Suppose two Bayesian agents each learn a generative model of the same environment. We will assume the two have converged on the predictive distribution, i.e. distribution over some observables in the environment, but may have different generative models containing different latent variables. Under what conditions can one agent guarantee that their latents are a function of the other agents latents? We give simple conditions under which such translation is guaranteed to be possible: the natural latent conditions. We also show that, absent further constraints, these are the most general conditions under which translatability is guaranteed. Crucially for practical application, our theorems are robust to approximation error in the natural latent conditions.

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.

To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.

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.

In this paper, we present a framework to understand the convergence of commonly used Q-learning reinforcement learning algorithms in practice. Two salient features of such algorithms are: (i)~the Q-table is recursively updated using an agent state (such as the state of a recurrent neural network) which is not a belief state or an information state and (ii)~policy regularization is often used to encourage exploration and stabilize the learning algorithm. We investigate the simplest form of such Q-learning algorithms which we call regularized agent-state-based Q-learning (RASQL) and show that it converges under mild technical conditions to the fixed point of an appropriately defined regularized MDP, which depends on the stationary distribution induced by the behavioral policy. We also show that a similar analysis continues to work for a variant of RASQL that learns periodic policies. We present numerical examples to illustrate that the empirical convergence behavior matches with the proposed theoretical limit.