# maximum likelihood and mse(mean square error)

thisis the equation of maxilum likelihood

Then this is maximum likelihgood when the distribution is Gaussian distribution.
I want to know to to derive the bottom equation from the left ?

• Don't post unsearchable images. Instead, typeset your full question (in MathJax). – David G. Stork Jun 11 '20 at 5:29

$$\frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{1}{2} \left(\frac{y - \hat{y}}{\sigma}\right)^2\right]$$
This is your $$P_{model}(y_i\ | \ x_i ; \theta)$$ for a single observation. Assuming each observation is independent, the likelihood (probability of observing $$y_1...y_m$$ together) is $$\prod_{i = 1}^m P_{model}(y_i\ | \ x_i ; \theta).$$ Taking the log of this whole expression gives your result.