1.4 Interpretability and Coefficients

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🪄 Step 1: Intuition & Motivation

Core Idea: We’ve trained our Logistic Regression model — great! 🎉 But what good is a model if we can’t explain what it’s saying?

This is where interpretability enters the scene. Each coefficient ($\beta_j$) in Logistic Regression tells a story — how a particular feature changes the odds of the outcome.

Unlike black-box models (like neural networks), Logistic Regression gives us an open window into its decision process — and that’s its superpower.

Simple Analogy: Imagine a courtroom. Each feature (age, income, education, etc.) is a witness, and the coefficients are how much each witness influences the verdict (the prediction). Some speak strongly (large $\beta$), some whisper softly (small $\beta$), and some even contradict expectations (negative $\beta$).

🌱 Step 2: Core Concept

Let’s uncover what those mysterious $\beta$ values actually mean.

What’s Happening Under the Hood?

Our logistic model predicts probability via the sigmoid function:

$$ P(y=1|x) = \frac{1}{1 + e^{-(\beta_0 + \beta_1x_1 + ... + \beta_nx_n)}} $$

Taking the log-odds (the “logit”) gives:

$$ \log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n $$

Here’s the key:

$\beta_j$ represents how much the log-odds of the outcome change for a one-unit increase in feature $x_j$, keeping all other features constant.

So:

Positive $\beta_j$ → increases the log-odds (more likely event)
Negative $\beta_j$ → decreases the log-odds (less likely event)

Why It Works This Way

In Linear Regression, a one-unit increase in $x_j$ changes $y$ by $\beta_j$.

But in Logistic Regression, $y$ is probabilistic — so $\beta_j$ changes log-odds, not raw outcomes.

This log-odds view gives us two benefits:

It linearizes probability relationships.
It allows us to use good old linear math, while keeping probabilities bounded between 0 and 1.

How It Fits in ML Thinking

Interpretability is crucial in many ML contexts — from credit scoring to medical diagnosis.

Logistic Regression is beloved because:

You can directly read off the influence of each variable.
The model’s decisions are explainable to humans — no “neural magic” involved.

That’s why it’s often used as a baseline model and as a transparent alternative when interpretability trumps accuracy.

📐 Step 3: Mathematical Foundation

Now let’s translate those log-odds into something more intuitive — odds ratios.

Log-Odds to Odds Ratio

$$ e^{\beta_j} = \text{odds ratio} $$

If $e^{\beta_j} > 1$: a one-unit increase in $x_j$ increases the odds of the outcome.
If $e^{\beta_j} < 1$: a one-unit increase in $x_j$ decreases the odds of the outcome.
If $e^{\beta_j} = 1$: $x_j$ has no effect.

Think of the odds ratio as a multiplier: if $e^{\beta_j} = 2$, it means “this feature doubles the odds” of the event (not the probability!).

Scaling Effects on Coefficients

Logistic Regression is scale-sensitive. If one feature is measured in kilometers and another in meters, their coefficients aren’t directly comparable.

That’s why we usually standardize features (mean = 0, variance = 1).

After scaling, each $\beta_j$ tells you the effect of a 1 standard deviation change in that feature — making coefficients easier to compare fairly.

The Multicollinearity Trap

When two or more features are highly correlated, the model struggles to assign proper credit.

Result?

Coefficients become unstable (signs may flip unexpectedly).
Standard errors inflate, reducing confidence in interpretation.

Example: Imagine including both “age” and “years of work experience” — they’re strongly correlated. The model might assign weird or contradictory coefficients just to balance their overlap.

Use Variance Inflation Factor (VIF) or regularization (like Ridge Regression) to control multicollinearity.

🧠 Step 4: Assumptions or Key Ideas

Each coefficient reflects the effect of its feature assuming all others remain constant.
Features must be on comparable scales to interpret $\beta$ meaningfully.
There should be no strong multicollinearity among predictors.

These ensure coefficients are interpretable and stable.

⚖️ Step 5: Strengths, Limitations & Trade-offs

Highly interpretable model — coefficients tell a clear story.
Helps identify feature importance in classification problems.
Connects cleanly to domain understanding (especially with odds ratios).

Interpretation can break down with correlated features.
Coefficients lose meaning if features are not scaled properly.
Nonlinear relationships can’t be captured.

You trade raw predictive power for clarity. Logistic Regression may not win accuracy competitions, but it wins trust — especially in fields like healthcare, finance, and risk modeling, where interpretability is king.

🚧 Step 6: Common Misunderstandings

🚨 Common Misunderstandings (Click to Expand)

❌ “Negative coefficients mean the feature is bad.” → It simply means higher values of that feature reduce the odds of the event — not that it’s harmful.
❌ “Coefficient = importance.” → Not always! A large coefficient on a rarely varying feature might have little impact overall.
❌ “Sign flips = bug.” → Often caused by multicollinearity or small data variance.

🧩 Step 7: Mini Summary

🧠 What You Learned: Coefficients in Logistic Regression represent how each feature affects the log-odds of the outcome.

⚙️ How It Works: Each $\beta_j$ adjusts the model’s predicted odds multiplicatively — through $e^{\beta_j}$, the odds ratio.

🎯 Why It Matters: Understanding coefficients helps explain and trust your model — a vital skill for ethical and reliable ML.

2.1 Understand Regularized Logistic Regression 1.3 Train Using Gradient Descent (From Scratch)