Bayes classifier expected classification error for multiclass case

Question

Assume a feature $x \in [a,b]$ and two classes $\omega_1, \omega_2$ with prior probabilities $P(\omega_1), P(\omega_2)$ and likelihood functions $p(x | \omega_1), p(x | \omega_2)$. Then, the expected classification error is defined as:

$$ P_e = \int\limits_{R_2}P(\omega_1)p(x|\omega_1)dx + \int\limits_{R_1}P(\omega_2)p(x|\omega_2)dx $$

where $R_1, R_2$ are the decision regions for classes $\omega_1, \omega_2$ respectively.

Now, my question is about the multiclass variant with $n$ classes. How will we proceed to calculate the expected classification error?

My approach would be to calculate $P_e$ as shown above for every possible pair of classes and then divide it by the number of classes (mean). That'll be:

$$ P_{e_{(mult)}} = \left(\sum \int\limits_{R_j}P(\omega_i)p(x|\omega_i)dx + \int\limits_{R_i}P(\omega_j)p(x|\omega_j)dx\right) / n, \quad i \neq j $$

Is this approach accurate when it comes to the expected classification error in the multiclass case?

Answer 1

The regions in first equation are in fact $\mathcal R-\mathcal R_1$ and $\mathcal R-\mathcal R_2$ respectively. In each integral we're calculating the probability of having a sample from class $k$ in the region $\mathcal R-\mathcal R_k$. For the multivariate case, you'll do this for each class, i.e. $$P_e=\sum_{k=1}^n \int_{\mathcal R-\mathcal R_k}P(w_k)p(x|w_k)dx$$

In your pairwise methodology, for each class $k$, all integrals are being calculated twice, so you'll divide by $2$ not $n$. It's equivalent to the above formula because $$\int_{\mathcal R-\mathcal R_k} P(w_k)p(x|w_k)dx=\sum_{j\neq k}\int_{\mathcal R_j}P(w_k)p(x|w_k)dx$$ considering $\mathcal R=\mathcal R_1\cup \mathcal R_2 \dots \mathcal R_n$.