Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
z and σ(z) dimensionless; 0<σ(z)<1 for finite z
| Symbol | Meaning | Unit |
|---|---|---|
| Real-valued logit or input score | dimensionless | |
| Logistic sigmoid output | dimensionless |
\sigma(z)=\frac{1}{1+e^{-z}}The function is the inverse-logit map used in binary logistic regression to turn a linear score into a modeled probability. Its probabilistic interpretation depends on the statistical model and calibration. For large positive or negative z, σ approaches 1 or 0 asymptotically but does not reach either at a finite input. Numerically stable implementations avoid overflow by using equivalent stable evaluation forms.
The function is the inverse-logit map used in binary logistic regression to turn a linear score into a modeled probability. Its probabilistic interpretation depends on the statistical model and calibration. For large positive or negative z, σ approaches 1 or 0 asymptotically but does not reach either at a finite input. Numerically stable implementations avoid overflow by using equivalent stable evaluation forms.
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
Logit from probability
Only for 0<p<1.
Use the example for Logistic sigmoid function to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
The relation applies only under its stated assumptions.
A model produces z=ln(3). What sigmoid output does it return?
σ(ln 3)=1/(1+1/3)=0.75.
In a suitably calibrated model, this represents 75% for the positive class.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: The function is the inverse-logit map used in binary logistic regression to turn a linear score into a modeled probability. Its probabilistic interpretation depends on the statistical model and calibration..
Formula cluster
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
For z=ln(3)≈1.099, σ(z)=1/(1+1/3)=0.75. The model assigns 75% probability to the positive label when its assumptions hold.
Logistic regression, binary classification and probabilistic outputs in neural networks.
Here is how to work through a typical Logistic sigmoid function (σ(z) = 1/(1+e⁻ᶻ)) task step by step:
Task
Use the example for Logistic sigmoid function to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
The logistic sigmoid maps a real-valued score monotonically to a value between zero and one.
The relation applies only under its stated assumptions.
Task
A model produces z=ln(3). What sigmoid output does it return?
Solution path
σ(ln 3)=1/(1+1/3)=0.75.
In a suitably calibrated model, this represents 75% for the positive class.
The logistic sigmoid maps a real input z to a value strictly between zero and one. At z=0 the output is 0.5; positive inputs approach 1 and negative inputs approach 0. It often follows a linear score to express a binary model prediction as a probability. The curve is smooth and monotonically increasing. The output can be read as a probability only under an appropriate logistic model. A score is not automatically calibrated; the model, training data and evaluation determine its interpretation. Evaluate calibration separately from discrimination and accuracy. The output also does not choose a decision threshold. Report the selected cutoff.
Substitute z=ln 3: e^(−z)=e^(−ln 3)=1/3. Thus σ(z)=1/(1+1/3)=3/4=0.75. Check with the inverse logit ln(p/(1−p)): for p=0.75 it is ln(0.75/0.25)=ln 3. The output is dimensionless and lies in the valid open interval (0,1). The output can be read as a probability only under an appropriate logistic model. A score is not automatically calibrated; the model, training data and evaluation determine its interpretation. Evaluate calibration separately from discrimination and accuracy. The output also does not choose a decision threshold. Report the selected cutoff.
No. The function mathematically maps a score into the interval from zero to one. The model and its assumptions connect that value to a conditional probability. Poor data, misspecification or distribution shift can harm calibration. Check reliability diagrams and held-out validation data rather than treating the sigmoid shape itself as evidence of accurate probabilities. The output can be read as a probability only under an appropriate logistic model. A score is not automatically calibrated; the model, training data and evaluation determine its interpretation. Evaluate calibration separately from discrimination and accuracy. The output also does not choose a decision threshold. Report the selected cutoff.
The derivative is σ′(z)=σ(z)(1−σ(z)). The steepest slope occurs at z=0, where σ=0.5 and the slope is 0.25. For very positive or negative inputs, the slope becomes small as the function saturates. This helps explain vanishing gradients in deep networks with repeated sigmoid activations, but is not a full account of training behavior. The output can be read as a probability only under an appropriate logistic model. A score is not automatically calibrated; the model, training data and evaluation determine its interpretation. Evaluate calibration separately from discrimination and accuracy. The output also does not choose a decision threshold. Report the selected cutoff.
For large positive z, e^(−z) is tiny and σ is near one; for very negative z, e^(−z) can overflow and σ is near zero. Floating-point arithmetic may then produce rounding or overflow problems. Libraries use stable implementations, often with separate forms for positive and negative inputs. For losses, a log-sigmoid routine is safer than computing p and then taking log(p). The output can be read as a probability only under an appropriate logistic model. A score is not automatically calibrated; the model, training data and evaluation determine its interpretation. Evaluate calibration separately from discrimination and accuracy. The output also does not choose a decision threshold. Report the selected cutoff.
Quanta exam set
Curated exam set for "Logistic sigmoid function":
Question (front)
Which formula describes Logistic sigmoid function?
Answer in your set
Question (front)
How do you rearrange Logistic sigmoid function for Logit from probability?
Answer in your set
Question (front)
Which common mistake happens with Logistic sigmoid function?
Answer in your set
+ 7 more cards: units, variables, derivation, example, exam task
The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for σ(z) = 1/(1+e⁻ᶻ): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential