Blog

For a suitable submartingale, Doob–Meyer separates predictable increasing drift from a martingale. The split depends on the information available, and does not remove every form of predictable variation.

Finite-dimensional projections can introduce spectral points absent from the full Koopman operator. A worked compression example separates spectral pollution, continuous spectrum, and eigenvalue sensitivity.

A conceptual map of generation: subword tokenization → learned embeddings → attention-based next-token prediction. Concrete diagrams show what each stage buys you and what it assumes.

Normalizing flows turn density estimation into calculus: invertible transforms + Jacobians give exact log-likelihoods. This post builds the change-of-variables view and why architectures like RealNVP make it tractable.

Pseudospectra quantify eigenvalue sensitivity at a chosen error tolerance. A worked matrix example explains their perturbation guarantee and why it does not automatically validate a Koopman approximation.

GARCH models conditional variance; adaptive conformal inference adjusts coverage over time. Their different targets explain what each can—and cannot—do at a regime boundary.