Here, I list some functions related to logistic regression that I get confused often.

Logit

Logit function is basically the log of odds:

\[\begin{align} \text{logit}(p) = \log \frac{p}{1-p} \end{align}\]

where $p$ is a probability.

Logistic function

Logistic function is a common example of sigmoid function, which produces a S-shaped curve (aka. sigmoid curve).

\[\begin{align} f(z) = \frac{e^{z}}{1 + e^{z}} = \frac{1}{1 + e^{-z}} \end{align}\]

Some simple algebra shows that the logistic function is the inverse of the logit function (we use $p$ instead of $f(z)$ to be more intuitive as it relates to probability):

\[\begin{align} p &= \frac{e^z}{1 + e^z} \\ p + p e^z &= e^z \\ e^z &= \frac{p}{1-p} \\ z &= \log \frac{p}{1-p} \\ \end{align}\]

Note, $f(z)$ is commonly interpreted as a probability as it ranges from 0 (when $z \rightarrow -\infty$) to 1 (when $z \rightarrow \infty$).

In logistic regression, we are basically using a linear function of features to estimate the $z$, the log odds.

plots for logit and logistic functions
Figure 1. Plots of logit and Logistic functions. Note the axes are reversed between logit and logistic plots as they are inverse function of each other (Notebook).

Softmax function

Softmax function is the extension of logistic function to more than two dimensions.

Suppose $\mathbf{z} = (z_1, \cdots, z_d) \in \mathbb{R}^d$.

\[\begin{align} f(z_k) = \frac{e^{z_k}}{\sum_{i=1}^{K} e^{z_k}} \end{align}\]

If d dimensions correspond to d categories, then each $f(z_k)$ can be interpreted as the probability of category $i$.

Note, adding a constant ($C$) to each $z_k$ won’t affect $f(z_k)$:

\[\begin{align} \frac{e^{z_k + C}}{\sum_{i=1}^{K} e^{z_k + C}} = \frac{e^C e^{z_k}}{e^C \sum_{i=1}^{K} e^{z_k}} = \frac{e^{z_k}}{\sum_{i=1}^{K} e^{z_k}} \end{align}\]

so for mathematical convenience, we can offset all components in $\mathbf{z}$ so that $z_1 = 0$. Then in the case of 2 dimension, softmax function simplifies to the logistic function.

Multinomial logit function

From the softmax function, we could derive the corresponding logit function for more than two dimensions.

\[\begin{align} p(z_k) &= \frac{e^{z_k}}{\sum_{i=1}^{K} e^{z_k}} \\ p(z_k) &= \frac{e^{z_k}}{e^{z_k} + \sum_{i \ne k}^{K} e^{z_k}} \\ p(z_k)e^{z_k} + p(z_k) \sum_{i \ne k}^{K} e^{z_k} &= e^{z_k} \\ e^{z_k} &= \frac{p(z_k) \sum_{i \ne k}^{K} e^{z_k} }{1 - p(z_k)} \\ z_k &= \log \frac{p(z_k) \sum_{i \ne k}^{K} e^{z_k} }{1 - p(z_k)} \end{align}\]

So $z_k$ is a log-odds weighted by $\sum_{i \ne k}^{K} e^{z_k} $.