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It is often the case in Statistics that one needs to compute sums of infinite series, especially in marginalising over discrete latent variables. This has become more relevant with the popularization of gradient-based techniques (e.g. Hamiltonian Monte Carlo) in the Bayesian inference context, for which discrete latent variables are hard or impossible to deal with. For many commonly used infinite series, custom algorithms have been developed which exploit specific features of each problem. General techniques, suitable for a large class of problems with limited input from the user are less established. We employ basic results from the theory of infinite series to investigate general, problem-agnostic algorithms to truncate infinite sums within an arbitrary tolerance $\varepsilon > 0$ and provide robust computational implementations with provable guarantees. We compare three tentative solutions to estimating the infinite sum of interest: (i) a "naive" approach that sums terms until the terms are below the threshold $\varepsilon$; (ii) a `bounding pair' strategy based on trapping the true value between two partial sums; and (iii) a `batch' strategy that computes the partial sums in regular intervals and stops when their difference is less than $\varepsilon$. We show under which conditions each strategy guarantees the truncated sum is within the required tolerance and compare the error achieved by each approach, as well as the number of function evaluations necessary for each one. A detailed discussion of numerical issues in practical implementations is also provided. The paper provides some theoretical discussion of a variety of statistical applications, including raw and factorial moments and count models with observation error. Finally, detailed illustrations in the form noisy MCMC for Bayesian inference and maximum marginal likelihood estimation are presented.

We propose a new framework to reconstruct a stochastic process $\left\{\mathbb{P}_{t}: t \in[0, T]\right\}$ using only samples from its marginal distributions, observed at start and end times $0$ and $T$. This reconstruction is useful to infer population dynamics, a crucial challenge, e.g., when modeling the time-evolution of cell populations from single-cell sequencing data. Our general framework encompasses the more specific Schr\"odinger bridge (SB) problem, where $\mathbb{P}_{t}$ represents the evolution of a thermodynamic system at almost equilibrium. Estimating such bridges is notoriously difficult, motivating our proposal for a novel adaptive scheme called the GSBflow. Our goal is to rely on Gaussian approximations of the data to provide the reference stochastic process needed to estimate SB. To that end, we solve the \acs{SB} problem with Gaussian marginals, for which we provide, as a central contribution, a closed-form solution and SDE-representation. We use these formulas to define the reference process used to estimate more complex SBs, and show that this does indeed help with its numerical solution. We obtain notable improvements when reconstructing both synthetic processes and single-cell genomics experiments.

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

Probabilistic databases (PDBs) model uncertainty in data in a quantitative way. In the established formal framework, probabilistic (relational) databases are finite probability spaces over relational database instances. This finiteness can clash with intuitive query behavior (Ceylan et al., KR 2016), and with application scenarios that are better modeled by continuous probability distributions (Dalvi et al., CACM 2009). We formally introduced infinite PDBs in (Grohe and Lindner, PODS 2019) with a primary focus on countably infinite spaces. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics. We argue that finite point processes are an appropriate model from probability theory for dealing with general probabilistic databases. This allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries.

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.

Markov Logic Networks (MLNs), which elegantly combine logic rules and probabilistic graphical models, can be used to address many knowledge graph problems. However, inference in MLN is computationally intensive, making the industrial-scale application of MLN very difficult. In recent years, graph neural networks (GNNs) have emerged as efficient and effective tools for large-scale graph problems. Nevertheless, GNNs do not explicitly incorporate prior logic rules into the models, and may require many labeled examples for a target task. In this paper, we explore the combination of MLNs and GNNs, and use graph neural networks for variational inference in MLN. We propose a GNN variant, named ExpressGNN, which strikes a nice balance between the representation power and the simplicity of the model. Our extensive experiments on several benchmark datasets demonstrate that ExpressGNN leads to effective and efficient probabilistic logic reasoning.

This paper studies the problem of domain division problem which aims to segment instances drawn from different probabilistic distributions. Such a problem exists in many previous recognition tasks, such as Open Set Learning (OSL) and Generalized Zero-Shot Learning (G-ZSL), where the testing instances come from either seen or novel/unseen classes of different probabilistic distributions. Previous works focused on either only calibrating the confident prediction of classifiers of seen classes (W-SVM), or taking unseen classes as outliers. In contrast, this paper proposes a probabilistic way of directly estimating and fine-tuning the decision boundary between seen and novel/unseen classes. In particular, we propose a domain division algorithm of learning to split the testing instances into known, unknown and uncertain domains, and then conduct recognize tasks in each domain. Two statistical tools, namely, bootstrapping and Kolmogorov-Smirnov (K-S) Test, for the first time, are introduced to discover and fine-tune the decision boundary of each domain. Critically, the uncertain domain is newly introduced in our framework to adopt those instances whose domain cannot be predicted confidently. Extensive experiments demonstrate that our approach achieved the state-of-the-art performance on OSL and G-ZSL benchmarks.

A fundamental computation for statistical inference and accurate decision-making is to compute the marginal probabilities or most probable states of task-relevant variables. Probabilistic graphical models can efficiently represent the structure of such complex data, but performing these inferences is generally difficult. Message-passing algorithms, such as belief propagation, are a natural way to disseminate evidence amongst correlated variables while exploiting the graph structure, but these algorithms can struggle when the conditional dependency graphs contain loops. Here we use Graph Neural Networks (GNNs) to learn a message-passing algorithm that solves these inference tasks. We first show that the architecture of GNNs is well-matched to inference tasks. We then demonstrate the efficacy of this inference approach by training GNNs on a collection of graphical models and showing that they substantially outperform belief propagation on loopy graphs. Our message-passing algorithms generalize out of the training set to larger graphs and graphs with different structure.

In this paper, we propose a listwise approach for constructing user-specific rankings in recommendation systems in a collaborative fashion. We contrast the listwise approach to previous pointwise and pairwise approaches, which are based on treating either each rating or each pairwise comparison as an independent instance respectively. By extending the work of (Cao et al. 2007), we cast listwise collaborative ranking as maximum likelihood under a permutation model which applies probability mass to permutations based on a low rank latent score matrix. We present a novel algorithm called SQL-Rank, which can accommodate ties and missing data and can run in linear time. We develop a theoretical framework for analyzing listwise ranking methods based on a novel representation theory for the permutation model. Applying this framework to collaborative ranking, we derive asymptotic statistical rates as the number of users and items grow together. We conclude by demonstrating that our SQL-Rank method often outperforms current state-of-the-art algorithms for implicit feedback such as Weighted-MF and BPR and achieve favorable results when compared to explicit feedback algorithms such as matrix factorization and collaborative ranking.