Does different the initial prior result in the same posterior of a Kalman filter as time goes to infinity, why?

Question

Let $p(x_0) \in \{p_i(x_0)\colon i\in {\mathcal I}\}=:{\mathcal P}$ be the prior of a discrete-time Kalman filter, where ${\mathcal P}$ is the family of nondegenerated Gaussian distributions. Then firstly, does the following limit exist (l.i.m. means covergence in the mean square sense) for every $i \in {\mathcal I}$?

\begin{equation} \lim_{t \to \infty} \|p_i(x_t|y_t,\ldots,y_1)-p_j(x_t|y_t,\ldots,y_1)\| = 0, \quad \forall i,j \in {\mathcal I} \end{equation} where $y_t$'s are observations. $p_i$ means using $p_i(x_0)$ as the prior. $\|\cdot\|$ denotes the function norm.

Thank you very much!

Answer 1

This answer is based on the book "Optimal and Robust Estimation" by Lewis, Xie and Popa. Let's say you have this a system with the dynamics: $$x_{k+1}=A_kx_k+B_ku_k+G_kw_k$$ Where $u_k$ is the input and $w_k$ is a Gaussian noise. Also, let the observation equation be: $$z_k=H_kx_k+v_k$$ Now again, $v_k$ is a Gaussian noise process. Your prior for the states would traditionaly be also Gaussian, i.e.: $$x_0\sim(\bar{x}_0,P_{x0})$$ You initialize the system like $$P_0=P_{x0},\;\hat{x}_0=\bar{x}_0$$ From now on, a minus as a superscript indicates the estimate without measurement correction, right? Well in the "a priori recursive formulation" the estimates for the mean of the states (without measurements) and the "Kalman Gain" $K_k$ are given by: $$x^{\mathbf{-}}_{k+1}=A(I-K_kH_k)x^{\mathbf{-}}_{k}+B_ku_k+A_kK_kz_k$$ $$K_k=P^{\mathbf{-}}_{k}H_k^T(H_kP^{\mathbf{-}}_{k}H^T_k+R_k)$$ $P^{\mathbf{-}}_{k}$, the estimate of the covariance in the step $k$, is calculated through: $$P_{k+1}^{\mathbf{-}}=A[P^{\mathbf{-}}_{k}-P^{\mathbf{-}}_{k}H_k^T(H_kP^{\mathbf{-}}_{k}H_k^T+R_k)^{-1}HP^{\mathbf{-}}_{k}]A_k^T+G_kQ_kG^T_k$$ Here, $R_k$ is the covariance matrix of $v_k$ and $Q_k$ the one of $w_k$. After each observation you perform the "measurment update". Now, under this formulation, we can go to relevan part of theorem 2.3 in section 2.5.3. In this theorem the processes are assumed to be stationary, so the dynamic matrices and the noise covariances are fixed (independent of $k$), in such conditions

Let $\sqrt{Q}$ be a square root of the process noise covariance so that $Q=\sqrt{Q}\sqrt{Q}^T\geq 0$, and let the measurement noise have $R>0$. Suppose $(A,G\sqrt{Q})$ is reachable. Then $(A,H)$ is detectable if and only if

1. There is a unique positive definite limiting solution $P$ to the states covariance equation (2.61 in the book), which is independent of the prior $P_0$. Furthermore, $P$ is the unique positive definite solution to the algebraic Riccati equation. ...

I hope this answers your question, dear Ryan.