I just want to get clear conceptually about the difference between detection and estimation in terms of a communication problem. Suppose I have a source and a destination. The source transmits binary symbols (1) $x_1$ and (0) $x_0$. Our job is to detect which symbol has been transmitted. As per detection theory, I can have two hypotheses:

$H_1:y=0+n,$ $H_2:y=1+n$ now if I know the noise distribution as $f(n)$, I can design a detection rule using log likelihood function $$f(y|x_0)>f(y|x_1): \text{The detected symbol is} 0$$ $$f(y|x_0)<f(y|x_1): \text{The detected symbol is} 1$$

Now can I also detect the symbol using ML estimation? For example, suppose I want to detect the parameter which can have either a value 0 or 1. Now the output is observation and If I set $\frac{\partial (y|x)}{\partial x}\bigg{\vert}_{x=\hat{x}(y)}=0$, where I have assumed no knowledge about the input probability distribution of the input symbols $x_0$ and $x_1$. The rule is to derive $\hat{x}(y)$ and find its Euclidean distance with $x_0$ and $x_1$ and assign the value to the symbol with smaller Euclidean distance value. Can detection also happen this way? I am confused as to why we can't detect using estimation procedures?


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