Simulation-based inference (SBI) offers a flexible and general approach to performing Bayesian inference: In SBI, a neural network is trained on synthetic data simulated from a model and used to rapidly infer posterior distributions for observed data. A key goal for SBI is to achieve accurate inference with as few simulations as possible, especially for expensive simulators. In this work, we address this challenge by repurposing recent probabilistic foundation models for tabular data: We show how tabular foundation models---specifically TabPFN---can be used as pre-trained autoregressive conditional density estimators for SBI. We propose Neural Posterior Estimation with Prior-data Fitted Networks (NPE-PFN) and show that it is competitive with current SBI approaches in terms of accuracy for both benchmark tasks and two complex scientific inverse problems. Crucially, it often substantially outperforms them in terms of simulation efficiency, sometimes requiring orders of magnitude fewer simulations. NPE-PFN eliminates the need for selecting and training an inference network and tuning its hyperparameters. We also show that it exhibits superior robustness to model misspecification and can be scaled to simulation budgets that exceed the context size limit of TabPFN. NPE-PFN provides a new direction for SBI, where training-free, general-purpose inference models offer efficient, easy-to-use, and flexible solutions for a wide range of stochastic inverse problems.

Understanding the generalization properties of large-scale models necessitates incorporating realistic data assumptions into the analysis. Therefore, we consider Principal Component Regression (PCR)—combining principal component analysis and linear regression—on data from a low-dimensional manifold. We present an analysis of PCR when the data is sampled from a spiked covariance model, obtaining fundamental asymptotic guarantees for the generalization risk of this model. Our analysis is based on random matrix theory and allows us to provide guarantees for high-dimensional data. We additionally present an analysis of the distribution shift between training and test data. The results allow us to disentangle the effects of (1) the number of parameters, (2) the data-generating model and, (3) model misspecification on the generalization risk. The use of PCR effectively regularizes the model and prevents the interpolation peak of the double descent. Our theoretical findings are empirically validated in simulation, demonstrating their practical relevance.

In conventional regression analysis, predictions are typically represented as point estimates derived from covariates. The Gaussian Process (GP) offer a kernel-based framework that predicts and quantifies associated uncertainties. However, kernel-based methods often underperform ensemble-based decision tree approaches in regression tasks involving tabular and categorical data. Recently, Recursive Feature Machines (RFMs) were proposed as a novel feature-learning kernel which strengthens the capabilities of kernel machines. In this study, we harness the power of these RFMs in a probabilistic GP-based approach to enhance uncertainty estimation through feature extraction within kernel methods. We employ this learned kernel for in-depth uncertainty analysis. On tabular datasets, our RFM-based method surpasses other leading uncertainty estimation techniques, including NGBoost and CatBoost-ensemble. Additionally, when assessing out-of-distribution performance, we found that boosting-based methods are surpassed by our RFM-based approach.

Kernel principal component analysis (kPCA) is a widely studied method to construct a low-dimensional data representation after a nonlinear transformation. The prevailing method to reconstruct the original input signal from kPCA—an important task for denoising—requires us to solve a supervised learning problem. In this paper, we present an alternative method where the reconstruction follows naturally from the compression step. We first approximate the kernel with random Fourier features. Then, we exploit the fact that the nonlinear transformation is invertible in a certain subdomain. Hence, the nameinvertible kernel PCA (ikPCA). We experiment with different data modalities and show that ikPCA performs similarly to kPCA with supervised reconstruction on denoising tasks, making it a strong alternative.