Logistic Regression Prediction

Implement the prediction step of logistic regression.

Given a feature matrix $X$ (without bias), a weight vector $w$ , and a bias $b$ , compute the probability for each sample using:

$P(y=1|x) = \sigma(x \cdot w + b) = \frac{1}{1 + e^{-(x \cdot w + b)}}$

Then classify each sample: if $P \geq 0.5$ , predict 1; otherwise predict 0.

Return a list of tuples [(probability, prediction), ...] with probabilities rounded to 4 decimal places.

Input:

X = [[1, 2], [3, 4], [-1, -2]]
w = [0.5, -0.3]
b = 0.1

Output:

[(0.5, 1), (0.5987, 1), (0.5498, 1)]

Reasoning:

We compute the dot product of each sample in $X$ $X$ with the weight vector $w$ $w$ and add the bias $b$ $b$ :
- For the first sample: $1 \cdot 0.5 + 2 \cdot -0.3 + 0.1 = 0.5 - 0.6 + 0.1 = 0$
- For the second sample: $3 \cdot 0.5 + 4 \cdot -0.3 + 0.1 = 1.5 - 1.2 + 0.1 = 0.4$
- For the third sample: $-1 \cdot 0.5 + -2 \cdot -0.3 + 0.1 = -0.5 + 0.6 + 0.1 = 0.2$
Then, we apply the sigmoid function $\sigma(x) = \frac{1}{1 + e^{-x}}$ $σ (x) = \frac{1}{1 + e ^{- x}}$ to each result:
- For the first sample: $\sigma(0) = \frac{1}{1 + e^{0}} = \frac{1}{1 + 1} = 0.5$
- For the second sample: $\sigma(0.4) = \frac{1}{1 + e^{-0.4}} \approx 0.5987$
- For the third sample: $\sigma(0.2) = \frac{1}{1 + e^{-0.2}} \approx 0.5498$
We classify each sample based on the predicted probability, with $P \geq 0.5$ resulting in a prediction of 1, and $P < 0.5$ resulting in a prediction of 0.
The final output is a list of tuples containing the predicted probabilities rounded to 4 decimal places and the corresponding predictions: $(0.5, 1), (0.5987, 1), (0.5498, 1)$