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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,

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It goes on to say that this can be simplified via Baye's Rule as follows, enter image description here

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?

AlfroJang80
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1 Answers1

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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 $x_j$.

blat
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  • Hmm. Why is the denominator independant of x_j? Surely, doesn't the probability that you receive a certain y_k depend on which x_j was sent? – AlfroJang80 Mar 01 '20 at 16:56
  • @AlfroJang80 It's independent, because $x_j$ is not present in this term. It's so to say the prior probability of observing $y_k$ without any knowledge about $x_j$. – blat Mar 02 '20 at 09:11