Markov Filtering

A type of Markov Inference for computing the current Belief State (probability distribution) given previous evidence. $\mathrm{P}\left({\mathrm{X}}_t\mid {\mathrm{e}}_{1:t}\right)$ We want to find this iteratively (via recursion). So we first define $f_{0:k}=P(S_k\mid e_{0:k}).$ Given the above and using the Bayes Rule with Normalization, the base case is $f_{0:0}=\alpha P(e_0\mid S_0)\cdot P(S_0).$ Now the recursive case is $f_{0:k}=\alpha P(e_k\mid S_k){\sum }_{s_{k-1}}P(S_k\mid s_{k-1})\cdot f_{0:(k-1)}$ Apply

• Use the base case to directly calculate the first probability distribution, make sure to normalize the values

• Expand the sum in the recursive case

• Expand the Random Variables $S_k$ in the recursive case with angled brackets

• Then fill in all probabilities

• Calculate and normalize the result

• Filtering 1

• Filtering 2