Markov Smoothing

A type of Markov Inference for estimating past states via all of the given evidence (essential for learning, why?). $\mathrm{P}\left({\mathrm{X}}_k\mid {\mathrm{e}}_{1:t}\right)\text{ for }0\le k<t$ It works similar to Filtering where you go through all of the evidence iteratively with recursion. Here however the recursion is backwards and goes from the current state to a past state. Of course the past state must not be the first state, so we also do Filtering to get the forward part.

$\mathrm{P}\left({\mathrm{X}}_k\mid {\mathrm{e}}_{1:t}\right)=\alpha \cdot f_{1:k}\cdot b_{k+1:t}$

The backward message

$b_{k+1:t}=P\left(e_{k+1:t}\mid X_k\right)$ $={\sum }_{{\mathrm{x}}_{k+1}}P\left({\mathrm{e}}_{k+1}\mid {\mathrm{x}}_{k+1}\right)\cdot P\left({\mathrm{e}}_{k+2:t}\mid {\mathrm{x}}_{k+1}\right)\cdot P\left({\mathrm{x}}_{k+1}\mid {\mathrm{X}}_k\right)$ We get $P\left(e_{k+1}\mid x_{k+1}\right)$ by the Sensor Model. And $\mathrm{P}\left({\mathrm{x}}_{k+1}\mid {\mathrm{X}}_k\right)$ by the Transition Model.

The recursive call $P\left({\mathrm{e}}_{k+2:t}\mid {\mathrm{x}}_{k+1}\right)$ goes one more step into the past.