The is no other model or hypothesis verification tool in Bayesian statistics that is as widely used as the Bayes factor. We focus on generative models that are likelihood-free and, therefore, render the computation of Bayes factors (marginal likelihood ratios) far from obvious. We propose a deep learning estimator of the Bayes factor based on simulated data from two competing models using the likelihood ratio trick. This estimator is devoid of summary statistics and obviates some of the difficulties with ABC model choice. We establish sufficient conditions for consistency of our Deep Bayes Factor estimator as well as its consistency as a model selection tool. We investigate the performance of our estimator on various examples using a wide range of quality metrics related to estimation and model decision accuracy. After training, our deep learning approach enables rapid evaluations of the Bayes factor estimator at any fictional data arriving from either hypothesized model, not just the observed data $Y_0$. This allows us to inspect entire Bayes factor distributions under the two models and to quantify the relative location of the Bayes factor evaluated at $Y_0$ in light of these distributions. Such tail area evaluations are not possible for Bayes factor estimators tailored to $Y_0$. We find the performance of our Deep Bayes Factors competitive with existing MCMC techniques that require the knowledge of the likelihood function. We also consider variants for posterior or intrinsic Bayes factors estimation. We demonstrate the usefulness of our approach on a relatively high-dimensional real data example about determining cognitive biases.

Kriging is a technique for multivariate interpolation of arbitrarily scattered data. It is originated from some mining-related applications, which is developed into the field of geostatistics by the pioneering work of Matheron (1963). It has now become a prominent technique for function approximation and uncertainty quantification in spatial statistics (Cressie, 2015), computer experiments (Santner et al., 2003), and machine learning (Rasmussen and Williams, 2006). Kriging can be briefly explained as follows.

Simulations of complex phenomena are crucial in the natural sciences and engineering for different scenarios, e.g., for gaining system understanding, prediction of future scenarios, risk assessment, or system design. However, often they are based on complex ordinary differential equations or partial differential equations which may not have closed-form solutions and may have to be solved using expensive numerical methods. To overcome computational overhead, the field of surrogate models (Zhu and Zabaras, 2018; Gramacy, 2020; Lavin et al., 2021) has emerged which provide fast approximations of computationally expensive simulation. Examples are polynomial chaos expansion (Wiener, 1938; Sudret, 2008; Oladyshkin and Nowak, 2012; Bürkner et al., 2023), Gaussian processes (Rasmussen and Williams, 2005) or neural networks (Goodfellow et al., 2016). Recently there has been a great interest in applying surrogate models in relevant areas, for example in hydrology (Mohammadi et al., 2018; Tarakanov and Elsheikh, 2019; Zhang et al., 2020), in fluid dynamics (Meyer et al., 2021), in climate prediction (Kuehnert et al., 2022), or in systems biology (Renardy et al., 2018; Alden et al., 2020).

In this work, we deepen on the use of normalizing flows for causal reasoning. Specifically, we first leverage recent results on non-linear ICA to show that causal models are identifiable from observational data given a causal ordering, and thus can be recovered using autoregressive normalizing flows (NFs). Second, we analyze different design and learning choices for causal normalizing flows to capture the underlying causal data-generating process. Third, we describe how to implement the do-operator in causal NFs, and thus, how to answer interventional and counterfactual questions. Finally, in our experiments, we validate our design and training choices through a comprehensive ablation study; compare causal NFs to other approaches for approximating causal models; and empirically demonstrate that causal NFs can be used to address real-world problems, where the presence of mixed discrete-continuous data and partial knowledge on the causal graph is the norm. The code for this work can be found at https://github.com/psanch21/causal-flows.

The integration of data and knowledge from several sources is known as data fusion. When data is only available in a distributed fashion or when different sensors are used to infer a quantity of interest, data fusion becomes essential. In Bayesian settings, a priori information of the unknown quantities is available and, possibly, present among the different distributed estimators. When the local estimates are fused, the prior knowledge used to construct several local posteriors might be overused unless the fusion node accounts for this and corrects it. In this paper, we analyze the effects of shared priors in Bayesian data fusion contexts. Depending on different common fusion rules, our analysis helps to understand the performance behavior as a function of the number of collaborative agents and as a consequence of different types of priors. The analysis is performed by using two divergences which are common in Bayesian inference, and the generality of the results allows to analyze very generic distributions. These theoretical results are corroborated through experiments in a variety of estimation and classification problems, including linear and nonlinear models, and federated learning schemes.

