Recursive Bayesian estimation

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In robotics[edit]




The true state x is assumed to be an unobserved Markov process, and the measurements z are the observed states of a Hidden Markov Model (HMM). The following picture presents a Bayesian Network of a HMM.

Hidden Markov Model


p(\textbf{x}_k|\textbf{x}_{k-1},\textbf{x}_{k-2},\dots,\textbf{x}_0) = p(\textbf{x}_k|\textbf{x}_{k-1} )


p(\textbf{z}_k|\textbf{x}_k,\textbf{x}_{k-1},\dots,\textbf{x}_{0}) = p(\textbf{z}_k|\textbf{x}_{k} )


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}).



 p(\textbf{x}_k|\textbf{z}_{1:k-1}) = \int p(\textbf{x}_k | \textbf{x}_{k-1}) p(\textbf{x}_{k-1} | \textbf{z}_{1:k-1} )  \, d\textbf{x}_{k-1}


 p(\textbf{x}_k|\textbf{z}_{1:k}) = \frac{p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{z}_{1:k-1})}{p(\textbf{z}_k|\textbf{z}_{1:k-1})} 

The denominator

p(\textbf{z}_k|\textbf{z}_{1:k-1}) = \int p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{z}_{1:k-1}) d\textbf{x}_{k}

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


Sequential Bayesian filtering[edit]


The method is named:

when we estimate the current value given past and current observations,
when estimating past values given present and past measures, and


External links[edit]