An example of this is question answering (QA): given a question from the user, the model may provide a brief answer, but it may also follow with supporting facts and explanations, which can vary in form and detail. The user can be satisfied by a wide variety of responses, irrespective of their style or (to some extent) the choice of supporting facts included. Free-form NLG poses significant challenges to uncertainty quantification: some aspects of generation are irrelevant to the task's purpose and best excluded from uncertainty quantification, but it often appears that we are unable to characterize them precisely. If left unaddressed, however, the model's variation in the irrelevant aspects may dominate in standard uncertainty measures such as token-level entropy (Kuhn et al., 2023), making them uninformative about the model's actual performance on the task. Starting from Kuhn et al. (2023), a recent line of work (Kuhn et al., 2023; Lin et al., 2024; Zhang et al., 2023; Aichberger et al., 2024) studied this issue and proposed measuring the "semantic uncertainty" of generation; "semantics" is defined as the equivalence class of textual responses that logically entail one another. Empirical improvements in downstream tasks evidenced their contributions and highlighted the importance of task-specific uncertainty quantification, but important conceptual and practical issues remain. From a practical perspective, semantic equivalence is estimated using machine learning models, resulting in imprecise estimates that do not necessarily define an equivalence relation.

A Bayesian nonparametric model known as the Wishart process has been shown to be effective in this situation, but its inference remains highly challenging. In this work, we introduce a Sequential Monte Carlo (SMC) sampler for the Wishart process, and show how it compares to conventional inference approaches, namely MCMC and variational inference. Using simulations we show that SMC sampling results in the most robust estimates and out-of-sample predictions of dynamic covariance. SMC especially outperforms the alternative approaches when using composite covariance functions with correlated parameters. We demonstrate the practical applicability of our proposed approach on a dataset of clinical depression (n = 1), and show how using an accurate representation of the posterior distribution can be used to test for dynamics on covariance.

We investigate how to select the number of communities for weighted networks without a full likelihood modeling. First, we propose a novel weighted degree-corrected stochastic block model (DCSBM), in which the mean adjacency matrix is modeled as the same as in standard DCSBM, while the variance profile matrix is assumed to be related to the mean adjacency matrix through a given variance function. Our method of selection the number of communities is based on a sequential testing framework, in each step the weighed DCSBM is fitted via some spectral clustering method. A key step is to carry out matrix scaling on the estimated variance profile matrix. The resulting scaling factors can be used to normalize the adjacency matrix, from which the testing statistic is obtained. Under mild conditions on the weighted DCSBM, our proposed procedure is shown to be consistent in estimating the true number of communities. Numerical experiments on both simulated and real network data also demonstrate the desirable empirical properties of our method.

We present a formulation of flow matching as variational inference, which we refer to as variational flow matching (VFM). Based on this formulation we develop CatFlow, a flow matching method for categorical data. CatFlow is easy to implement, computationally efficient, and achieves strong results on graph generation tasks. In VFM, the objective is to approximate the posterior probability path, which is a distribution over possible end points of a trajectory. We show that VFM admits both the CatFlow objective and the original flow matching objective as special cases. We also relate VFM to score-based models, in which the dynamics are stochastic rather than deterministic, and derive a bound on the model likelihood based on a reweighted VFM objective. We evaluate CatFlow on one abstract graph generation task and two molecular generation tasks. In all cases, CatFlow exceeds or matches performance of the current state-of-the-art models.

A common objective in the analysis of tabular data is estimating the conditional distribution (in contrast to only producing predictions) of a set of "outcome" variables given a set of "covariates", which is sometimes referred to as the "density regression" problem. Beyond estimation on the conditional distribution, the generative ability of drawing synthetic samples from the learned conditional distribution is also desired as it further widens the range of applications. We propose a flow-based generative model tailored for the density regression task on tabular data. Our flow applies a sequence of tree-based piecewise-linear transforms on initial uniform noise to eventually generate samples from complex conditional densities of (univariate or multivariate) outcomes given the covariates and allows efficient analytical evaluation of the fitted conditional density on any point in the sample space. We introduce a training algorithm for fitting the tree-based transforms using a divide-and-conquer strategy that transforms maximum likelihood training of the tree-flow into training a collection of binary classifiers--one at each tree split--under cross-entropy loss. We assess the performance of our method under out-of-sample likelihood evaluation and compare it with a variety of state-of-the-art conditional density learners on a range of simulated and real benchmark tabular datasets. Our method consistently achieves comparable or superior performance at a fraction of the training and sampling budget. Finally, we demonstrate the utility of our method's generative ability through an application to generating synthetic longitudinal microbiome compositional data based on training our flow on a publicly available microbiome study.