This paper studies the joint community detection and phase synchronization problem on the \textit{stochastic block model with relative phase}, where each node is associated with an unknown phase angle. This problem, with a variety of real-world applications, aims to recover the cluster structure and associated phase angles simultaneously. We show this problem exhibits a \textit{``multi-frequency''} structure by closely examining its maximum likelihood estimation (MLE) formulation, whereas existing methods are not originated from this perspective. To this end, two simple yet efficient algorithms that leverage the MLE formulation and benefit from the information across multiple frequencies are proposed. The former is a spectral method based on the novel multi-frequency column-pivoted QR factorization. The factorization applied to the top eigenvectors of the observation matrix provides key information about the cluster structure and associated phase angles. The second approach is an iterative multi-frequency generalized power method, where each iteration updates the estimation in a matrix-multiplication-then-projection manner. Numerical experiments show that our proposed algorithms significantly improve the ability of exactly recovering the cluster structure and the accuracy of the estimated phase angles, compared to state-of-the-art algorithms.

Despite its exceptional soft tissue contrast, Magnetic Resonance Imaging (MRI) faces the challenge of long scanning times compared to other modalities like X-ray radiography. Shortening scanning times is crucial in clinical settings, as it increases patient comfort, decreases examination costs and improves throughput. Recent advances in compressed sensing (CS) and deep learning allow accelerated MRI acquisition by reconstructing high-quality images from undersampled data. While reconstruction algorithms have received most of the focus, designing acquisition trajectories to optimize reconstruction quality remains an open question. This thesis explores two approaches to address this gap in the context of Cartesian MRI. First, we propose two algorithms, lazy LBCS and stochastic LBCS, that significantly improve upon G\"ozc\"u et al.'s greedy learning-based CS (LBCS) approach. These algorithms scale to large, clinically relevant scenarios like multi-coil 3D MR and dynamic MRI, previously inaccessible to LBCS. Additionally, we demonstrate that generative adversarial networks (GANs) can serve as a natural criterion for adaptive sampling by leveraging variance in the measurement domain to guide acquisition. Second, we delve into the underlying structures or assumptions that enable mask design algorithms to perform well in practice. Our experiments reveal that state-of-the-art deep reinforcement learning (RL) approaches, while capable of adaptation and long-horizon planning, offer only marginal improvements over stochastic LBCS, which is neither adaptive nor does long-term planning. Altogether, our findings suggest that stochastic LBCS and similar methods represent promising alternatives to deep RL. They shine in particular by their scalability and computational efficiency and could be key in the deployment of optimized acquisition trajectories in Cartesian MRI.

In the field of surrogate modeling, polynomial chaos expansion (PCE) allows practitioners to construct inexpensive yet accurate surrogates to be used in place of the expensive forward model simulations. For black-box simulations, non-intrusive PCE allows the construction of these surrogates using a set of simulation response evaluations. In this context, the PCE coefficients can be obtained using linear regression, which is also known as point collocation or stochastic response surfaces. Regression exhibits better scalability and can handle noisy function evaluations in contrast to other non-intrusive approaches, such as projection. However, since over-sampling is generally advisable for the linear regression approach, the simulation requirements become prohibitive for expensive forward models. We propose to leverage transfer learning whereby knowledge gained through similar PCE surrogate construction tasks (source domains) is transferred to a new surrogate-construction task (target domain) which has a limited number of forward model simulations (training data). The proposed transfer learning strategy determines how much, if any, information to transfer using new techniques inspired by Bayesian modeling and data assimilation. The strategy is scrutinized using numerical investigations and applied to an engineering problem from the oil and gas industry.

Recent work has shown that energy-based language modeling is an effective framework for controllable text generation because it enables flexible integration of arbitrary discriminators. However, because energy-based LMs are globally normalized, approximate techniques like Metropolis-Hastings (MH) are required for inference. Past work has largely explored simple proposal distributions that modify a single token at a time, like in Gibbs sampling. In this paper, we develop a novel MH sampler that, in contrast, proposes re-writes of the entire sequence in each step via iterative prompting of a large language model. Our new sampler (a) allows for more efficient and accurate sampling from a target distribution and (b) allows generation length to be determined through the sampling procedure rather than fixed in advance, as past work has required. We perform experiments on two controlled generation tasks, showing both downstream performance gains and more accurate target distribution sampling in comparison with single-token proposal techniques.

In this paper, we propose a latent-variable generative model called mixture of dynamical variational autoencoders (MixDV AE) to model the dynamics of a system composed of multiple moving sources. A DV AE model is pre-trained on a single-source dataset to capture the source dynamics. Then, multiple instances of the pre-trained DV AE model are integrated into a multi-source mixture model with a discrete observation-to-source assignment latent variable. The posterior distributions of both the discrete observation-to-source assignment variable and the continuous DV AE variables representing the sources content/position are estimated using a variational expectation-maximization algorithm, leading to multi-source trajectories estimation. We illustrate the versatility of the proposed MixDV AE model on two tasks: a computer vision task, namely multi-object tracking, and an audio processing task, namely single-channel audio source separation. Experimental results show that the proposed method works well on these two tasks, and outperforms several baseline methods.