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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.

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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.

xyz
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  • Can you explain this maybe with a concrete example? It's not really sinking in. expected risk function Given a fixed theta, integrate over the probability density function f(x) to compute the risk. – bottledatthesource Jun 30 '18 at 05:32
  • @bottledatthesource please do not use answers for asking questions. If you have a question, post it as a question. – Tim Jun 30 '18 at 09:46

1 Answers1

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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 $\int L(y,f(x)) p(x,y) dy dx$.

Ale
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