Where is the error in this naive derivation of the Kalman filter?

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?

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