Logit Space

1. What is Logit?

The logit function is the log-odds transformation of a probability $p$:

$\text{logit}(p) = \ln\left(\frac{p}{1-p}\right)$

• Input: probability $p \in (0,1)$
• Output: real number $(-∞, +∞)$

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2. What is Logit Space?

• Logit space refers to the domain where probabilities are expressed as log-odds instead of raw probabilities.
• In this space, probabilities are “unbounded”:
  • $p = 0.5 \;\;\Rightarrow\;\; \text{logit}(0.5) = 0$
  • $p \to 1 \;\;\Rightarrow\;\; \text{logit}(p) \to +\infty$
  • $p \to 0 \;\;\Rightarrow\;\; \text{logit}(p) \to -\infty$

This is useful because linear models can operate naturally in $(-∞, +∞)$ but not in $(0,1)$.

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3. Why Use Logit Space?

1. Linearization: Logistic regression models log-odds as a linear function of inputs:
   • $\text{logit}(p) = \beta_0 + \beta_1 x_1 + \dots + \beta_n x_nl$​
   • This makes estimation with linear methods possible.
2. Unbounded Range: Probabilities are limited to [0,1], but logit values can stretch across all real numbers, making optimization easier.
3. Interpretability:
   • Each coefficient in logistic regression = change in log-odds per unit change in predictor.
   • Exponentiating gives odds ratios (easier to interpret).

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4. Example

Suppose probability of purchase = 0.8.

• Odds = $0.8/0.2 = 4$.
• Logit = $\ln(4) \approx 1.386$.

So, in logit space, this customer has a log-odds of 1.386.

If another predictor increases logit by +0.7, the new log-odds = 2.086.
Converting back to probability:

$p = \frac{1}{1+e^{-2.086}} \approx 0.89$

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5. Applications

• Logistic regression: Works in logit space.
• Bayesian statistics: Priors on probabilities often modeled in logit space.
• Neural networks: Raw outputs before applying sigmoid are in logit space (often called logits).

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In short:
Logit space is the transformed domain where probabilities $(0,1)$ are expressed as log-odds $(-∞,+∞)$. Logistic regression operates in this space because it makes probability modeling linear and mathematically convenient.