The framework comes with three practical challenges: (1) it requires the causal directed acyclic graph (DAG), together with an SCM, (2) it is statistically ill-posed since it probes regression models in regions of low probability density, (3) it relies on Shapley values which are computationally expensive to find. In this paper, we propose simplified, efficient methods of root cause analysis when the task is to identify a unique root cause instead of quantitative contribution analysis. Our proposed methods run in linear order of SCM nodes and they require only the causal DAG without counterfactuals. Furthermore, for those use cases where the causal DAG is unknown, we justify the heuristic of identifying root causes as the variables with the highest anomaly score.

We introduce a novel generative model for the representation of joint probability distributions of a possibly large number of discrete random variables. The approach uses measure transport by randomized assignment flows on the statistical submanifold of factorizing distributions, which also enables to sample efficiently from the target distribution and to assess the likelihood of unseen data points. The embedding of the flow via the Segre map in the meta-simplex of all discrete joint distributions ensures that any target distribution can be represented in principle, whose complexity in practice only depends on the parametrization of the affinity function of the dynamical assignment flow system. Our model can be trained in a simulation-free manner without integration by conditional Riemannian flow matching, using the training data encoded as geodesics in closed-form with respect to the e-connection of information geometry. By projecting high-dimensional flow matching in the meta-simplex of joint distributions to the submanifold of factorizing distributions, our approach has strong motivation from first principles of modeling coupled discrete variables. Numerical experiments devoted to distributions of structured image labelings demonstrate the applicability to large-scale problems, which may include discrete distributions in other application areas. Performance measures show that our approach scales better with the increasing number of classes than recent related work.

The limitations of purely neural learning have sparked an interest in probabilistic neurosymbolic models, which combine neural networks with probabilistic logical reasoning. As these neurosymbolic models are trained with gradient descent, we study the complexity of differentiating probabilistic reasoning. We prove that although approximating these gradients is intractable in general, it becomes tractable during training. Furthermore, we introduce WeightME, an unbiased gradient estimator based on model sampling. Under mild assumptions, WeightME approximates the gradient with probabilistic guarantees using a logarithmic number of calls to a SAT solver. Lastly, we evaluate the necessity of these guarantees on the gradient. Our experiments indicate that the existing biased approximations indeed struggle to optimize even when exact solving is still feasible.

Random effects meta-analysis model is an important tool for integrating results from multiple independent studies. However, the standard model is based on the assumption of normal distributions for both random effects and within-study errors, making it susceptible to outlying studies. Although robust modeling using the $t$ distribution is an appealing idea, the existing work, that explores the use of the $t$ distribution only for random effects, involves complicated numerical integration and numerical optimization. In this paper, a novel robust meta-analysis model using the $t$ distribution is proposed ($t$Meta). The novelty is that the marginal distribution of the effect size in $t$Meta follows the $t$ distribution, enabling that $t$Meta can simultaneously accommodate and detect outlying studies in a simple and adaptive manner. A simple and fast EM-type algorithm is developed for maximum likelihood estimation. Due to the mathematical tractability of the $t$ distribution, $t$Meta frees from numerical integration and allows for efficient optimization. Experiments on real data demonstrate that $t$Meta is compared favorably with related competitors in situations involving mild outliers. Moreover, in the presence of gross outliers, while related competitors may fail, $t$Meta continues to perform consistently and robustly.

We address the problem of uncertainty quantification for graph-structured data, or, more specifically, the problem to quantify the predictive uncertainty in (semi-supervised) node classification. Key questions in this regard concern the distinction between two different types of uncertainty, aleatoric and epistemic, and how to support uncertainty quantification by leveraging the structural information provided by the graph topology. Challenging assumptions and postulates of state-of-the-art methods, we propose a novel approach that represents (epistemic) uncertainty in terms of mixtures of Dirichlet distributions and refers to the established principle of linear opinion pooling for propagating information between neighbored nodes in the graph. The effectiveness of this approach is demonstrated in a series of experiments on a variety of graph-structured datasets.