Logistic regression — when a linear score isn't a probability

Background

A linear model outputs a raw score $\hat{y} = \mathbf{w}^\top \mathbf{x} + b$ that can be any real number — $5$, $-0.3$, or $\pi$. That is fine for "dollars", but a class probability must live in $[0, 1]$. This mismatch is the core reason plain linear regression fails at classification, and the whole motivation for the sigmoid that follows.

Problem statement

Implement valid_probabilities(scores) that returns a boolean mask flagging which raw scores are already in the valid probability range $[0, 1]$ (inclusive).

Input

• scores — array-like of real-valued linear-model outputs.

Output

Returns a boolean np.ndarray the same length as scores: True where $0 \le \text{score} \le 1$, else False.

Examples

Example 1

Input:  scores = [5, -0.3, 0.6, 0.4, 1.0, 3.14159]
Output: [False False  True  True  True False]

Explanation: only $0.6$, $0.4$, and $1.0$ are usable as probabilities; $5$, $-0.3$, and $\pi$ escape $[0, 1]$ — exactly the failure the lesson highlights.

Example 2

Input:  scores = [0.0, 1.0]
Output: [ True  True]

Explanation: the endpoints $0$ and $1$ are valid probabilities (the check is inclusive).

Constraints

• Treat the range as inclusive on both ends: $0$ and $1$ count as valid.
• Convert scores to a NumPy array and build the mask with a single vectorised comparison — no Python loop.

Notes

• This is the diagnostic that motivates logistic regression: a linear score is unbounded, so we need a squashing function (the sigmoid) to guarantee every output lands in $[0, 1]$.