Singular statistical models arise whenever different parameter values induce the same distribution, leading to non-identifiability and a breakdown of classical asymptotic theory. While existing approaches analyze these phenomena in parameter space, the resulting descriptions depend heavily on parameterization and obscure the intrinsic statistical structure of the model. In this paper, we introduce an invariant framework based on \emph{observable charts}: collections of functionals of the data distribution that distinguish probability measures. These charts define local coordinate systems directly on the model space, independent of parameterization. We formalize \emph{observable completeness} as the ability of such charts to detect identifiable directions, and introduce \emph{observable order} to quantify higher-order distinguishability along analytic perturbations. Our main result establishes that, under mild regularity conditions, observable order provides a lower bound on the rate at which Kullback-Leibler divergence vanishes along analytic paths. This connects intrinsic geometric structure in model space to statistical distinguishability and recovers classical behavior in regular models while extending naturally to singular settings. We illustrate the framework in reduced-rank regression and Gaussian mixture models, where observable coordinates reveal both identifiable structure and singular degeneracies. These results suggest that observable charts provide a unified and parameterization-invariant language for studying singular models and offer a pathway toward intrinsic formulations of invariants such as learning coefficients.

A plethora of methods was developed for solving this problem, including the celebrated do-calculus [Pearl, 1995]. In practice, these results are not always applicable since they require a fully specified causal diagram as input, which is usually not available.

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First, a random patch of the image is selected and resized to224 224 with a random horizontal flip, followed byacolor distortion, consisting ofarandom sequence ofbrightness, contrast, saturation, hue adjustments, and anoptional grayscale conversion. FinallyGaussian blur and solarization are appliedtothepatches. Optimization We use theLARS optimizer [70] with a cosine decay learning rate schedule [71], without restarts, over1000epochs, with awarm-up period of10epochs. Wesetthebase learning rate to 0.2, scaled linearly [72] with the batch size (LearningRate = 0.2 BatchSize/256). Forthetargetnetwork,the exponential moving average parameterτ starts fromτbase = 0.996and is increased to one during training.