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I'm having trouble understanding the notation $p(y|H_1)$ in the following example:

What we have is a variable $Y$ that we're trying to estimate. Our hypotheses are $H_0$ and $H_1$ which both have apriori probabilities $P_0$ and $P_1$ respectively.

Now my book says that the relationship between the hypotheses and the observed quantity $Y$ is given in the form of a probabilistic "measurement model": $$P_{Y|H}(y|H_0)\quad \& \quad P_{Y|H}(y|H_1)$$

Now what do these signify, in english, or expanded probability notation?

My best bet is that $P(y|H_0) = P(y = 0|H_0) + P(y=1|H_0)$, considering that $Y$ is binary, can take only 0 and 1.

From this the book then arrives at the optimal decision rule which contains this expression.

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  • $\begingroup$ That is incorrect - it is not the sum of the two probabilities. $P(y|H_0)$ is considered to be a function of $y$, in the same way that you probably have used the generic notation $f(y)$ for any kind of mathematics. The value that the function returns depends only on $y$. $P(y|H_1)$ is a different function from $P(y|H_0)$. $\endgroup$ – Dean Nov 21 '17 at 0:28
  • $\begingroup$ Oh ok! So what does this quantity mean in english? It should be something like (******) given that the hypothesis $H_0$ is true, correct? $\endgroup$ – Rami Awar Nov 21 '17 at 8:02
  • $\begingroup$ Presumably the probability that $Y=y$ given that the event $H_0$ occurs (i.e. that the null hypothesis $H_0$ is true) $\endgroup$ – Henry Nov 21 '17 at 12:12
  • $\begingroup$ @Henry Do you want to mark your comment as the answer? I got it now, thanks. $\endgroup$ – Rami Awar Nov 21 '17 at 14:45
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As requested in comments:

Presumably $P_{Y\mid H}(y\mid H_0)$ represents the probability that $Y=y$ given that the event $H_0$ occurs (i.e. that the null hypothesis $H_0$ is true)

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