Many problems require a probability estimate as output. Logistic
regression is an extremely efficient mechanism for calculating
probabilities. Practically speaking, you can use the returned
probability in either of the following two ways:

"As is"

Converted to a binary category.

Let's consider how we might use the probability "as is." Suppose we
create a logistic regression model to predict the probability that a
dog will bark during the middle of the night. We'll call that
probability:

p(bark | night)

If the logistic regression model predicts a p(bark | night) of 0.05,
then over a year, the dog's owners should be startled awake approximately
18 times:

In many cases, you'll map the logistic regression output into the solution
to a binary classification problem, in which the goal is to correctly
predict one of two possible labels (e.g., "spam" or "not spam"). A later
module
focuses on that.

You might be wondering how a logistic regression model can ensure
output that always falls between 0 and 1. As it happens,
a sigmoid function, defined as follows, produces output having
those same characteristics:

$$y = \frac{1}{1 + e^{-z}}$$

The sigmoid function yields the following plot:

Figure 1: Sigmoid function.

If z represents the output of the linear layer of a model trained
with logistic regression, then sigmoid(z) will yield a value (a probability)
between 0 and 1. In mathematical terms:

$$y' = \frac{1}{1 + e^{-(z)}}$$

where:

y' is the output of the logistic regression model for a particular example.

z is b + w_{1}x_{1} + w_{2}x_{2} + ... w_{N}x_{N}

The w values are the model's learned weights, and b is the bias.

The x values are the feature values for a particular example.

Note that z is also referred to as the log-odds because the inverse of the
sigmoid states that z can be defined as the log of the probability of
the "1" label (e.g., "dog barks") divided by the probability of the
"0" label (e.g., "dog doesn't bark"):

$$ z = log(\frac{y}{1-y}) $$

Here is the sigmoid function with ML labels:

Figure 2: Logistic regression output.

Click the dropdown arrow to see a sample logistic regression inference calculation.

Suppose we had a logistic regression model with three features that
learned the following bias and weights:

b = 1

w_{1} = 2

w_{2} = -1

w_{3} = 5

Further suppose the following feature values for a given example:

x_{1} = 0

x_{2} = 10

x_{3} = 2

Therefore, the log-odds:

$$b + w_1x_1 + w_2x_2 + w_3x_3$$

will be:

(1) + (2)(0) + (-1)(10) + (5)(2) = 1

Consequently, the logistic regression prediction for this particular
example will be 0.731: