Text Classification with Logistic Regression

Text Classification with Logistic Regression

The sigmoid turns a score into a probability

Scoring three reviews in NumPy

Compute $\sigma(-1)$ and round to three decimal places.

A binary sentiment classifier has weights $\mathbf{w} = [-1, 1, 0.5]$ applied to the feature vector $\mathbf{x} = [3, 1, 2]$ and bias $b = -0.5$. Compute the logit $z = \mathbf{w} \cdot \mathbf{x} + b$.

A binary classifier outputs $\hat{y} = 0.71$ as $P(y{=}1 \mid x)$. With the standard threshold, which class does the classifier predict, and which condition explains it?

Compute the logit of $p = 0.8$, that is $\ln \frac{p}{1-p}$, and round to three decimal places.

A test set stacks $m = 6$ documents as rows of $\mathbf{X}$, each with $n = 6$ features. With a weight vector $\mathbf{w}$ and bias vector $\mathbf{b}$ (one entry per document), the vectorized form is $\hat{\mathbf{y}} = \sigma(\mathbf{X}\mathbf{w} + \mathbf{b})$. How many probability values does $\hat{\mathbf{y}}$ contain? Answer with the number.