The sigmoid function, often represented as $\sigma(x) = \frac{1}{1 + e^{-x}}$, is widely used in machine learning due to its several key properties:

1. Output Range: The sigmoid function squashes its input to a range between 0 and 1. This makes it ideal for models where outputs need to represent probabilities, such as in binary classification problems.

2. Smooth Gradient: The sigmoid function has a smooth derivative, making it differentiable at all points. This characteristic is useful for gradient-based optimization algorithms (like backpropagation in neural networks) since it provides a clear path for weight updates.

3. Non-linear Activation: It introduces non-linearity into the model, enabling the neural network to learn and model complex data patterns, which a linear function cannot capture.

4. Interpretability: The output of the sigmoid function is often interpreted as a probability score, which is useful in classification tasks to determine the likelihood of a certain class (e.g., 0 or 1).

5. Historical Popularity: While other activation functions like ReLU are often preferred in deeper neural networks due to issues like vanishing gradients in sigmoid, it is still used in the final layer of binary classifiers or logistic regression models.

Would you like to explore more about how the sigmoid function affects model training or its alternatives?

Here are some follow-up questions to expand on this:

1. How does the sigmoid function compare to ReLU in deep learning?
2. What is the "vanishing gradient" problem with the sigmoid function?
3. Where is the sigmoid function used in logistic regression?
4. How does the derivative of the sigmoid function help in backpropagation?
5. Can the sigmoid function be used in multiclass classification problems?

Tip: The sigmoid function can slow down learning when used in deep networks due to its gradient becoming very small for extreme values of input (vanishing gradient). ReLU can help mitigate this issue.