Question about computing Bayes Error - with or without loss function?

Question

I am new to Bayesian Decision Theory and don't understand the following concept:

So from what I understood, the Bayes error is used to report the performance of a Bayes classifier in terms of the probability of making and error. From the conditional error probabilities conditional error http://sebastianraschka.com/_my_resources/images/equations/cond_error.png

we can obtain the total probability of making an error (the probability to mis-classify).

error http://sebastianraschka.com/_my_resources/images/equations/error.png

Now, if my Bayes classifier was designed to minimize the overall risk, I have a loss function that gives penalties to certain decisions.

conditional risk http://sebastianraschka.com/_my_resources/images/equations/cond_risk.png overall risk http://sebastianraschka.com/_my_resources/images/equations/overall_risk.png

So, if my classifier includes such a loss function when I optimize my classifier for minimum overall risk, shouldn't be the Bayes error also include the loss function term?

Hope you can help me here, because I think I am missing something here ...

EDIT:

I'll try to express my problem using an 2D-classification problem:

Let's assume I have two pdfs (e.g., p(x|c1) and p(x|c2) ) with slight overlap. And mis-classifying a pattern as c2 where it truly belongs to c1 is more costly than vice-versa.

In this case I would assign a higher loss to "classify pattern x as c1 when it is truly c2" than "classify pattern x as c2 when it is truly c1" in order to calculate and minimize the overall risk.

I would therefore increase the probability to classify a pattern x as c2 over c1 due to the minimum risk optimization. Isn't this something I have to also include in p(error)?

Answer 1

You should be incorporating it, yes. The loss function will affect your policy on how to classify a pattern. Let's call that policy $\pi$, and say it consists of the probability you assign to each class. You thus have a set of probabilities $p(\textrm{choose } c_i|x,\pi) $, the probability of assigning pattern $x$ to class $c_i$ under policy $\pi$.

$P(\textrm{error}|x)$, as you've written it, has separate cases depending on your choice of classification: for example, your first equation has two cases, depending on whether you choose class $\omega_1$ or $\omega_2$.

However, instead of $P(\textrm{error} )$, what you really want to know is $P(\textrm{error}|\pi)$, i.e. the chance of an error given your policy on classification. So you also need to sum over all of these possible cases. For your example, you'd have $$\begin{align*}P(\textrm{error}|\pi) &= \int_{-\infty}^{\infty} P(\textrm{error} |x,\pi) p(x) dx \\&= \int_{-\infty}^{\infty} \sum_{i=1}^2 P(\textrm{error} |x,\textrm{choose } c_i) p(\textrm{choose } c_i|x,\pi) p(x) dx.\end{align*} $$ You could also write $P(\textrm{error} |R) ,$ and condition on the loss function $R$ directly, but your choice of $\pi$ given $R$ is usually deterministic, so it won't make a difference.