# Logit Function

A Logit Function is an algebraic function of the form [math]\displaystyle{ \operatorname{logit}(p)=\ln\left( \frac{p}{1-p} \right) = \ln(p)-\ln(1-p). \!\, }[/math]

**Context:**- It is an Inverse Function to a Logistic Function.
- It can range from being a Binomial Logit/Binary Logit to being a Multinomial Logit.

**Example(s):**- [math]\displaystyle{ f(x)= 5.4 \times \ln\left( \frac{x}{1-x}\right) }[/math]

**Counter-Example(s):****See:**Logistic Regression, Conditional Logit, Nested Logit, Mixed Logit, Exploded Logit, Ordered Logit, Inverse Function, Sigmoid Function, Logarithm, Logarithmic Unit, Bit, Odds Ratio, Additive Function.

## References

### 2019

- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/logit Retrieved:2019-4-12.
- In statistics, the
**logit**function or the**log-odds**is the logarithm of the odds*p*/(1 −*p*) where is the probability. . It is a type of function that creates a map of probability values from [math]\displaystyle{ [0, 1] }[/math] to [math]\displaystyle{ [-\infty, +\infty] }[/math] . It is the inverse of the sigmoidal "logistic" function or logistic transform used in mathematics, especially in statistics. In deep learning, the term**logits layer**is popularly used for the last neuron layer of neural network for classification task which produces raw prediction values as real numbers ranging from [math]\displaystyle{ [-\infty, +\infty] }[/math] .

- In statistics, the

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/logit#Definition Retrieved:2015-1-28.
- The
**logit**of a number*p*between 0 and 1 is given by the formula::[math]\displaystyle{ \operatorname{logit}(p)=\log\left( \frac{p}{1-p} \right) =\log(p)-\log(1-p)=-\log\left( \frac{1}{p} - 1\right). \!\, }[/math]

The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used. The choice of base corresponds to the choice of logarithmic unit for the value: base 2 corresponds to a bit, base e to a nat, and base 10 to a ban (dit, hartley); these units are particularly used in information-theoretic interpretations.

The "logistic" function of any number [math]\displaystyle{ \alpha }[/math] is given by the inverse-logit:

:[math]\displaystyle{ \operatorname{logit}^{-1}(\alpha) = \frac{1}{1 + \operatorname{exp}(-\alpha)} = \frac{\operatorname{exp}(\alpha)}{ \operatorname{exp}(\alpha) + 1} }[/math]

If

*p*is a probability, then*p*/(1 −*p*) is the corresponding odds; the logit of the probability is the logarithm of the odds. Similarly, the difference between the logits of two probabilities is the logarithm of the odds ratio (*R*), thus providing a shorthand for writing the correct combination of odds ratios only by adding and subtracting::[math]\displaystyle{ \operatorname{log}(R)=\log\left( \frac{{p_1}/(1-p_1)}{{p_2}/(1-p_2)} \right) =\log\left( \frac{p_1}{1-p_1} \right) - \log\left(\frac{p_2}{1-p_2}\right)=\operatorname{logit}(p_1)-\operatorname{logit}(p_2). \!\, }[/math]

- The