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Sigmoid and threshold

Logistic regression turns any number into a probability between 0 and 1 with the sigmoid function. Above a chosen threshold the model says yes. Lowering the threshold finds more real cases, but also gives more false alarms.

\sigma(z) = \frac{1}{1 + e^{-z}}

the sigmoid function

\hat y = 1 \iff \sigma(z) \ge t

decision with threshold t

Symbols

$\sigma(z)$	probability
$z$	linear combination of the features
$t$	threshold, often 0.5

Example

$\sigma(0) = 0.5$ and $\sigma(2) \approx 0.88$ .

With threshold 0.5, $z = 2$ is classified as yes.

The threshold is a choice, not a law of nature. Choose it from what the errors cost.

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← Precision and recall

Part of Machine Learning and Data Analysis: Classification and evaluation.