Recursive Bayesian estimation: Difference between revisions

Revision as of 17:22, 21 March 2007

Revision as of 02:18, 22 May 2007

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{{Merge|Sequential_bayesian_filtering|date=February 2007}}

'''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 ==

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

:<math>p(\textbf{x}_k|\textbf{x}_0,...\dots,\textbf{x}_{k-1}) = p(\textbf{x}_k|\textbf{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(\textbf{z}_k|\textbf{x}_0,...\dots,\textbf{x}_{k}) = p(\textbf{z}_k|\textbf{x}_{k} )</math>

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

:<math>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})</math>

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.)

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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.