# #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 ðŸ™‚

## 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!

Get this bounty!!!

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