Phase 4: The Mathematical Truth: Linear vs Logistic Regression
How to beautifully explain the mathematical truth behind Linear and Logistic regression
(Still under construction)
1. The Theoretical World: The Full Data-Generating Process
In the idealized mathematical world, we assume the existence of a Data-Generating Process (DGP) —
a mechanism that maps all causal inputs in the universe to the variable of interest \(Y\):
\[ Y = f(X_1, X_2, X_3, \ldots) \]
Each \(X_j\) represents a different characteristic or influence (education, experience, intelligence, etc.),
and together they fully determine the value of \(Y\).
If we had complete access to this \(f(\cdot)\) and to all relevant \(X_j\)’s, the process would be deterministic:
given the inputs, we could predict \(Y\) exactly.
Formally, the joint distribution of all variables is
\[ (Y, X_1, X_2, X_3, \ldots) \sim F_{Y, X_1, X_2, X_3, \ldots} \]
and the DGP defines the conditional distribution
\[ F_{Y \mid X_1, X_2, X_3, \ldots}(y \mid x_1, x_2, x_3, \ldots) \]
which would be perfectly degenerate (a Dirac delta) if the world were truly deterministic.
2. The Linear Regression Case: Deterministic DGP, Incomplete Knowledge
2.1. From Theory to Practice
In the real world, we rarely observe all the variables that influence \(Y\).
Suppose the true DGP is
\[ Y = 20 + 3X_1 + 2X_2 \]
where \(X_1\) is years of education and \(X_2\) is years of experience.
If we knew both \(X_1\) and \(X_2\), the model would be deterministic:
for \(X_1=10, X_2=5\), we know \(Y=20+3(10)+2(5)=50\).
But imagine we only measure \(X_1\).
We must then approximate the conditional behavior of \(Y\) given \(X_1\):
\[ Y = \beta_0 + \beta_1 X_1 + \varepsilon \]
The term \(\varepsilon\) now captures the influence of \(X_2\) (and any other unobserved factors):
\[ \varepsilon = 2X_2 \]
and since \(X_2\) varies randomly in the population, \(\varepsilon\) is random to us.
This motivates the probabilistic model
\[ Y \mid X_1 = x_1 \sim \mathcal{N}(\beta_0 + \beta_1 x_1, \sigma^2) \]
where the conditional variance \(\sigma^2\) reflects epistemic randomness — randomness due to our ignorance about unobserved determinants.
In other words, the world might be deterministic, but our model is not.
2.2. Conceptual Picture: “Slicing” the Joint Distribution
The true DGP defines a high-dimensional surface in the space of all variables.
By conditioning on only one variable (\(X_1\)), we take a slice through that surface:
\[ F_{Y \mid X_1}(y \mid x_1) \]
This slice still contains variation from the omitted axes (\(X_2, X_3, \ldots\)).
That variation appears as the random scatter of points around the regression line.
Hence, in linear regression:
Randomness arises because our model conditions on an incomplete subset of the true DGP.
If we had access to all the relevant \(X_j\)’s and the true functional form \(f\),
the process would become deterministic and \(Y\) would be known with certainty.
3. The Logistic Regression Case: A Probabilistic DGP
Now consider a fundamentally different scenario.
Let \(Y\) represent whether a person gets hired (\(Y=1\)) or not hired (\(Y=0\)).
We still have predictors such as education \(X_1\) and experience \(X_2\).
Even if we could measure every possible factor, two people with identical \((X_1, X_2)\) might still face different outcomes.
The process of hiring, medical recovery, or clicking an ad is inherently random at the individual level.
Here, the DGP itself is stochastic:
\[ Y \mid X_1, X_2 \sim \text{Bernoulli}(p(X_1, X_2)) \]
where
\[ p(X_1, X_2) = P(Y=1 \mid X_1, X_2) \]
This means that the theoretical DGP is not a deterministic function \(f(\cdot)\) but a probability law describing how likely each outcome is.
