# Kalman filter: state estimate

Assume we have a linear state-space model: $$z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + w_{k}$$

If I understand correctly, having observations $$z_{0}, \dots, z_{k}$$, in filtering problem the goal is to get the estimate $$\hat{x}_{k|k}$$ of $$E[x_{k}|z_{k}, \dots, z_{0}]$$.

The question: why do we call $$\hat{x}_{k|k}$$ the state estimate if state is itself random?

In wiki, it is even more confusing:

In what follows, the notation $$\hat{\mathbf{x}}_{n|m}$$ represents the estimate of $$x$$ at time $$n$$ given observations up to and including at time $$m \leq n$$.

Each measurement is associated with a certain time index and this then also fixes the associated estimate of the state with respect to an index. For example $$\hat{x}_{k|k} = E[x_{k}|z_{k}, \dots, z_{0}]$$ is a different problem compared to $$\hat{x}_{k+1|k} = E[x_{k+1}|z_{k}, \dots, z_{0}]$$. Namely, in that case the measurement $$z_{k+1}$$ is not used, so during that time step one can only predict. For example if $$w_k\sim\mathcal{N}(0,W)$$ then the expected value of the prediction would yield $$\hat{x}_{k+1|k} = F\,\hat{x}_{k|k}$$.
Also note that the state itself is a stochastic process, which is does not have to be completely "random" since it can be modeled by $$x_{k} = F\,x_{k-1} + w_{k}$$. Only if $$F=0$$ would the state inherit all the stochastic properties from $$w_k$$.
• @ABK How do you define "random"? Namely, each $v_k$ and $w_k$ are usually a sample from a certain distribution, independent of time. And frequently is assumed that this distribution is zero mean Gaussian white noise. – Kwin van der Veen Nov 27 '19 at 12:19
• @ABK as a couter example would you consider $x_k=k$ random? Also, it would be a possibility that $w_k=0$ for all $k$, in which case $x_k$ would be deterministic if you would know the initial conditions. – Kwin van der Veen Nov 27 '19 at 12:35