Bayesian expected loss (Risk) I'm reading the book "Pattern Classifications" from Duda, Hart & Stork. In chapter 2, they discuss about Bayesian expected loss (risk) but they never mention what happens when there's a rejection cost. This is what I think is correct, but I don't know:
$\omega_1, \omega_2, \dots, \omega_c$: states of nature.
$\alpha_1, \alpha_2, \dots, \alpha_a$: possible actions.
$\lambda(\alpha_i|\omega_j)=\lambda_{ij}$: loss when I decide $\alpha_i$ and the true state of nature is $\omega_j$
So, I observe a particular $x$ and choose action $\alpha_i$. If the true state of nature is $\omega_j$, I have a loss $\lambda(\alpha_i|\omega_j)$. The probability that the true state of nature is $\omega_j$ is $P(\omega_j|x)$.
So, the expected loss (risk) when I decide $\alpha_i$ is:
$$R(\alpha_i|x) = \sum_{j=1}^{c} \lambda(\alpha_i|\omega_j) P(\omega_j|x)$$
And the overall risk is:
$$R = \int R(\alpha(x)|x) p(x) dx$$
To minimize $R$, I have to choose the $\alpha(x)$ that minimizes $R(\alpha(x)|x)$ for all $x$.
In the case that I have $c$ classes, I have:
$$R(\alpha_1|x) = \lambda_{11}P(\omega_1|x)+\lambda_{12}P(\omega_2|x)+\cdots+\lambda_{1c}P(\omega_c|x) 
\\
R(\alpha_2|x) = \lambda_{21}P(\omega_1|x)+\lambda_{22}P(\omega_2|x)+\cdots+\lambda_{2c}P(\omega_c|x) \\
\vdots \\
R(\alpha_c|x) = \lambda_{c1}P(\omega_1|x)+\lambda_{c2}P(\omega_2|x)+\cdots+\lambda_{cc}P(\omega_c|x) $$
I have a cost of rejection $\lambda_r$. So now I say (tell me if this is correct):
I decide $\omega_1$ if $R(\alpha_1|x)$ < minimum$\{R(\alpha_2|x), \dots, R(\alpha_c|x) \} + \lambda_r$
I decide $\omega_2$ if $R(\alpha_2|x)$ < minimum$\{R(\alpha_1|x), R(\alpha_3|x), \dots, R(\alpha_c|x) \} + \lambda_r$
$\vdots$
I decide $\omega_c$ if $R(\alpha_1|x)$ < minimum$\{R(\alpha_1|x), \dots, R(\alpha_{c-1}|x) \} + \lambda_r$
Is it OK to put the $\lambda_r$ just like that?

As I understand well, your cost if a fixed value $\lambda_r$, basically what you can do in a Bayesian setting is to consider rejection as a new action, let us call it $\alpha_{c+1}$ whose value is constant ($\lambda_r$), meaning that its cost is
$$R(\alpha_{c+1}|x)=\lambda_r$$
whatever the probability distribution over $x$. You then just have to compare this values to all the others, and choose rejection if it is the lowest.
