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Random freezes, missing sound and display glitches usually trace back to one bad driver. Find and replace yours safely.Free scan · under a minuteOrdinary linear regression predicts a numeric response; logistic regression predicts the probability of a class. Maximum-entropy classification is a related way to build conditional class probabilities, and under common feature-based formulations it has the same log-linear form as logistic regression. The word “regression” in logistic regression describes its mathematical lineage—not an unbounded numeric output.
What “regression” means in these methods
Regression is a broad family of methods for relating inputs, or predictors, to a response. In ordinary linear regression, the response is typically numeric: for example, estimating a house price from its features. A model’s purpose may be to explain relationships and support inference, to predict accurately, or both; those goals are not identical. The University of British Columbia’s [Stat 406 introduction to learning and regression](https://www.stat.ubc.ca/~pleiss/teaching/Stat406_2026W2/) frames supervised learning as predicting a response from covariates and distinguishes inference from prediction.
Logistic regression uses a linear combination of input features too, but transforms that score into a class probability. It is therefore a classification model, not ordinary linear regression with a different name. “Regression” alone does not imply a numeric target in every method; the useful distinction here is between ordinary linear regression and logistic regression.
How ordinary linear regression differs from logistic regression
In a standard linear regression setup, the model represents the response with a linear predictor, often written as ŷ = w·x + b. Its output is a numeric value and is not inherently confined to a range such as zero to one. Ordinary least squares commonly fits the coefficients by minimizing squared residuals.
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For a binary outcome, logistic regression instead models the conditional probability that the response belongs to one class:
P(Y=1 | x) = σ(w·x + b), where σ(z) = 1 / (1 + e−z).
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The linear score can take any real value, but the logistic, or sigmoid, function maps it to a probability between zero and one. A decision rule—often comparing the probability with a threshold—can then assign a class. The University of British Columbia’s [Stat 406 classification and logistic regression lecture](https://www.stat.ubc.ca/~pleiss/teaching/Stat406_2026W2/) presents logistic regression as a classification method fitted through logistic loss.
What the coefficients mean
For binary logistic regression, the model is linear in log odds, not in probability:
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log(P(Y=1 | x) / (1 − P(Y=1 | x))) = w·x + b.
Holding other features fixed, increasing a feature by one unit changes the modeled log odds by its coefficient. It does not mean the probability increases by that same amount. The probability change depends on the starting score and other feature values because the sigmoid is nonlinear. Hang Li’s 2024 book Machine Learning Methods, Chapter 6, defines the binary model for outcomes 0 and 1 and discusses its log-linear form.
How logistic regression is fitted
Binary logistic regression is commonly estimated by maximum likelihood: choose coefficients that make the observed labels most likely under the model. Equivalently, the fitting procedure can minimize average logistic loss, also called binary cross-entropy. The equivalence is described in the [UBC Stat 406 lecture on logistic regression](https://www.stat.ubc.ca/~pleiss/teaching/Stat406_2026W2/). Optimization methods can include gradient-based or quasi-Newton approaches; the chosen method is a computational means of finding the coefficients, not a different model definition.
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For more than two classes, multinomial logistic regression assigns a probability to each class using a normalized exponential, or softmax, transformation. The probabilities across the available classes sum to one. This multiclass version should not be confused with the binary sigmoid formula: the target and probability normalization differ.
What maximum entropy means in classification
Maximum entropy is a general principle for selecting a probability distribution. Given specified constraints, it selects the distribution with the greatest entropy among distributions satisfying those constraints. Entropy measures uncertainty in a distribution; maximizing it avoids adding structure beyond what the constraints require.
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In classification, constraints can express desired relationships between features and class outcomes. The resulting conditional model has a log-linear, exponential form: feature contributions enter a linear score, and a normalizing term turns the scores into valid class probabilities. Hang Li’s Machine Learning Methods, Chapter 6 (Tsinghua University Press, 2024), describes maximum-entropy models and logistic regression as log-linear models with similar forms. Maximum-entropy classification is commonly fitted through likelihood or regularized-likelihood optimization.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Why maximum entropy and logistic regression are closely related
In common feature-based classification formulations, the maximum-entropy solution is a conditional log-linear probability model. Logistic regression is also a conditional log-linear model: in the binary case it uses a sigmoid, and in the multiclass case it uses a softmax. With corresponding feature definitions and constraints, the models can take the same mathematical form and be fitted through equivalent likelihood-based objectives.
That relationship has a boundary: “maximum entropy” names a principle or modeling framework, while “logistic regression” names a particular family of conditional probability models. They are not interchangeable labels for every possible formulation. Whether a maximum-entropy model is equivalent to logistic regression depends on how its constraints and features are specified.
Comparison at a glance
| Method | Typical target | What it models | Form and fitting | Key caution |
|---|---|---|---|---|
| Ordinary linear regression | Numeric response | A conditional response, often its mean | Linear predictor; ordinary setup commonly uses least squares | Explanatory inference and predictive accuracy are distinct goals. |
| Logistic regression | Binary or categorical class | Conditional class probability | Sigmoid for binary outcomes or softmax for multiclass; commonly maximum likelihood or equivalent logistic loss | Coefficients are linear in log odds for the binary model, not direct probability increments. |
| Maximum-entropy classification | Categorical class | Conditional class probabilities subject to feature constraints | Exponential/log-linear model with a normalizer; commonly likelihood-based fitting | Its relationship to logistic regression depends on the model specification and constraints. |
The comparison reflects the standard formulations described in Li’s 2024 textbook and UBC Stat 406’s September 17, 2026 lectures on [regression](https://www.stat.ubc.ca/~pleiss/teaching/Stat406_2026W2/) and [classification](https://www.stat.ubc.ca/~pleiss/teaching/Stat406_2026W2/).
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- Choose ordinary linear regression when the response you need to estimate is numeric and a linear-response model is appropriate for your purpose.
- Choose logistic regression when the target is a class and you want conditional class probabilities from a familiar, interpretable log-linear model. For two classes, interpret coefficients on the log-odds scale; for multiple classes, use the multinomial formulation.
- Choose a maximum-entropy classification formulation when the modeling task is naturally expressed as finding the highest-entropy conditional distribution that satisfies specified feature constraints. In common formulations, this leads to a model closely related or equivalent to logistic regression; verify the constraints and parameterization before treating the names as synonymous.
For any of the three, decide whether the priority is interpretation or prediction, and evaluate the model against that goal. A method’s name or probability output alone does not establish predictive performance.
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