Furthermore, in continuous state-action settings these methods must make coarse approximations to be computationally tractable (e.g.

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models.

This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for

Causal inference is no exception.

For general non-Gaussian distributions, the covariance and precision matrices do not encode the independence structure of the variables, as they do for the multivariate Gaussian. This paper builds on previous work to show that for a class of non-Gaussian distributions -- those derived from diagonal transformations of a Gaussian -- information about the conditional independence structure can still be inferred from the precision matrix, provided the data meet certain criteria, analogous to the Gaussian case. We call such transformations of the Gaussian as the generalized nonparanormal. The functions that define these transformations are, in a broad sense, arbitrary. We also provide a simple and computationally efficient algorithm that leverages this theory to recover conditional independence structure from the generalized nonparanormal data. The effectiveness of the proposed algorithm is demonstrated via synthetic experiments and applications to real-world data.

Rigorous statistical methods, including parameter estimation with accompanying uncertainties, underpin the validity of scientific discovery, especially in the natural sciences. With increasingly complex data models such as deep learning techniques, uncertainty quantification has become exceedingly difficult and a plethora of techniques have been proposed. In this case study, we use the unifying framework of approximate Bayesian inference combined with empirical tests on carefully created synthetic classification datasets to investigate qualitative properties of six different probabilistic machine learning algorithms for class probability and uncertainty estimation: (i) a neural network ensemble, (ii) neural network ensemble with conflictual loss, (iii) evidential deep learning, (iv) a single neural network with Monte Carlo Dropout, (v) Gaussian process classification and (vi) a Dirichlet process mixture model. We check if the algorithms produce uncertainty estimates which reflect commonly desired properties, such as being well calibrated and exhibiting an increase in uncertainty for out-of-distribution data points. Our results indicate that all algorithms are well calibrated, but none of the deep learning based algorithms provide uncertainties that consistently reflect lack of experimental evidence for out-of-distribution data points. We hope our study may serve as a clarifying example for researchers developing new methods of uncertainty estimation for scientific data-driven modeling.

We develop a Bayesian framework for variable selection in linear regression with autocorrelated errors, accommodating lagged covariates and autoregressive structures. This setting occurs in time series applications where responses depend on contemporaneous or past explanatory variables and persistent stochastic shocks, including financial modeling, hydrological forecasting, and meteorological applications requiring temporal dependency capture. Our methodology uses hierarchical Bayesian models with spike-and-slab priors to simultaneously select relevant covariates and lagged error terms. We propose an efficient two-stage MCMC algorithm separating sampling of variable inclusion indicators and model parameters to address high-dimensional computational challenges. Theoretical analysis establishes posterior selection consistency under mild conditions, even when candidate predictors grow exponentially with sample size, common in modern time series with many potential lagged variables. Through simulations and real applications (groundwater depth prediction, S&P 500 log returns modeling), we demonstrate substantial gains in variable selection accuracy and predictive performance. Compared to existing methods, our framework achieves lower MSPE, improved true model component identification, and greater robustness with autocorrelated noise, underscoring practical utility for model interpretation and forecasting in autoregressive settings.