Recursive Bayesian estimation: Difference between revisions

Revision as of 01:46, 1 May 2008

Revision as of 02:41, 1 May 2008

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:<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 up to time ''t'' is

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

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

:<math> p(\textbf{x}_k|\textbf{Z}_{k}) = \frac{p(\textbf{zZ}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1})}{p(\textbf{zZ}_k|\textbf{Z}_{k-1})} </math>

= \alpha\,p(\textbf{Z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1})

</math>

The denominator

:<math>p(\textbf{zZ}_k|\textbf{Z}_{k-1}) = \int p(\textbf{zZ}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1}) d\textbf{x}_k</math>

is constant relative to <math>x</math>, so we can always substitute it for a coefficient <math>\alpha</math>, which can usually be ignored in practice. The numerator can be calculated and then simply normalized, since its integral must be unitary.

is a less significant normalisation term.

== Applications ==