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→Model: No major style guide (APA, MLS, CMS) permits such colon usage. All indicate that when used in a sentence a colon must be preceded by an independent clause. |
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:<math>p(\textbf{z}_k|\textbf{x}_k,\textbf{x}_{k-1},\dots,\textbf{x}_{0}) = 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>
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Sequential Bayesian filtering is the extension of the Bayesian estimation for the case when the observed value changes in time. It is a method to estimate the real value of an observed variable that evolves in time.
There are several variations:
;filtering: when estimating the ''current'' value given past and current observations,
;[[smoothing problem|smoothing]]: when estimating ''past'' values given past and current observations, and
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