Conditionally linear-Gaussian state-space model

Assume AC. Let $Z_t$ take values in a standard-Borel space $({\mathsf{Z}}_t,{\mathcal{Z}}_t)$ and let $U_t\in {\mathbb{R}}^{d_t}$ for a finite horizon $0\le t\le T$, where each $d_t$ is a positive integer. For $1\le t\le T$, let $Y_t\in {\mathbb{R}}^{p_t}$ with each $p_t$ a positive integer; observations begin at time $1$. The nonlinear component has initial probability ${\mu }_0$ and transition probability kernels $M_t(z_{t-1},d{z}_t)$; its transition does not depend on $U_{t-1}$. Given the entire nonlinear path $z_{0:T}$, write ${\mathsf{Z}}_{0:t}={\prod }_{s=0}^t{\mathsf{Z}}_s$ with its finite-product sigma-algebra and specify Borel maps $m_0:{\mathsf{Z}}_0\to {\mathbb{R}}^{d_0},P_0:{\mathsf{Z}}_0\to {\mathbb{R}}^{d_0\times d_0},$ and, for $1\le t\le T$, $A_t:{\mathsf{Z}}_{0:t}\to {\mathbb{R}}^{d_t\times d_{t-1}},Q_t:{\mathsf{Z}}_{0:t}\to {\mathbb{R}}^{d_t\times d_t},C_t:{\mathsf{Z}}_{0:t}\to {\mathbb{R}}^{p_t\times d_t},R_t:{\mathsf{Z}}_{0:t}\to {\mathbb{R}}^{p_t\times p_t}\mathrm{.}$ Require $P_0(z_0)$ and $Q_t(z_{0:t})$ to be symmetric positive semidefinite, meaning $v^{T}Pv\ge 0$ for every matching vector $v$, and $R_t(z_{0:t})$ to be symmetric positive definite, meaning $v^T{R}_{t}v>0$ for every nonzero $v$. Conditional on $Z_{0:T}=z_{0:T}$, the linear component and observations have the finite linear-Gaussian model, with the transition and observation equations for $1\le t\le T$: $U_0\sim N(m_0(z_0),P_0(z_0)),U_t=A_t(z_{0:t})U_{t-1}+{\xi }_t,Y_t=C_t(z_{0:t})U_t+{\epsilon }_t,$ where the centered Gaussian noises ${\xi }_t\sim N(0,Q_t(z_{0:t}))$ and ${\epsilon }_t\sim N(0,R_t(z_{0:t}))$ are mutually independent over time and independent of $U_0$, under that conditional law. For every fixed path $z_{0:T}$ these are exactly the hypotheses of Linear-Gaussian state-space models with independent noises with deterministic matrices.

This hierarchical conditionally linear-Gaussian model is the input class for the Rao–Blackwellized filter below: it samples the nonlinear path and keeps the conditional Gaussian law of $U_t$ analytically. It does not include a nonlinear transition whose law depends on $U_{t-1}$, nor does it assert that the marginal nonlinear process remains Markov after conditioning on all observations. Such models need their own marginal transition and backward kernels. A state-space filtering or smoothing law is a data-indexed regular conditional-distribution version as in Filtering, predictive, and smoothing laws in a finite-horizon state-space model; no pointwise uniqueness on null data histories is imposed.