To compute the vectorized gradient of the cost function for the logistic model, we can follow the process based on the information you provided. Let's break down the steps:

Given Information:

1. Logistic model: $g(-2.5 + 0.5X)$
2. Feature vector $X = [1, 2, 3]$
3. Label $Y = [0, 0, 0]$ (all zero indicates the negative class)
4. Initial bias $b_0$ is likely to be initialized as 0 (although not explicitly stated, this is common).

Steps to Evaluate:

1. Sigmoid Function:

The sigmoid function for logistic regression is: $\sigma(z) = \frac{1}{1 + e^{-z}}$ Where $z = -2.5 + 0.5 \times X$. The sigmoid will map the linear combination of inputs to a probability between 0 and 1.

2. Compute Cost:

The cost function for logistic regression is defined as: $J(\theta) = -\frac{1}{m} \sum_{i=1}^{m} \left[ y^{(i)} \log(h_{\theta}(x^{(i)})) + (1 - y^{(i)}) \log(1 - h_{\theta}(x^{(i)})) \right]$ Where $h_{\theta}(x^{(i)}) = \sigma(\theta^T X)$.

3. Gradient of the Cost Function:

The gradient of the cost function with respect to the parameters $\theta$ is given by: $\nabla J(\theta) = \frac{1}{m} X^T \left( \sigma(X \theta) - Y \right)$ This computes the direction and magnitude by which we adjust the weights to minimize the cost.

Let's go step by step and compute the result using the given vectors $X$, $Y$, and the function $-2.5 + 0.5X$. I'll now calculate the vectorized gradient.The vectorized gradient of the cost function for the given logistic model is approximately 0.43. This gradient indicates the direction and magnitude by which the model parameters should be adjusted to minimize the cost.

Would you like more details on any part of the process, or do you have any other questions?

Here are 5 questions to expand on this topic:

1. How does the gradient descent algorithm use the gradient to update parameters?
2. How does the sigmoid function shape influence the model's output?
3. How would the gradient change if the labels $Y$ were different?
4. How can regularization techniques like L2 be applied to this model?
5. What happens when we increase the number of features in the feature vector $X$?

Tip: In logistic regression, the gradient always points in the direction of steepest increase. Hence, parameters are updated in the opposite direction of the gradient.