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I was going through the following text and actually i could not resources to understand the same online.

If anyone could explain or point me out to any resource to understand the same that would be helpful.

enter image description here

To get Risk function corresponding to loss function ; its is multiplied to PMF (I am not sure, i could be wrong.)

Why is that multiplied.

And in the formula why are we integrating w.r.t dy?

PS: Explanation or any resource explaining the formula would be helpful.

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  • $\begingroup$ Can you explain this maybe with a concrete example? It's not really sinking in. ![expected risk function](i.stack.imgur.com/Zw4Gc.png) Given a fixed theta, integrate over the probability density function f(x) to compute the risk. $\endgroup$ – bottledatthesource Jun 30 '18 at 5:32
  • $\begingroup$ @bottledatthesource please do not use answers for asking questions. If you have a question, post it as a question. $\endgroup$ – Tim Jun 30 '18 at 9:46
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The risk function is the expected loss. In your case the loss function is a random variable that depends on the variable you want to predict (target variable) $Y \in \mathcal{Y}$ and on some function $f(\cdot)$ of the random variable $X \in \mathcal{X}$ (the data you may observe). You are dealing with uncertainty on the realization of both $Y$ and of $X$.

For continuous random variables to compute the expectation of $L(Y,f(X))$ you need the joint density function of $(X,Y)$, hence the expectation is

$ E[L(Y,f(X))] = \int_{\mathcal{X}}\int_{\mathcal{Y}} L(y,f(x)) p(x,y) dy dx $

where the notation is simplified as $\int L(y,f(x)) p(x,y) dy dx$.

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