I'm taking a communication theory course and I have some confusion regarding the maximum a posteriori rule.
In my notes it says that,
Consider a communication system where the transmit symbols are x from a choice of J possible symbols. The received symbols are y, from a choice of K possible symbols. The maximum a posteriori rule says that the receiver makes the following estimate of x,
It goes on to say that this can be simplified via Baye's Rule as follows,
That's my confusion. I've tried applying Bayes' Rule, however I seem to have an extra P(yk) on the bottom line. Is it somehow indpendant?
We have $$ \operatorname{argmax}_{x_j} P(x_j, y_k) = \operatorname{argmax}_{x_j} \frac{P(x_j, y_k)}{P(y_k)}$$ as the denominator does not depend on $x_j$.