in analyzing wireless, traffic, and financial networks. Many classical statistical models often fall short in handling the complexity and high non-linearity present in time-series data. Recent advances in deep learning allow for better modelling of spatial and temporal dependencies. While most of these models focus on obtaining accurate point forecasts, they do not characterize the prediction uncertainty. In this work, we consider the time-series data as a random realization from a nonlinear state-space model and target Bayesian inference of the hidden states for probabilistic forecasting. We use particle flow as the tool for approximating the posterior distribution of the states, as it is shown to be highly effective in complex, high-dimensional settings. Thorough experimentation on several real world time-series datasets demonstrates that our approach provides better characterization of uncertainty while maintaining comparable accuracy to the state-of-the art point forecasting methods.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.

Multivariate time series forecasting is extensively studied throughout the years with ubiquitous applications in areas such as finance, traffic, environment, etc. Still, concerns have been raised on traditional methods for incapable of modeling complex patterns or dependencies lying in real word data. To address such concerns, various deep learning models, mainly Recurrent Neural Network (RNN) based methods, are proposed. Nevertheless, capturing extremely long-term patterns while effectively incorporating information from other variables remains a challenge for time-series forecasting. Furthermore, lack-of-explainability remains one serious drawback for deep neural network models. Inspired by Memory Network proposed for solving the question-answering task, we propose a deep learning based model named Memory Time-series network (MTNet) for time series forecasting. MTNet consists of a large memory component, three separate encoders, and an autoregressive component to train jointly. Additionally, the attention mechanism designed enable MTNet to be highly interpretable. We can easily tell which part of the historic data is referenced the most.