LaTeX
p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}Variables
- z_i: score for class i
- p_i: normalized probability-like value
How to use this formula
Converts a vector of real scores into positive values that sum to one.
Important notes
- Subtract max(z) before exponentiating for numerical stability.
- Softmax output depends on all scores simultaneously.
Quick example
Equal scores produce equal probabilities.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Subtract max(z) before exponentiating for numerical stability.
- For the Softmax Function, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Softmax Function, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- For the Softmax Function, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
Equal scores produce equal probabilities.
Common mistakes
- When copying Softmax Function, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Softmax Function, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use Softmax Function in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
p_i=\frac{e^{z_i}}{\sum_j e^{z_j}}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Softmax Function used for?
Converts a vector of real scores into positive values that sum to one.
Can I copy this formula as LaTeX?
Yes. Copy p_i=\frac{e^{z_i}}{\sum_j e^{z_j}} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.