Generalized Linear Models (GLMs): Linear Regression
🪄 Step 1: Intuition & Motivation
Core Idea: Linear Regression is like a friendly but limited artist — it can only draw straight lines through data that behaves linearly and follows a normal (Gaussian) error pattern. But what if your target isn’t continuous — maybe it’s binary (yes/no), count-based (0, 1, 2, …), or skewed (time-to-event)? That’s when Generalized Linear Models (GLMs) step in — a more flexible family that keeps Linear Regression’s soul but adapts it to different types of data and distributions.
Simple Analogy: Think of Linear Regression as a Swiss Army knife with just one blade (the straight line). GLMs add more blades — one for probabilities (Logistic Regression), one for counts (Poisson Regression), and one for time (Exponential models). Same handle, different tools.
🌱 Step 2: Core Concept
What’s Happening Under the Hood?
GLMs extend Linear Regression by relaxing two assumptions:
- The response variable ($y$) doesn’t have to be Gaussian (normal).
- The relationship between inputs ($X$) and expected output ($E[y]$) doesn’t have to be purely linear.
Instead, GLMs introduce:
- A link function: connects the mean of the response ($E[y]$) to a linear predictor ($X\beta$).
- An exponential family distribution: specifies the nature of $y$ (Normal, Bernoulli, Poisson, etc.).
The general GLM structure is:
$$ g(E[y]) = X\beta $$Here:
- $g(\cdot)$ → link function (transformation)
- $E[y]$ → expected value of response variable
- $X\beta$ → linear predictor (just like in Linear Regression)
Why It Works This Way
Linear Regression assumes $y$ is continuous and unbounded — but probabilities, counts, or rates don’t fit that pattern.
GLMs fix this by:
- Applying a transformation ($g$) that maps valid ranges (like 0–1 for probabilities) to the entire real line.
- Using distributions that reflect the real data behavior (e.g., Bernoulli for binary outcomes).
This keeps the math stable and the predictions realistic.
How It Fits in ML Thinking
They show how linear models can adapt to different data types without losing interpretability — a concept later generalized in deep learning via activation functions.
In essence, the link function in GLMs plays the same role as activation functions in neural networks.
📐 Step 3: Mathematical Foundation
Generalized Linear Model Equation
Where:
- $E[y]$ → expected (mean) response
- $X\beta$ → linear predictor
- $g(\cdot)$ → link function (transformation ensuring valid predictions)
Each GLM type chooses:
- A distribution for $y$ (from the exponential family).
- A link function that maps $E[y]$ appropriately.
Examples of Common GLMs
| Model | Response Type | Distribution | Link Function $g(\cdot)$ | Typical Use |
|---|---|---|---|---|
| Linear Regression | Continuous | Normal (Gaussian) | Identity ($g(\mu) = \mu$) | Predict continuous outcomes |
| Logistic Regression | Binary | Bernoulli | Logit ($g(\mu) = \log(\frac{\mu}{1 - \mu})$) | Classification (Yes/No) |
| Poisson Regression | Count | Poisson | Log ($g(\mu) = \log(\mu)$) | Event counts (e.g., #clicks) |
| Gamma Regression | Positive Continuous | Gamma | Inverse ($g(\mu) = 1/\mu$) | Rates or durations |
for example, logit → probabilities stay between 0 and 1; log → counts stay positive.
🧠 Step 4: Key Ideas and Assumptions
1️⃣ Exponential Family:
GLMs assume the response $y$ follows a distribution from the exponential family (Normal, Bernoulli, Poisson, Gamma, etc.).
2️⃣ Link Function:
The link connects linear predictor ($X\beta$) to the mean of $y$.
3️⃣ Linearity in Parameters:
Even though relationships can be nonlinear in $y$, they remain linear in coefficients ($\beta$), keeping estimation simple and interpretable.
4️⃣ Independence of Observations:
Like in Linear Regression, each data point’s error is assumed independent.
⚖️ Step 5: Strengths, Limitations & Trade-offs
- Extends linear models to non-Gaussian outcomes.
- Interpretable coefficients (effect of predictors).
- Theoretical elegance + practical flexibility.
- Connects naturally to ML concepts like activations and loss functions.
- Sensitive to incorrect choice of link function or distribution.
- Computationally heavier than OLS (requires iterative estimation).
- Assumes correct specification of $g(E[y])$ — misspecification leads to bias.
They generalize regression to fit reality better — not by abandoning linearity, but by adapting it.
🚧 Step 6: Common Misunderstandings
🚨 Common Misunderstandings (Click to Expand)
“GLMs are nonlinear models.”
No — they’re linear in parameters ($\beta$), even if the relationship between $X$ and $y$ is nonlinear.“Logistic Regression isn’t linear.”
It is — it’s a GLM with a logit link, meaning $logit(p) = X\beta$.“Choosing the wrong link doesn’t matter much.”
It does — the link controls how errors behave and how interpretable coefficients are.
🧩 Step 7: Mini Summary
🧠 What You Learned: GLMs extend Linear Regression to handle diverse response types by combining link functions with different probability distributions.
⚙️ How It Works: Transform the mean of the response with a suitable link ($g(E[y]) = X\beta$) and estimate $\beta$ iteratively.
🎯 Why It Matters: GLMs form the foundation of many advanced models — from Logistic Regression to Poisson and Gamma regressions — keeping linear simplicity while embracing real-world complexity.