Editing Kalman filter (section)

== Relationship to recursive Bayesian estimation ==
The Kalman filter can be presented as one of the simplest [[dynamic Bayesian network]]s. The Kalman filter calculates estimates of the true values of states recursively over time using incoming measurements and a mathematical process model. Similarly, [[recursive Bayesian estimation]] calculates [[density estimation|estimates]] of an unknown [[probability density function]] (PDF) recursively over time using incoming measurements and a mathematical process model.<ref>{{cite journal|first1=C. Johan |last1=Masreliez|first2= R D|last2= Martin |year=1977|title=Robust Bayesian estimation for the linear model and robustifying the Kalman filter|journal= IEEE Transactions on Automatic Control|volume= 22|issue= 3|pages= 361–371|doi=10.1109/TAC.1977.1101538|author-link1= C. Johan Masreliez}}</ref>

In recursive Bayesian estimation, the true state is assumed to be an unobserved [[Markov process]], and the measurements are the observed states of a hidden Markov model (HMM).
[[File:HMM Kalman Filter Derivation.svg|class=skin-invert-image|center|hidden markov model]]

Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
:<math>p(\mathbf{x}_k\mid \mathbf{x}_0,\dots,\mathbf{x}_{k-1}) = p(\mathbf{x}_k\mid \mathbf{x}_{k-1})</math>

Similarly, the measurement at the ''k''-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
:<math>p(\mathbf{z}_k\mid\mathbf{x}_0,\dots,\mathbf{x}_k) = p(\mathbf{z}_k\mid \mathbf{x}_k )</math>

Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
:<math>p\left(\mathbf{x}_0, \dots, \mathbf{x}_k, \mathbf{z}_1, \dots, \mathbf{z}_k\right) = p\left(\mathbf{x}_0\right)\prod_{i=1}^k p\left(\mathbf{z}_i \mid \mathbf{x}_i\right)p\left(\mathbf{x}_i \mid \mathbf{x}_{i-1}\right)</math>

However, when a Kalman filter is used to estimate the state '''x''', the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep. This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set.

This results in the ''predict'' and ''update'' phases of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (''k''&nbsp;−&nbsp;1)-th timestep to the ''k''-th and the probability distribution associated with the previous state, over all possible <math>x_{k-1}</math>.

:<math>p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right) = \int p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) p\left(\mathbf{x}_{k-1} \mid \mathbf{Z}_{k-1}\right)\, d\mathbf{x}_{k-1}</math>

The measurement set up to time ''t'' is
:<math>\mathbf{Z}_t = \left\{\mathbf{z}_1, \dots, \mathbf{z}_t\right\}</math>

The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
:<math>p\left(\mathbf{x}_k \mid \mathbf{Z}_k\right) = \frac{p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right)}{p\left(\mathbf{z}_k \mid \mathbf{Z}_{k-1}\right)}</math>

The denominator
:<math>p\left(\mathbf{z}_k \mid \mathbf{Z}_{k-1}\right) = \int p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) p\left(\mathbf{x}_k \mid \mathbf{Z}_{k-1}\right)\, d\mathbf{x}_k</math>

is a normalization term.

The remaining probability density functions are
:<math>\begin{align}
      p\left(\mathbf{x}_k \mid \mathbf{x}_{k-1}\right) &= \mathcal{N}\left(\mathbf{F}_k\mathbf{x}_{k-1}, \mathbf{Q}_k\right) \\
          p\left(\mathbf{z}_k \mid \mathbf{x}_k\right) &= \mathcal{N}\left(\mathbf{H}_k\mathbf{x}_k, \mathbf{R}_k\right) \\
  p\left(\mathbf{x}_{k-1} \mid \mathbf{Z}_{k-1}\right) &= \mathcal{N}\left(\hat{\mathbf{x}}_{k-1}, \mathbf{P}_{k-1}\right)
\end{align}</math>

The PDF at the previous timestep is assumed inductively to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for <math>\mathbf{x}_k</math>  given the measurements <math>\mathbf{Z}_k</math> is the Kalman filter estimate.