# AdaBoost - why using such alpha function?

I'm reading the paper where AdaBoost was invented (link), and I couldn't understand why they have chosen the formula α_t = 1/2 * ln((1-ε_t) / ε_t).

What is the motivation behind that specific formula? Why not using something that feels more natural like ε_t?

Recall that the final hypothesis after $$T$$ rounds is $$h_T(x)=sign\left(\sum\limits_{i=1}^T \alpha_t h_t(x)\right)$$, i.e. $$\alpha_t$$ is the weight of $$h_t$$ in $$h_T$$. If $$\epsilon_t$$ is high (near one) you want to answer the opposite of $$h_t$$, so you want $$\alpha_t$$ to be negative and very large in absolute value. If on the other hand $$\epsilon_t$$ is very low then you want $$\alpha_t$$ to be very large. The worst case is $$\epsilon_t=\frac{1}{2}$$, in which case $$h_t$$ is of no use to you.
The function $$\log\frac{1-\epsilon_t}{\epsilon_t}$$ satisfies those properties. This magnitude is known as log odds (where the probability considered is $$p=1-\epsilon_t$$, the success probability). Intuitively, the odds ratio tells you how often an event with probability $$p$$ occurs, in our case if e.g. $$\epsilon_t=1/3$$ then $$\frac{1-\epsilon_t}{\epsilon_t}=2$$, i.e. $$h_t$$ is expected to succeed with ratio $$2:1$$.