Variational Filtering, Rebuilt From the Linear Case

Thu, 30 Apr 2026 13:15:00 +0530

This week started by rebuilding the variational Bayesian filtering experiments around a scalar linear-Gaussian state-space model. The goal was not to win on a toy problem. The goal was to make the mechanics testable before asking nonlinear questions:

\[ z_t = z_{t-1} + w_t,\quad w_t \sim \mathcal{N}(0, Q) \]\[ y_t = x_t z_t + v_t,\quad v_t \sim \mathcal{N}(0, R) \]

The filter carried a strict online marginal \(q^F_t(z_t)\) plus an edge/backward conditional \(q^B_t(z_{t-1} \mid z_t)\). That made the posterior edge factor explicit:

Calibration on mlbot.blog

Variational Filtering, Rebuilt From the Linear Case