The Inference Report

Understanding when neural networks learn interpretable representations is central to mechanistic interpretability. We study this through identifiability: when can supervised learning recover the true latent variables underlying the data? We introduce two data generating processes distinguished by their causal direction where either labels cause latents (generative) or latents cause labels (discriminative). When the representation has at least as many dimensions as the latent space, we prove that the optimal encoders are linear in the latents. Conversely, when dimensions are insufficient, identifiability generically fails and encoders become nonlinear. We then show that additional constraints yield recovery of individual latent variables, not just linear mixtures. In the generative setting, non-negative activations and energy efficiency force each neuron to encode exactly one latent, providing a precise account of selective grandmother cell-like responses. In the discriminative setting, when latents are sparse and the true measurement matrix satisfies the spark condition, the learned encoder inherits compressed sensing structure, enabling sparse dictionary learning to perfectly recover individual latents. This constitutes, to our knowledge, the first identifiability result for undercomplete nonlinear independent component analysis. Our framework unifies nonlinear ICA, compressed sensing, and neural network interpretability, delineating exactly when representations transition from nonlinear distortions to linear representations and, under additional constraints or steps, to clean, axis-aligned codes. Theoretical predictions are validated on synthetic benchmarks and on activations drawn from vision models, large language models, and macaque inferotemporal cortex.

We propose the Time-partitioning Gaussian Process Mixture Model (TPGPMM), a Bayesian nonparametric framework for identifying persistent regimes and modelling nonstationary dependence in time series. The model combines an adaptive Pitman--Yor prior on contiguous segmentations with regime-specific Gaussian processes, without imposing Markovian within-regime dependence. Learning the Pitman--Yor discount parameter allows adaptation to homogeneous and heterogeneous regime lengths. A split--merge Metropolis-within-Gibbs sampler with exact Gibbs boundary updates jointly learns change points and model hyperparameters, quantifying uncertainty over segmentations and within-regime dynamics. With an RBF kernel and a fixed, conservative observation-noise level, TPGPMM favours sparse segmentations that distinguish persistent shifts from transient shocks. Simulations demonstrate recovery of the regime structure. Applications to the CBOE Volatility Index (VIX) and the 10-year United States Treasury yield identify regimes aligned with major market and monetary-policy events. Variants that estimate observation noise improve predictive accuracy relative to the main specification, but can produce less sparse or less event-aligned segmentations. A variant with regime-specific means fits better than a fitted stationary Gaussian process and forecasts about as accurately. These results highlight a trade-off between interpretable regime identification and predictive accuracy.

Estimating causal effects on right-censored time-to-event outcomes is challenging under unmeasured confounding. Proximal causal inference addresses unmeasured confounding through proxy variables, but existing proximal survival methods rely largely on restrictive parametric bridge models. We introduce Proximal Survival Bridge learning (PSB), a flexible doubly robust proximal framework for censored outcomes. Building on existing proximal identification of treatment-specific survival curves, the framework yields outcome-bridge (OB), treatment-bridge (TB), and doubly robust (DR) estimators, with two flexible bridge learners: PSB-K, a kernel estimator with a low-rank time decomposition, and PSB-N, a neural estimator trained by a projected maximum-moment-restriction loss. Unlike existing flexible proximal learners for scalar outcomes, PSB directly handles censoring and survival-curve targets. On a standard SEM benchmark, a proximal semi-synthetic ACTG study, and the real-world Framingham study, PSB-K performs strongest on SEM and ACTG, whereas PSB-N is the most stable on Framingham. Additional results with modern causal survival baselines, weakened or invalid proxies, and heavier censoring assess robustness across challenging settings.

This paper investigates the statistical behavior of Stochastic Gradient Descent (SGD) with constant step size under the framework of iterated random functions. Compared to previous studies establishing the convergence of SGD in probability measure, e.g., Wasserstein distance, our approach directly provides the convergence in Euclidean distance by showing the Geometric Moment Contraction (GMC) of SGD. This new convergence can address the non-stationarity of SGD due to fixed initial points and can provide a more refined asymptotic analysis of SGD. Specifically, we prove quenched central limit theorem, invariance principle and $α$-stable convergence for averaged SGD (ASGD) regardless of initialization. Furthermore, we provide a novel perspective to understand the impact of step sizes in SGD by studying its derivative with respect to the step size. The existence of stationary solutions for the derivative processes is shown, enabling the use of interpolation to recover the full-trajectory of SGD. Subsequently, we utilize multiple step sizes and show an enhanced Richardson-Romberg extrapolation with improved bias representation, which brings ASGD estimates closer to the global optimum. Finally, we propose a new online inference method and a bias-reduced variant for the extrapolated ASGD. Empirical confidence intervals are constructed and the coverage probabilities approach the nominal level as $n$ grows.

Deep learning models, particularly Long Short-Term Memory networks, are increasingly used for operational streamflow forecasting, with ensemble spread providing a signal of predictive uncertainty. As climate change increases the frequency of hydroclimatic extremes, these models will increasingly operate under flow conditions that are rare in their historical records. We therefore evaluate whether ensemble spread remains informative under increasing flows, examining both its magnitude relative to error and its ability to rank errors. Prediction error is approximately $2\times$ the corrected spread at ordinary flows and up to $4\times$ at the highest flows, with the discrepancy increasing with flow (log-log slope $1.30$, 95% CI: $[1.20,1.41]$). Under-dispersion is strongest in observed extreme flows beyond the training record, reaching an $18\times$ discrepancy. This is pervasive and not explainable by conditional mean bias. Area Under the Sparsification Error, a standard distribution-free metric of uncertainty ranking, describing how well a model knows when it is uncertain, indicates near-oracle ranking for ensemble spread ($0.071$), yet ordering by predicted discharge alone nearly matches it. Informative uncertainty ranking can coexist with severe under-dispersion. In deployment, error ranking cannot be verified, yet the spread is available, and it is relied on as the uncertainty signal under increasingly extreme conditions, while systematically understating risk, most severely for the extreme events that matter most.

Generative modeling across related distributions raises a question: how can shared structure support accurate generation for a target distribution with limited observations? We develop a statistical framework for diffusion transfer that analyzes a shared pooled score together with a target-specific offset. Under Besov regularity of the pooled density and a stable hierarchical composition model for the offset, we derive convergence rates that separate the costs of learning these components and identify when a simpler offset yields a faster risk-bound rate than separate fitting. Under suitable boundary and regularity assumptions, we prove that the offset remains bounded and jointly regular as diffusion time approaches zero. These results allow the lower time cutoff to vanish polynomially and provide total-variation guarantees for diffusion models as density estimators. Synthetic simulations and image benchmarks provide a proof of concept for the framework.