Recursive Bayesian estimation: Difference between revisions

Revision as of 01:39, 20 November 2005

Revision as of 15:20, 22 January 2006

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:<math>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})</math>

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 uptoup 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 ''k''th and the probability distribution associated with the previous state, with the true state at (''k'' - 1) integrated out.

:<math> 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} </math>

The measurement set uptoup to time ''t'' is

:<math> \textbf{Z}_{t} = \left \{ \textbf{z}_{1},...,\textbf{z}_{t} \right \} </math>