If is a probability, then is the corresponding odds; the of the probability is the logarithm of the odds, i.e. 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 shannon, base to a “nat”, and base 10 to a hartley; these units are particularly used in information-theoretic interpretations. For each choice of base, the logit function takes values between negative and positive infinity. The “logistic” function of any number is given by the inverse-: The difference between the s of two probabilities is the logarithm of the odds ratio, thus providing a shorthand for writing the correct combination of odds ratios only by adding and subtracting:
History
There have been several efforts to adapt linear regression methods to a domain where the output is a probability value,, instead of any real number. In many cases, such efforts have focused on modeling this problem by mapping the range to and then running the linear regression on these transformed values. In 1934 Chester Ittner Bliss used the cumulative normaldistribution function to perform this mapping and called his model probit an abbreviation for "probability unit";. However, this is computationally more expensive. In 1944, Joseph Berkson used log of odds and called this function logit, abbreviation for "logistic unit" following the analogy for probit. Log odds was used extensively by Charles Sanders Peirce.. G. A. Barnard in 1949 coined the commonly used term log-odds; the log-odds of an event is the logit of the probability of the event.
The inverse-logit function is also sometimes referred to as the expit function.
In plant disease epidemiology the logit is used to fit the data to a logistic model. With the Gompertz and Monomolecular models all three are known as Richards family models.
The log-odds function of probabilities is often used in state estimation algorithms because of its numerical advantages in the case of small probabilities. Instead of multiplying very small floating point numbers, log-odds probabilities can just be summed up to calculate the joint probability.
Comparison with probit
Closely related to the function are the probit function and probit model. The and are both sigmoid functions with a domain between 0 and 1, which makes them both quantile functions – i.e., inverses of the cumulative distribution function of a probability distribution. In fact, the is the quantile function of the logistic distribution, while the is the quantile function of the normal distribution. The function is denoted, where is the CDF of the normal distribution, as just mentioned: As shown in the graph on the right, the and functions are extremely similar when the function is scaled, so that its slope at matches the slope of the. As a result, probit models are sometimes used in place of logit models because for certain applications the implementation is easier.
