Here we consider the problem of denoising features associated to complex data, modeled as signals on a graph, via a smoothness prior. This is motivated in part by settings such as single-cell RNA where the data is very high-dimensional, but its structure can be captured via an affinity graph. This allows us to utilize ideas from graph signal processing. In particular, we present algorithms for the cases where the signal is perturbed by Gaussian noise, dropout, and uniformly distributed noise. The signals are assumed to follow a prior distribution defined in the frequency domain which favors signals which are smooth across the edges of the graph. By pairing this prior distribution with our three models of noise generation, we propose Maximum A Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data and provide algorithms for computing the M.A.P. Finally, we demonstrate the algorithms' ability to effectively restore signals from white noise on image data and from severe dropout in single-cell RNA sequence data.

Bayesian causal discovery aims to infer the posterior distribution over causal models from observed data, quantifying epistemic uncertainty and benefiting downstream tasks. However, computational challenges arise due to joint inference over combinatorial space of Directed Acyclic Graphs (DAGs) and nonlinear functions. Despite recent progress towards efficient posterior inference over DAGs, existing methods are either limited to variational inference on node permutation matrices for linear causal models, leading to compromised inference accuracy, or continuous relaxation of adjacency matrices constrained by a DAG regularizer, which cannot ensure resulting graphs are DAGs. In this work, we introduce a scalable Bayesian causal discovery framework based on a combination of stochastic gradient Markov Chain Monte Carlo (SG-MCMC) and Variational Inference (VI) that overcomes these limitations. Our approach directly samples DAGs from the posterior without requiring any DAG regularization, simultaneously draws function parameter samples and is applicable to both linear and nonlinear causal models. To enable our approach, we derive a novel equivalence to the permutation-based DAG learning, which opens up possibilities of using any relaxed gradient estimator defined over permutations. To our knowledge, this is the first framework applying gradient-based MCMC sampling for causal discovery. Empirical evaluation on synthetic and real-world datasets demonstrate our approach's effectiveness compared to state-of-the-art baselines.

Generating samples from a complex (e.g., non-Gaussian, high-dimensional) probability distribution is a core computational challenge in diverse applications, ranging from computational statistics and machine learning to molecular simulation. A recurring setting is where the density ρ of the target distribution is specified up to a normalizing constant--for example, in Bayesian modeling, where ρ represents the posterior density. Here, evaluations of the score log ρ are often available as well, even for complex statistical models [Villa et al., 2021]. Alternatively, many new methods enable effective score estimation from data, without explicit density estimation; examples include score estimation from time series observations in chaotic dynamical systems [Chandramoorthy and Wang, 2022, Ni, 2020] and score-based modeling of image distributions [Song et al., 2020b,a]. In these settings, transport or "flow"-driven algorithms for generating samples have seen extensive success. The central idea is to construct a transport map from a simple, prescribed source distribution to the target distribution of interest. One class of transport approaches, e.g., as represented by variational inference with normalizing flows, involves constructing a parametric class of invertible maps and minimizing some statistical divergence between the pushforward (see Section 2) of the source by a member of this class and the target. A different, essentially nonparametric, class of transport approaches are based on particle systems, e.g., Stein variational gradient descent (SVGD)

Estimating the prevalence of a medical condition, or the proportion of the population in which it occurs, is a fundamental problem in healthcare and public health. Accurate estimates of the relative prevalence across groups -- capturing, for example, that a condition affects women more frequently than men -- facilitate effective and equitable health policy which prioritizes groups who are disproportionately affected by a condition. However, it is difficult to estimate relative prevalence when a medical condition is underreported. In this work, we provide a method for accurately estimating the relative prevalence of underreported medical conditions, building upon the positive unlabeled learning framework. We show that under the commonly made covariate shift assumption -- i.e., that the probability of having a disease conditional on symptoms remains constant across groups -- we can recover the relative prevalence, even without restrictive assumptions commonly made in positive unlabeled learning and even if it is impossible to recover the absolute prevalence. We conduct experiments on synthetic and real health data which demonstrate our method's ability to recover the relative prevalence more accurately than do baselines, and demonstrate the method's robustness to plausible violations of the covariate shift assumption. We conclude by illustrating the applicability of our method to case studies of intimate partner violence and hate speech.

Simulation-based inference (SBI) is constantly in search of more expressive algorithms for accurately inferring the parameters of complex models from noisy data. We present consistency models for neural posterior estimation (CMPE), a new free-form conditional sampler for scalable, fast, and amortized SBI with generative neural networks. CMPE combines the advantages of normalizing flows and flow matching methods into a single generative architecture: It essentially distills a continuous probability flow and enables rapid few-shot inference with an unconstrained architecture that can be tailored to the structure of the estimation problem. Our empirical evaluation demonstrates that CMPE not only outperforms current state-of-the-art algorithms on three hard low-dimensional problems, but also achieves competitive performance in a high-dimensional Bayesian denoising experiment and in estimating a computationally demanding multi-scale model of tumor spheroid growth.

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.