Do larger models learn increasingly similar representations of the same world? The Platonic Representation Hypothesis (PRH) predicts that they should, but similar can mean native cosine geometry---nearest-neighbour structure induced by cosine similarity before any learned alignment---or linear recoverability: correspondence revealed by a supervised linear map. Using matched Legacy Survey and Hyper Suprime-Cam (HSC) embeddings across five model families, we find that Ridge alignment strengthens the relationship between model size and cross-dataset neighbourhood correspondence relative to native cosine geometry. The effect appears in all five families and persists when both representations are reduced to dimension, so it is not simply that wider embeddings give alignment more dimensions to work with. The learned maps remain far from rotations or uniform scaling, and Sparse Autoencoder (SAE) and Block-Sparse Featuriser (BSF) features give no advantage over the same alignment applied to dense embeddings. Model scale thus improves linear recoverability more clearly than native cosine-neighbourhood correspondence, without establishing convergence toward a shared native metric geometry.
With only one confirmed example, namely the Earth, the abundance of life across the universe remains deeply uncertain, and local evidence about Earth's particular circumstances can significantly shape our knowledge of life elsewhere. Such extrapolation is difficult not only because of this sample size of one, but also due to observer selection effects discussed under the Anthropic Principle. In this paper, we describe and apply a framework called Full Indexical Conditioning (FIC) for performing Bayesian inference in this situation. Inspired by, but distinct from, Full Non-Indexical Conditioning, FIC conditions on all evidence available to the reasoner while considering that this evidence could have been otherwise. We address the reference class problem associated with such reasoning, demonstrating that substantive conclusions are unaffected by the choice of reference class provided it is sufficiently large. We apply FIC to several problems in astrobiology, including the Doomsday argument, the early abiogenesis time, and our location around a G-type rather than M-type star, clarifying the foundations of these arguments. We conclude that the early abiogenesis time does not rule out pessimistic possibilities for life's abundance, and that while life around M stars may be uncommon, its temporal distribution could extend far into the future. Ultimately, FIC offers a clear conceptual framework, grounded in straightforward probability theory, for tackling problems involving anthropic selection effects.
The evidence, $\mathcal{Z} = p(x | \mathcal{M})$, of a data-generating model $\mathcal{M}$ is a powerful tool in Bayesian frameworks: it enables model inference, model misspecification tests, and goodness-of-fit assessment as an alternative to standard discrepancy statistics, e.g. $χ^2$. While versatile, the evidence is notoriously difficult to compute, even when the likelihood is available. In both likelihood-based and simulation-based inference, estimating $\mathcal{Z}$ provides an immediate solution to high-dimensional goodness-of-fit or model misspecification tests: using $\mathcal{Z}(x)$ as the test statistic induces a minimum-volume level-$α$ test. Such tests are essential to assess model fidelity, which is necessary to trust simulation-based parameter inference. Existing learned approaches largely target Bayes factors, which only measure evidence ratios. Instead, we provide accurate, fast, and amortized estimates of the absolute evidence. We present ENFR: Evidence Networks with Flow-matched References. Building on the Evidence Network framework of Jeffrey & Wandelt (2024), the proposed method first learns Bayes factors with respect to a flow-matched reference, from which the absolute evidence is recovered. We show improvement in estimation of $\mathcal{Z}$ over the flow-based baseline in all applications considered and the best performance of all density estimation methods considered in our benchmarks. We furthermore introduce and provide the proof for an architecture that jointly learns the individual evidences of a set of models $\{\mathcal{M}_i\}$. We demonstrate the performance and scaling behavior of these approaches on data from a variety of scientific applications.
Many of the most important datasets in the physical sciences live natively on the sphere, and these data often carry no preferred position or orientation. Using methods that do not reflect this symmetry leads to far higher simulation requirements, which can be catastrophic for scientific analyses (forcing reliance on cheap, imperfect models that bias results). We present a novel spherical implementation of a gauge equivariant mesh architecture that is specifically designed for the Hierarchical Equal Area isoLatitude Pixelization (\healpix{}) of the sphere. The $\mathrm{SO}(3)$-equivariant convolution is built from steerable filters, parallel transport and spin-weighted features. The architecture supplies the exact symmetry, so the network does not spend training data learning it. The convolution operations are constructed to give provably exact equivariance, given a band-limit. Our full CNNs can achieve SO(3)-invariance error at per-cent-level and lower, which is controlled through well-understood hyperparameters. On benchmarks, our technique gives the most accurate dense gradient regression and competitive cosmic microwave background foreground removal and denoising. The improved simulation efficiency matters most in simulation-based inference (SBI), where high-fidelity simulations are expensive and neural compressors require large training sets, so a fixed budget forces analyses with less accurate simulators. In cosmology lensing SBI, our technique yields the tightest posteriors at matched simulation count. This approach is a novel method that will enable new convolutional neural network analyses, including generative and inference tasks, for processes and data on the sphere.