We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Simulation-based inference (SBI) enables Bayesian analysis when the likelihood is intractable but model simulations are available. Recent advances in statistics and machine learning, including Approximate Bayesian Computation and deep generative models, have expanded the applicability of SBI, yet these methods often face challenges in moderate to high-dimensional parameter spaces. Motivated by the success of gradient-based Monte Carlo methods in Bayesian sampling, we propose a novel SBI method that integrates score matching with Langevin dynamics to explore complex posterior landscapes more efficiently in such settings. Our approach introduces tailored score-matching procedures for SBI, including a localization scheme that reduces simulation costs and an architectural regularization that embeds the statistical structure of log-likelihood scores to improve score-matching accuracy. We provide theoretical analysis of the method and illustrate its practical benefits on benchmark tasks and on more challenging problems in moderate to high dimensions, where it performs favorably compared to existing approaches.

We investigate partially identifiable queries in a class of causal models. We focus on acyclic Structural Causal Models that are quasi-Markovian (that is, each endogenous variable is connected with at most one exogenous confounder). We look into scenarios where endogenous variables are observed (and a distribution over them is known), while exogenous variables are not fully specified. This leads to a representation that is in essence a Bayesian network where the distribution of root variables is not uniquely determined. In such circumstances, it may not be possible to precisely compute a probability value of interest. We thus study the computation of tight probability bounds, a problem that has been solved by multilinear programming in general, and by linear programming when a single confounded component is intervened upon. We present a new algorithm to simplify the construction of such programs by exploiting input probabilities over endogenous variables. For scenarios with a single intervention, we apply column generation to compute a probability bound through a sequence of auxiliary linear integer programs, thus showing that a representation with polynomial cardinality for exogenous variables is possible. Experiments show column generation techniques to be superior to existing methods.

Parametric partial differential equations (PDEs) are fundamental mathematical tools for modeling complex physical systems, yet their numerical evaluation across parameter spaces remains computationally intensive when using conventional high-fidelity solvers. To address this challenge, we propose a novel physical law-corrected prior Gaussian process (LC-prior GP) surrogate modeling framework that effectively integrates data-driven learning with underlying physical constraints to flexibly handle multi-coupled variables defined on complex geometries. The proposed approach leverages proper orthogonal decomposition (POD) to parameterize high-dimensional PDE solutions via their dominant modes and associated coefficients, thereby enabling efficient Gaussian process (GP) surrogate modeling within a reduced-dimensional coefficient space. A key contribution lies in the incorporation of physical laws together with a limited number of parameter samples to correct the GP posterior mean, thus avoiding reliance on computationally expensive numerical solvers. Furthermore, interpolation functions are constructed to describe the mapping from the full parameter space to the physics-based correction term. This mapping is subsequently backpropagated to constrain the original GP surrogate, yielding a more physically consistent conditional prior. To handle irregular geometries, the radial basis function-finite difference (RBF-FD) method is incorporated during training set computation, with its inherent differentiation matrices providing both computational efficiency and numerical accuracy for physical constraint optimization. The effectiveness of the proposed method is demonstrated through numerical experiments involving a reaction-diffusion model, miscible flooding models, and Navier-Stokes equations with multi-physics coupling defined on irregular domains.

While calibration of probabilistic predictions has been widely studied, this paper rather addresses calibration of likelihood functions. This has been discussed, especially in biometrics, in cases with only two exhaustive and mutually exclusive hypotheses (classes) where likelihood functions can be written as log-likelihood-ratios (LLRs). After defining calibration for LLRs and its connection with the concept of weight-of-evidence, we present the idempotence property and its associated constraint on the distribution of the LLRs. Although these results have been known for decades, they have been limited to the binary case. Here, we extend them to cases with more than two hypotheses by using the Aitchison geometry of the simplex, which allows us to recover, in a vector form, the additive form of the Bayes' rule; extending therefore the LLR and the weight-of-evidence to any number of hypotheses. Especially, we extend the definition of calibration, the idempotence, and the constraint on the distribution of likelihood functions to this multiple hypotheses and multiclass counterpart of the LLR: the isometric-log-ratio transformed likelihood function. This work is mainly conceptual, but we still provide one application to machine learning by presenting a non-linear discriminant analysis where the discriminant components form a calibrated likelihood function over the classes, improving therefore the interpretability and the reliability of the method.

This article focuses on covariance estimation for multi-view data. Popular approaches rely on factor-analytic decompositions that have shared and view-specific latent factors. Posterior computation is conducted via expensive and brittle Markov chain Monte Carlo (MCMC) sampling or variational approximations that underestimate uncertainty and lack theoretical guarantees. Our proposed methodology employs spectral decompositions to estimate and align latent factors that are active in at least one view. Conditionally on these factors, we choose jointly conjugate prior distributions for factor loadings and residual variances. The resulting posterior is a simple product of normal-inverse gamma distributions for each variable, bypassing MCMC and facilitating posterior computation. We prove favorable increasing-dimension asymptotic properties, including posterior contraction and central limit theorems for point estimators. We show excellent performance in simulations, including accurate uncertainty quantification, and apply the methodology to integrate four high-dimensional views from a multi-omics dataset of cancer cell samples.

