$($a$)$ Describe and compare Maximum A Posteriori $($MAP$)$ and maximum likelihood $($ML$)$ decision rules in digital communication.

$($b$)$ Consider a mathematical model of discrete memoryless channel $($DMC$)$ using $3$ possible input messages $x=\{1,2,3\}$ and $3$ possible output symbols $Y=\{a,b,c\}$ as shown below. It is characterized by the following matrix

$P(x,Y)=([0\mathrm{.}18,0\mathrm{.}05,0\mathrm{.}02],[0\mathrm{.}09,0\mathrm{.}25,0\mathrm{.}02],[0\mathrm{.}03,0\mathrm{.}2,0\mathrm{.}16])$

$($i$)$ Compute the channel transition matrix $P(\frac{Y}{x}).$

$($ii$)$ Given the received symbol $Y=c,$ what is the decision of the optimum receiver $($using the MAP rule$)$ regarding which message was transmitted?