The logistic function

To prevent the model from producing values outside the [0, 1] interval, we must model p(x) using a function that only gives outputs between 0 and 1 over the entire domain of x. The logistic function meets this requirement and always produces an S-shaped curve (see notebook examples), and so, regardless of the value of X, we will obtain a sensible prediction:

Here, the vector x includes a 1 for the intercept captured by the first component of . We can transform this expression to isolate the part that looks like a linear regression to arrive at:

The quantity p(x)/[1−p(x)] is called the oddsan alternative way to express probabilities that may be familiar from gambling, and can take on any value odds between 0 and ∞, where low values also imply low probabilities and high values imply high probabilities. 

The logit is also called log-odds (since it is the logarithm of the odds). Hence, the logistic regression represents a logit that is linear in x and looks a lot like the preceding linear regression.

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