# How to calculate the Bayesian Risk Classifier

I'm not exactly sure how to calculate the Bayesian risk Classifier $$L(r^*)$$ for $$Y\in\{ 0,1 \}$$.

For this scenario, assume:

$$X\in\mathbb{X}=[0,1],Y\in\{ 0,1 \}$$

$$\pi_y=P(Y=y)=1/2$$ for $$y\in{0,1}$$

Conditional distributions $$[X|Y=y]$$ characterised by:

$$f(x|Y=0)=2-2x$$ and $$f(x|Y=1)=2x$$.

I know the classification loss function for a generic classifier $$r:\mathbb{X}\longrightarrow\{0,1\}$$ is $$\ell(x,y,r(x)=[[r(x)\neq y]]$$ where $$\ell:\mathbb{X}\times\{0,1\}\times\{0,1\}$$.

I'm also aware that associated risk is $$L(r)=E([[r(X)\neq Y]])$$ which is equivalent to $$P(r(X)\neq Y)$$ and that $$L(r)\geq L(r^*)$$.

In the binary classification case, $$L(r^*)=E(min\{\tau_1(X),1-\tau_1(X)\}=1/2-1/2E(|2\tau_1(X)-1|)$$, but I'm not exactly sure how to go from that to the $$Y\in\{ 0,1 \}$$ case with the stated PDFs.

• The question is unclear. What do you need to find? Do you need to show that the Bayes classifier $r^*(x)=\begin{cases}1\quad \tau_1(x)>1/2\\0\quad otherwise\end{cases}$ is optimal? Oct 27 at 8:27