#StackBounty: #classification #expected-value #risk Binary Classification : Prove that $mathbb{E}_{mathcal{D}_n}left[R_e(h)right] =…

Bounty: 50

this is my first question here 🙂

Problem Statement

Let $h in mathcal{H}$ be a hypothesis to some class of binary classifiers $mathcal{H}$. Show that
$$mathbb{E}_{mathcal{D}_n}left[R_e(h)right] = R(h)$$
where the expectation on the LHS is over all possible training datasets $mathcal{D}_n$ of size $n$.

  1. $R_e(h)$ is the empirical risk of the algorithm over a given dataset $mathcal{D}_n$. It is defined as

$$R_e(h) = frac1nsum_{i=1}^{n}mathcal{L}(x_i, h(x_i))$$

  1. Here $mathcal{L}$ is the loss function for the binary classification
    problem defined as
    $$mathcal{L}(x,h) =
    begin{cases}
    1, & s(x) not= h(x) \
    0, & text{otherwise}
    end{cases}
    $$

  2. $s(x)$ is the system we are trying to model

  3. $R(h)$ is the true risk of the hypothesis $h$

My work

$$R_e(h) = frac1nsum_{i=1}^{n}mathcal{L}(X_i, h(x_i))$$
$$mathbb{E}{mathcal{D}_n}left[R_e(h)right] = int{mathcal{D}n}{R_e(h)p(mathcal{D}_n)}$$
$$ = frac{1}{n}int
{mathcal{D}n}{sum{x_i in mathcal{D}_n}mathcal{L}(x_i, h)p(mathcal{D}_n)}$$

Since I want to manipulate this to convert it to $R(h) = int_{x}{mathcal{L}(x,h)p(x)dx}$, I though of group all $x_i$ out of the above equation. But then I couldn’t find a way to get the term $p(x)$ into the picture and this is where I am stuck.

I am looking for progressive hints that will help me solve this myself. Thanks!


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