# Negative log-likelihood for normal distribution
  nll = -np.sum(-0.5*np.log(2*np.pi*sigma**2) - (y-predicted_y)**2 / (2*sigma**2))

[ nll = -\sum_{i=1}^{n} \left[ -\frac{1}{2}\log(2\pi\sigma^2) - \frac{(y_i-\hat{y_i})^2}{2\sigma^2} \right] ]

Where:

( n ) is the number of observations.
( \log ) denotes the natural logarithm.
( \pi ) represents the mathematical constant pi.
( \sigma ) is the standard deviation.
( y_i ) represents the actual value for the i-th observation.
( \hat{y_i} ) represents the predicted value for the i-th observation.
The sum is over all ( i ) from 1 to ( n ).