Recursive Bayesian estimation: Difference between revisions

Revision as of 03:11, 22 January 2008

Revision as of 01:40, 1 May 2008

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

Because of the Markov assumption, the probability of the current true state isgiven conditionallythe independentimmediately ofprevious allone earlieris statesconditionally givenindependent of the immediatelyother previousearlier statestates.

:<math>p(\textbf{x}_k|\textbf{x}_0,\dots_{k-1},\textbf{x}_{k-12},\dots,\textbf{x}_0) = p(\textbf{x}_k|\textbf{x}_{k-1} )</math>

Similarly, the measurement at the ''k''-th timestep is dependent only upon the current state, andso is conditionally independent of all other states given the current state.

:<math>p(\textbf{z}_k|\textbf{x}_0_k,\textbf{x}_{k-1},\dots,\textbf{x}_{k0}) = 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: