Log-likelihood function


### Becuz log monotonically increasing argmax easy
should maximize likelihood function, so negative value when it is logged
$l(\theta)$ is log likelihood and $-l(\theta)$ is negative log likelihood 
$$L(\theta) = L(\theta; X, \vec{y}) = p(\vec{y}|X;\theta)$$
$$l(\theta) = log{L(\theta)} = \Sigma_{i=1}^np(y^{(i)} |x{(i)};\theta)$$
it measures how well the parameters fit the observed data. The notation used to represent the likelihood function is L(θ), where θ represents the parameters of the model, and X and y represent the data. The likelihood function is defined as the conditional probability of the observed data given the values of the parameters of the model: $L(\theta) = p(y|X;\theta)$.


j function usually means negative log likelihood