Generating synthetic datasets that accurately reflect real-world observational data is critical for evaluating causal estimators, but remains a challenging task. Existing generative methods offer a solution by producing synthetic datasets anchored in the observed data (source data) while allowing variation in key parameters such as the treatment effect and amount of confounding bias. However, existing methods typically require users to provide point estimates of such parameters (rather than distributions) and fixed estimates (rather than estimates that can be improved with reference to the source data). This denies users the ability to express uncertainty over parameter values and removes the potential for posterior inference, potentially leading to unreliable estimator comparisons. We introduce simulation-based inference for causal evaluation (SBICE), a framework that models generative parameters as uncertain and infers their posterior distribution given a source dataset. Leveraging techniques in simulation-based inference, SBICE identifies parameter configurations that produce synthetic datasets closely aligned with the source data distribution. Empirical results demonstrate that SBICE improves the reliability of estimator evaluations by generating more realistic datasets, which supports a robust and data-consistent approach to causal benchmarking under uncertainty.

Human cognition is profoundly shaped by the environments in which it unfolds. Yet, it remains an open question whether learning and decision making can be explained as a principled adaptation to the statistical structure of real-world tasks. We introduce ecologically rational analysis, a computational framework that unifies the normative foundations of rational analysis with ecological grounding. Leveraging large language models to generate ecologically valid cognitive tasks at scale, and using meta-learning to derive rational models optimized for these environments, we develop a new class of learning algorithms: Ecologically Rational Meta-learned Inference (ERMI). ERMI internalizes the statistical regularities of naturalistic problem spaces and adapts flexibly to novel situations, without requiring hand-crafted heuristics or explicit parameter updates. We show that ERMI captures human behavior across 15 experiments spanning function learning, category learning, and decision making, outperforming several established cognitive models in trial-by-trial prediction. Our results suggest that much of human cognition may reflect adaptive alignment to the ecological structure of the problems we encounter in everyday life.

This work demonstrates a proof-of-principle for using uncertainty-aware architectures, in combination with active learning techniques and an in-the-loop physics simulation code as a data labeller, to construct efficient datasets for data-driven surrogate model generation. Building off of a previous proof-of-principle successfully demonstrating training set reduction on static pre-labelled datasets, using the ADEPT framework, this strategy was applied again to the plasma turbulent transport problem within tokamak fusion plasmas, specifically the QuaLiKiz quasilinear electrostatic gyrokinetic turbulent transport code. While QuaLiKiz provides relatively fast evaluations, this study specifically targeted small datasets to serve as a proxy for more expensive codes, such as CGYRO or GENE. The newly implemented algorithm uses the SNGP architecture for the classification component of the problem and the BNN-NCP architecture for the regression component, training models for all turbulent modes (ITG, TEM, ETG) and all transport fluxes ($Q_e$, $Q_i$, $Γ_e$, $Γ_i$, and $Π_i$) described by the general QuaLiKiz output. With 45 active learning iterations, moving from a small initial training set of $10^{2}$ to a final set of $10^{4}$, the resulting models reached a $F_1$ classification performance of ~0.8 and a $R^2$ regression performance of ~0.75 on an independent test set across all outputs. This extrapolates to reaching the same performance and efficiency as the previous ADEPT pipeline, although on a problem with 1 extra input dimension. While the improvement rate achieved in this implementation diminishes faster than expected, the overall technique is formulated with components that can be upgraded and generalized to many surrogate modeling applications beyond plasma turbulent transport predictions.

The quasi-repetitive nature of construction work and the resulting lack of generalizability in programming construction robots presents persistent challenges to the broad adoption of robots in the construction industry. Robots cannot achieve generalist capabilities as skills learnt from one domain cannot readily transfer to another work domain or be directly used to perform a different set of tasks. Human workers have to arduously reprogram their scene-understanding, path-planning, and manipulation components to enable the robots to perform alternate work tasks. The methods presented in this paper resolve a significant proportion of such reprogramming workload by proposing a generalizable learning architecture that directly teaches robots versatile task-performance skills through crowdsourced online natural language instructions. A Large Language Model (LLM), a standardized and modularized hierarchical modeling approach, and Building Information Modeling-Robot sematic data pipeline are developed to address the multi-task skill transfer problem. The proposed skill standardization scheme and LLM-based hierarchical skill learning framework were tested with a long-horizon drywall installation experiment using a full-scale industrial robotic manipulator. The resulting robot task learning scheme achieves multi-task reprogramming with minimal effort and high quality.