The Kalman filter is a method of predicting the future state of a linear state space system based on the previous ones.
A linear, discrete-time, stationary, state-space model is a pair of real valued stochastic processes $\{X_t \}_{t \in \mathbb{N}},\{Y_t\}_{t \in \mathbb{N}}$ that obey the recursive equations $$ \begin{cases} X_{t+1} = F X_t + v_t \ &t=0,1,2\dots \\ Y_t = H X_t + w_t \ &t=0,1,2\dots \end{cases} $$ where:
- $F \in \mathbb{R}, H \in \mathbb{R}$
- $v_t, w_t$ are random variables (additive noise) which admit a PDF (Probability Density Function);
The noise terms are zero-mean and: $$ \forall t_1 \neq t_2 : \ v_{t_1} \perp v_{t_2}, w_{t_1} \perp w_{t_2} \\ \forall t_1, t_2 : \ v_{t_1} \perp w_{t_2} \\ \forall t : \ E[v_t^2] = Q, E[w_t^2] = R $$ where the simbol $X \perp Y$ means that $X$ and $Y$ are independent and $Q,R$ are assumed to be positive real numbers. The initial condition of the recursion $x_0$ is a fixed real number.
Setting $Y^t = \{ Y_0, \dots , Y_t \}$The Kalman filter tells us that
$$E[X_t | Y^{t-1}] = \alpha E[X_{t-1} | Y^{t-2}] + \beta Y_{t-1} $$
where $\alpha$ and $\beta$ are carefully chosen and depend on the parameters of the linear model under consideration.
My question:
If I where to compute $E[X_t | Y^{t-1}]$ I would notice that $X_t = F X_{t-1} + v_{t-1} $ and $X_{t-1} = Y_{t-1}/ H - w_{t-1}/H$ so
$$X_t = F X_{t-1} + v_{t-1} = F (Y_{t-1}/ H - w_{t-1}/H) + v_{t-1} $$
and because $w_{t-1}, v_{t-1}$ are zero mean indipendent random variables
$$E[X_t | Y^{t-1}] = E[ F (Y_{t-1}/ H - w_{t-1}/H) + v_{t-1} | Y^{t-1}] = E[F Y_{t-1}/ H | Y^{t-1}] = F Y_{t-1}/ H $$
(I have also assumed $w_t$ is symmetric) then obviously by doing the same calculations $E[X_{t+1} | Y^{t}] = F Y_{t}/ H$.
There is no way to combine these two conditional expectations to obtain the recursive formula proven by Kalman. Where is my mistake?