## The Bayesian logistic regression model

The name logistic regression comes from the fact that the dependent variable of the regression is a logistic function. It is one of the widely used models in problems where the response is a binary variable (for example, fraud or not-fraud, click or no-click, and so on).

A logistic function is defined by the following equation:

It has the particular feature that, as *y* varies from to , the function value varies from 0 to 1. Hence, the logistic function is ideal for modeling any binary response as the input signal is varied.

The inverse of the logistic function is called *logit*. It is defined as follows:

In logistic regression, *y* is treated as a linear function of explanatory variables *X*. Therefore, the logistic regression model can be defined as follows:

Here, is the set of basis functions and are the model parameters as explained in the case of linear regression in Chapter 4, *Machine Learning Using Bayesian Inference*. From the definition of GLM in Chapter...