Suppose we use a decision tree to predict if a bank customer can pay back a credit. So it is a two class classification problem. Now we can make two mistakes:
- $\alpha$ error: The customer can back the credit, but we predict he can't.
- $\beta$ error: The customer can't pay back the credit, but we predict he can.
Now we know that $\beta$ errors are 123.4 times as expensive as $\alpha$ errors. But we only have a given set of data. In this set we have $n_1=10000$ customers who paid back the credit and $n_2 = 100$ customers who didn't.
How can the training of the decision tree be adjusted to account for the fact that $\beta$ errors are more expensive?
(Note: This is a theoretical question to learn about decision trees. I know about other classifiers like neural networks and I know of ensembles. However, I only want to know about decision trees here.)