Recursive Bayesian estimation

Recursive Bayesian estimation is a general probabilistic approach for estimating an unknown probability density function recursively over time using incomming measurements and a mathematical process model.

Model

The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a hidden Markov model.

Because of the Markov assumption, the true state is conditionally independent all earlier states given the immediately previous state.

$p({\textbf {x}}_{k}|{\textbf {x}}_{0},...,{\textbf {x}}_{k-1})=p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})$

Similarly the measurement a the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.

$p({\textbf {z}}_{k}|{\textbf {x}}_{0},...,{\textbf {x}}_{k})=p({\textbf {z}}_{k}|{\textbf {x}}_{k})$

Using these assumptions the probability distribution over all states of the HMM can be written simply as:

$p({\textbf {x}}_{0},...,{\textbf {x}}_{k},{\textbf {z}}_{1},...,{\textbf {z}}_{k})=p({\textbf {x}}_{0})\prod _{i=1}^{k}p({\textbf {z}}_{i}|{\textbf {x}}_{i})p({\textbf {x}}_{i}|{\textbf {x}}_{i-1})$

However, when the Kalman filter to estimate the state x the probability distribution of interest is that associated with the current states conditioned on the measurements upto the current timestep. (This is achieved by marginalising out the previous states and dividing by the probability of the measurement set.)

This leads to the predict and update steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is product of the probability distribution associated with the transition from the (k - 1)th timestep to the kth and the probability distribution associated with the previous state, with the true state at (k - 1) integrated out.

$p({\textbf {x}}_{k}|{\textbf {Z}}_{k-1})=\int p({\textbf {x}}_{k}|{\textbf {x}}_{k-1})p({\textbf {x}}_{k-1}|{\textbf {Z}}_{k-1})\,d{\textbf {x}}_{k-1}$

The measurement set upto time t is

${\textbf {Z}}_{t}=\left\{{\textbf {z}}_{1},...,{\textbf {z}}_{t}\right\}$

The probability distribution of updated is proportional to the product of the measurement likelihood and the predicted state.

$p({\textbf {x}}_{k}|{\textbf {Z}}_{k})={\frac {p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {Z}}_{k-1})}{p({\textbf {z}}_{k}|{\textbf {Z}}_{k-1})}}$

The denominator

$p({\textbf {z}}_{k}|{\textbf {Z}}_{k-1})=\int p({\textbf {z}}_{k}|{\textbf {x}}_{k})p({\textbf {x}}_{k}|{\textbf {Z}}_{k-1})d{\textbf {x}}_{k}$

is an unimportant normalisation term.

Applications

• Kalman filter, a optimal recursive Bayesian filter for linear models subject to white noise