We specify the probability function \(p(\cdot)\) using a logistic form:
\[ p(x_1, x_2) = \frac{e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}}{1 + e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2}} \]
3.1. Example: The Hiring Probability
Let
\[ \beta_0 = -8, \quad \beta_1 = 0.5, \quad \beta_2 = 0.2 \]
For a candidate with 12 years of education (\(X_1=12\)) and 5 years of experience (\(X_2=5\)):
\[ p = \frac{e^{-8 + 0.5(12) + 0.2(5)}}{1 + e^{-8 + 0.5(12) + 0.2(5)}} = \frac{e^{-1}}{1 + e^{-1}} \approx 0.27 \]
Thus, even knowing all the predictors and parameters,
\[ P(Y=1 \mid X_1=12, X_2=5) = 0.27 \]
means that among 100 identical candidates (same \(X_1, X_2\)),
around 27 would be hired and 73 would not — but we cannot know which ones.
Each individual outcome is generated by an independent Bernoulli(0.27) draw.
| Candidate | Education \(X_1\) | Experience \(X_2\) | \(p_i\) | Realized \(Y_i\) |
|---|---|---|---|---|
| 1 | 12 | 5 | 0.27 | 0 |
| 2 | 12 | 5 | 0.27 | 1 |
| 3 | 12 | 5 | 0.27 | 0 |
| 4 | 12 | 5 | 0.27 | 0 |
| 5 | 12 | 5 | 0.27 | 1 |
Even with complete knowledge of the DGP, \(Y\) remains random.
This randomness is ontological — it comes from the probabilistic nature of the DGP itself,
not from ignorance about missing variables.
4. The Fundamental Distinction
| Feature | Linear Regression | Logistic Regression |
|---|---|---|
| Nature of DGP | Deterministic function of all relevant \(X_j\) | Intrinsically probabilistic (Bernoulli process) |
| Conditional distribution | \(Y \mid X \sim \mathcal{N}(\mu(X), \sigma^2)\) | \(Y \mid X \sim \text{Bernoulli}(p(X))\) |
| Randomness arises from | Omitted or unobserved variables (epistemic) | The event-generation mechanism itself (ontological) |
| What the model approximates | Conditional mean \(E[Y\mid X] = \mu(X)\) | Success probability \(P(Y=1\mid X) = p(X)\) |
| If we knew all of \(X\) and the true DGP | \(Y\) becomes deterministic | \(Y\) remains random |
| Limiting case (\(p=0\) or \(1\)) | Perfect prediction possible | Becomes deterministic, but no longer logistic |
| Conceptual analogy | “The world is deterministic but we can’t see it all.” | “The world itself flips a weighted coin.” |
Hence:
🔹 Linear regression models a deterministic world we only partially see.
🔹 Logistic regression models a probabilistic world whose outcomes remain random even when all causes are known.
5. Appendix — Conditioning as Projection on Subspaces (Optional)
In the theoretical world, the joint distribution \(F_{Y,X_1,X_2,\ldots}\) defines a high-dimensional space of relationships between all variables. Conditioning on a subset of them — for instance, only \(X_1\) — corresponds to taking a projection of that space onto a lower-dimensional subspace.
In linear algebraic terms, if we represent \(Y\) and the \(X_j\)’s as elements of a vector space of random variables with an inner product \(\langle A,B \rangle = E[AB]\), then the conditional expectation \(E[Y \mid X_1]\) is precisely the orthogonal projection of \(Y\) onto the subspace spanned by \(X_1\).
The residual \(\varepsilon = Y - E[Y \mid X_1]\) is orthogonal to that subspace:
\[ E[\varepsilon X_1] = 0 \]
This provides a geometric interpretation of “taking a slice” of the DGP: we restrict our attention to one axis (one subspace) of the full theoretical space, and the remaining variation appears as randomness in the orthogonal complement.
Summary Insight:
Linear regression transforms deterministic complexity into stochastic simplicity
by conditioning on incomplete information.
Logistic regression models inherent probabilistic events,
where randomness persists even when all causes are known.