# Softmax regression: Intuition about why distribution of $y$ is in terms of $e^{\theta^Tx}$ as opposed to just $\theta^Tx$

I'm going through Andrew Ng's lecture notes on Machine Learning and I just learnt about softmax regression there.

We see that, for softmax regression, the conditional distribution of $y$ given $x$ is given as:

This formula contains terms of form $e^{\theta^Tx}$. I was just wondering if there is an intuitive explanation for this? Or, why isn't the derived formula for probability simpler like:

$$\frac{\theta^Tx}{\sum_j\theta_{j}^Tx}$$

And is there an intuitive explanation for what that would mean?

$$\frac{ e^{\Theta_{i}^T x }}{ \sum_{j} e^{\Theta_{j}^T x } }$$
• And why using e instead of arbitary number > 1? – mrgloom Sep 1 '17 at 15:42
There is an intuitive definition. I tried to explain the softmax in this answer. To put it simply, you are interpreting the unbounded $\theta_i^Tx$ as log-odds, and the softmax converts them to probabilities in $[0,1]$. Your formula has no such interpretation.