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In the following example, I have a question about the meaning of the letter $y$ below the $\max$ symbol.

I know that $\max P(y\,|\,x)$ is simply to find the maximum value of $P(y\,|\,x)$, probability of $y$ given $x$.

What is the role of the letter $y$ below the $\max$ symbol?

$$\max_y P(y\,|\,x)$$

soupless
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dreamer58
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2 Answers2

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The $y$ under $\max$ indicates that you want to know how large you can make the expression by choosing $y$ (but $x$ is fixed). So the value of $\max\limits_y P(y\mid x)$ depends on what $x$ is (in other words, it is a function of $x$). This is typically different from the overall maximum, i.e. if you were allowed to choose both $x$ and $y$.

Especially Lime
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  • Thank you for your answer, but I'm confused by the 2nd sentence. So, does the value of the expression depend on what $y$ is? (i.e. it is a function of $y$, not $x$) – dreamer58 Aug 06 '21 at 06:34
  • @dreamer58 No. Once you know $x$, the value of $y$ is whatever makes $P(y\mid x)$ as large as possible. – Especially Lime Aug 06 '21 at 07:05
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It says you want to maximize over $y$.

So what should $y$ be, given a fixed $x$, so that the function $P(y|x)$ is maximized.

Steven01123581321
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