Logistic regression — likelihood of the whole dataset

Background

Assuming the examples are independent, the probability of observing the entire labelled dataset is the product of each example's label-likelihood:

$$\mathcal{L} = \prod_{i=1}^{n} p_i^{\,y_i}\,(1 - p_i)^{\,1 - y_i} = \prod_{i=1}^{n} \begin{cases} p_i & y_i = 1 \\ 1 - p_i & y_i = 0 \end{cases}$$

Maximum likelihood estimation chooses the weights that make this product as large as possible. The lesson's point: for logistic regression this objective (via its log) is convex, so gradient descent finds the single global optimum.

Problem statement

Implement dataset_likelihood(y, p) returning the product over all examples of the probability assigned to each observed label.

Input

• y — array-like of true labels (0/1).
• p — array-like of predicted probabilities $\mathbb{P}(y=1)$, same length.

Output

Returns a Python float: the joint likelihood $\mathcal{L} \in [0, 1]$.

Examples

Example 1

Input:  y = [1, 0], p = [0.9, 0.8]
Output: 0.18

Explanation: class-1 contributes $0.9$, class-0 contributes $1-0.8=0.2$, and $0.9 \times 0.2 = 0.18$.

Example 2 — a perfect model scores 1

Input:  y = [1, 0, 1], p = [1.0, 0.0, 1.0]
Output: 1.0

Explanation: every observed label was given probability $1$, so the product is $1$.

Constraints

• Build each example's label-likelihood (p if y==1 else 1-p), then take the product.
• Return a plain float.

Notes

• Real implementations sum log-likelihoods instead of multiplying probabilities, because a product of many small numbers underflows to 0. That log form is binary cross-entropy — the next lesson.