Which is the decision rule in a gda classifier? On the text book there is the following formula for the prediction rule, but I don’t understand where it comes from:

The textbooks says it had been derived from
I suppose pi and theta are the parameters of the model but I really don’t get where the log comes from
 A: The chapter you're referring is about discriminant analysis with Gaussian mixtures, and $\mathbf{\pi}$ vector is the mixture probabilities, i.e. class priors. Also, $\theta_c$ represents all the parameters (i.e. mean vector and covariance matrix, and priors) for class $c$. We're interested in classifying a samples as best as we can, and in this framework, this is done through choosing the class maximizing the posterior: $$p(y=c|x,\theta)=\frac{p(x|y=c,\theta)p(y=c|\theta)}{p(x|\theta)}$$
while maximizing this expression wrt $c$, denominator has no effect and you can omit it. So, you maximize
$p(x|y=c,\theta)p(y=c|\theta)$. And you can also maximize the log of the expression since it's monotonically increasing. Also, considering that $p(x|y=c,\theta)=p(x|\theta_c)$ (because if class is given as $c$, we're only interested in parameters related to class $c$), and $p(y=c|\theta)=p(y=c|\pi)$ since probability of being class $c$ w/o data, i.e. prior, depends on only the prior probabilities and not mean/covariance of different mixture components.
So, the optimization objective becomes the following:
$$\log p(x|\theta_c) + \log p(y=c|\pi)$$
And you choose the argmax $c$ of it to decide the class of a given $x$.
