Recursive Bayesian estimation

Recursive Bayesian estimation is a general probabilistic approach for estimating an unknown probability density function recursively over time using incoming 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 (HMM).


 * [[Image:HMMKalmanFilterDerivation.png|463-239|Hidden Markov Model]]

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


 * $$p(\textbf{x}_k|\textbf{x}_0,\dots,\textbf{x}_{k-1}) = p(\textbf{x}_k|\textbf{x}_{k-1} )$$

Similarly the measurement at the kth timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.


 * $$p(\textbf{z}_k|\textbf{x}_0,\dots,\textbf{x}_{k}) = p(\textbf{z}_k|\textbf{x}_{k} )$$

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


 * $$p(\textbf{x}_0,\dots,\textbf{x}_k,\textbf{z}_1,\dots,\textbf{z}_k) = p(\textbf{x}_0)\prod_{i=1}^k p(\textbf{z}_i|\textbf{x}_i)p(\textbf{x}_i|\textbf{x}_{i-1})$$

However, when using the Kalman filter to estimate the state x, the probability distribution of interest is associated with the current states conditioned on the measurements up to 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 up to time t is
 * $$ \textbf{Z}_{t} = \left \{ \textbf{z}_{1},\dots,\textbf{z}_{t} \right \} $$

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 a less significant normalisation term.

Applications

 * Kalman filter, a recursive Bayesian filter for multivariate normal distributions
 * Particle filter, a sequential Monte Carlo (SMC) based technique, which models the PDF using a set of discrete points
 * Grid-based estimators, which subdivide the PDF into a discrete grid