Lab 1: Loglikelihood formula fix

lab1/lab-notes.md

@@ -104,7 +104,7 @@ Confusion matrix o misclassification error e framtana.
 
 3. The functions where implemented using the following formulas:
 
-   - _Loglikelihood_: $$\log P(T | \theta, \sigma) = -\frac{n}{2} \log(2 \pi \sigma^2) - \frac{1}{2 \sigma^2} \sum_{i=1}^{n} (T_i - \mathbf{X}_i \boldsymbol{\theta})^2$$
+   - _Loglikelihood_: $$\log P(T | \theta, \sigma) = \log P(y |\mathbf{X}, \theta, \sigma) = -\frac{n}{2} \log(2 \pi \sigma^2) - \frac{1}{2 \sigma^2} \sum_{i=1}^{n} (y_i - \mathbf{X}_i \boldsymbol{\theta})^2$$
    - _Ridge_: $$\mathcal{L}_{\text{ridge}}(\theta, \sigma, \lambda) = - \log P(T | \theta, \sigma) + \lambda \sum_{j=1}^{p} \theta_j^2$$
    - _RidgeOpt_: $$\hat{\theta}, \hat{\sigma} = \arg \min_{\theta, \sigma} \mathcal{L}_{\text{ridge}}(\theta, \sigma, \lambda)$$
    - _DF_: $$\text{df}(\lambda) = \text{tr}\left( X \left( X^T X + \lambda I \right)^{-1} X^T \right)$$