Fixed-interval smoothing law in a linear-Gaussian model

Assume AC and fix a finite-horizon linear-Gaussian state-space model (Linear-Gaussian state-space models with independent noises) with states $X_0,\dots ,X_T$, observations $Y_1,\dots ,Y_T$, and Borel observation space ${\mathsf{Y}}_{1:T}={\mathbb{R}}^{p_1}\times \cdots \times {\mathbb{R}}^{p_T}$, with the empty product a singleton when $T=0$. For $0\le k\le T$, a fixed-interval smoothing kernel for $X_k$ is an everywhere-defined probability kernel ${\psi }_{k\mathrm{\mid }T}:{\mathsf{Y}}_{1:T}⟶{\mathbb{R}}^{d_k}$ whose composition with the full observation vector $Y_{1:T}$ is a regular conditional distribution of $X_k$ given $\sigma (Y_{1:T})$. Thus it is the single-time marginal smoothing law after all observations through time $T$ are supplied. The full-path smoothing kernel and its version convention are defined in Filtering, predictive, and smoothing laws in a finite-horizon state-space model.

This definition does not assert that such a kernel exists or is unique at observation values in a null set. The theorem on this page constructs a Gaussian version under its explicit forecast and innovation assumptions.