LaTex Module for QA

$\mathcal{Z} = \int_{-\infty}^{\infty} \exp\left(-\frac{1}{2} x^T \mathbf{\Sigma}^{-1} x \right) dx$

First, the model computes a value $z$, which is a linear sum of the input features $\mathbf{x}_i = (x_{i1}, x_{i2}, ..., x_{id})$ weighted by the model’s parameters (weights or coefficients) $\mathbf{w} = (w_1, w_2, ..., w_d)$ plus a bias term $b$. This is represented as:$$z_i = \mathbf{w}^T \mathbf{x}_i + b = w_1 x_{i1} + w_2 x_{i2} + \dots + w_d x_{id} + b$$$z_i$ directly models the ‘log-odds‘ (also known as the logit) of the positive class ($y=1$) occurring. The ‘log-odds‘ represent the logarithm of the ratio between the probability of the event happening and the probability of it not happening:$$z_i = \log\left(\frac{P(y_i=1 | \mathbf{x}_i)}{1 - P(y_i=1 | \mathbf{x}_i)}\right)$$This relationship highlights the assumption of ‘Linearity of Log Odds‘ mentioned in the Fundamentals of AI module.Second, to obtain the actual probability $p_i = P(y_i=1 | \mathbf{x}_i)$, the model applies the ‘sigmoid function‘, $\sigma$, to the log-odds value $z_i$. The ‘sigmoid function‘, as described previously, maps any real-valued input $z_i$ to a value between 0 and 1:$$p_i = \sigma(z_i) = \frac{1}{1 + e^{-z_i}} = \frac{1}{1 + e^{-(\mathbf{w}^T \mathbf{x}_i + b)}}$$This probability $p_i$ represents the model’s confidence that the input instance $\mathbf{x}_i$ belongs to the positive class ($y=1$).