Problem $4.$ A robot navigates in a corridor with $8$ segments as follows.

When the position of a robot is uncertain, the degree of belief about its position is expressed probabilistically, referred to as $\backslash (\backslash$operatorname$\{$Bel$\}($x$)=$f$($x$)\backslash ).$ Suppose the initial state $($position$)$ of the robot is given by the following belief representation:

Prior Belief: $\backslash (\backslash$operatorname$\{$Bel$\}($x$)=$f$($x$)\backslash )$

The robot takes a measurement $\backslash ($ z $\backslash )$ about its position by a sensor$.$ However, due to sensor errors, the measured value cannot be fully trusted. Instead, the evidence must be considered probabilistically by the following likelihood model.

For example, if the robot takes a measurement $\backslash ($ z$=2\backslash ),$ the likelihood is calculated as

Likelihood: $\backslash ($ f$($z$=2\backslash$mid x$)\backslash )$

Using this result, obtain the updated belief $($posterior belief$)\backslash ($ f$($x $\backslash$mid z$=2\backslash ))$ by combining the sensor measurement. $($Use the Bayes filter.$)$