Markov Smoothing

A type of Markov Inference for estimating past states via all of the given evidence (essential for learning, why?).

P(Xk∣e1:t) for 0≤k<t\mathrm{P}\left(\mathrm{X}_k \mid \mathrm{e}_{1: t}\right) \text { for } 0 \leq 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.

P(Xk∣e1:t)=α⋅f1:k⋅bk+1:t\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

bk+1:t=P(ek+1:t∣Xk)b_{k+1: t}=P\left(e_{k+1: t} \mid X_k\right) =∑xk+1P(ek+1∣xk+1)⋅P(ek+2:t∣xk+1)⋅P(xk+1∣Xk)=\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(ek+1∣xk+1)P\left(e_{k+1} \mid x_{k+1}\right)

by the Sensor Model. And

P(xk+1∣Xk)\mathrm{P}\left(\mathrm{x}_{k+1} \mid \mathrm{X}_k\right)

by the Transition Model.

The recursive call

P(ek+2:t∣xk+1)P\left(\mathrm{e}_{k+2: t} \mid \mathrm{x}_{k+1}\right)

goes one more step into the past.