広島大学 先進理工系科学研究科 情報科学プログラム 2022年8月実施 専門科目II 問題6

Author

祭音Myyura

Description

Let the sigmoid function be

(1) Show that the following equation holds:

(2) Show that the derivative of the sigmoid function can be written as

(3) The inverse function of the sigmoid function is called the logit function. Show that the logit function can be written as

where .

Kai

Throughout, denotes the natural logarithm.

(1)

Start from the definition:

Then

Multiply numerator and denominator by :

(2)

Differentiate the definition:

Using the chain rule:

Now compute , from (1), we have

hence

(3)

Let . Then

We want to solve for in terms of .

First invert the fraction:

So

Take the natural logarithm of both sides:

so

Therefore the inverse function of is

which is defined for (and extends to at the endpoints ).