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I'm attempting to derive the Kalman filter using basic facts about normally distributed random variables. Can anyone complete this derivation? Here's what I have so far (there could be some mistakes):


Let's assume that \begin{align} x_{k+1} &= \Phi_k x_k + w_k. \end{align} We don't know with certainty the value of $x_k$. Rather, we view $x_k$ as a normally distributed random variable with a known mean $\mu_k$ and known variance $\Sigma_k$. The noise vector $w_k$ is also not known with certainty -- rather, $w_k$ is a normally distributed random variable with mean $0$ and variance $\Omega_k$. The matrix $\Phi_k$ is assumed to be known. We assume $w_k$ and $x_k$ are independent.

What do we know about the value of $x_{k+1}$? From probability theory (see this question), we know that the random variable $x_{k+1}$ is normally distributed with mean $\Phi_k \mu_k$ and variance $\Phi_k \Sigma_k \Phi_k^T + \Omega_k$. So far, this is the most we can say about $x_{k+1}$.

Next, what if we are given (as a measurement from a sensor) the value of the random variable \begin{equation} z_{k+1} = H_{k+1} x_{k+1} + v_{k+1}, \end{equation} where the random variable $v_{k+1}$ is normally distributed with mean $0$ and variance $\Gamma_{k+1}$ and is independent of $x_{k+1}$. We assume the matrix $H_{k+1}$ is known.

First of all, note that $z_{k+1}$ is normally distributed with mean $H_{k+1} \Phi_k \mu_k$ and variance $H_{k+1}(\Phi_k \Sigma_k \Phi_k^T + \Omega_k) H_{k+1}^T + \Gamma_k$. When we are given the value of $z_{k+1}$, we must update our beliefs about the value of $x_{k+1}$

Question: What are the (conditional) mean and variance of $x_{k+1}$ given the value of $z_{k+1}$?

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  • $\begingroup$ It looks to me like your work so far is correct. In your first phase, you have found $p(x_{k+1} \mid z_{1:k})$, where $z_{1:k}$ refers to all past measurements. With the measurement likelihood $p(z_{k+1} \mid x_{k+1})$, we have a joint distribution $p(x_{k+1}, z_{k+1} \mid z_{1:k}) = p(z_{k+1} \mid x_{k+1}) p(x_{k+1} \mid z_{1:k})$, which will also be normal. Now you should be able to apply standard techniques to this joint normal distribution to find $p(x_{k+1} \mid z_{1:k+1})$, which is your answer. $\endgroup$ – mikkola May 30 '16 at 19:03

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