What is the link between the logit and the probability of a binary event? - Cross Validated most recent 30 from stats.stackexchange.com 2019-09-22T16:50:46Z https://stats.stackexchange.com/feeds/question/156419 https://creativecommons.org/licenses/by-sa/4.0/rdf https://stats.stackexchange.com/q/156419 3 What is the link between the logit and the probability of a binary event? Quantopik https://stats.stackexchange.com/users/7947 2015-06-10T23:42:06Z 2015-06-11T00:20:46Z <p>Reading about logistic regression model, I wondered about the link existing between the logit (or $log\frac{p}{(1-p)}$) and the probability of an event defined as binary by assumption and modeled by using the logistic regression model.</p> <p>I know that transformation has a numerical result that is constrained to the interval $[0, 1]$, and, consequentially, can be directly thought as a probability cause of the latter reason, but, I wonder if it does exist a mathematical explanation to this.</p> <p>Can someone answer to the question by providing a mathematical proof or reference about?</p> https://stats.stackexchange.com/questions/156419/-/156422#156422 8 Answer by A. Donda for What is the link between the logit and the probability of a binary event? A. Donda https://stats.stackexchange.com/users/17023 2015-06-11T00:20:46Z 2015-06-11T00:20:46Z <p>In section 4.2 of <a href="http://research.microsoft.com/en-us/um/people/cmbishop/PRML/" rel="noreferrer"><em>Pattern Recognition and Machine Learning</em></a> (Springer 2006), Bishop shows that the logit arises naturally as the form of the posterior probability distribution in a Bayesian treatment of two-class classification. He then goes on to show that the same holds for discretely distributed features, as well as a subset of the family of exponential distributions. For multi-class classification the logit generalizes to the normalized exponential or softmax function. Following this, the value of the logit or softmax can therefore actually be interpreted as a probability in a variety of settings, but not as the frequentist probability of an event, but as the Bayesian probability of an underlying cause (class) given the data.</p>