# Why is posterior density proportional to prior density times likelihood function?

According to Bayes’ theorem, $P(y|\theta)P(\theta) = P(\theta|y)P(y)$. But according to my econometric text, it says that $P(\theta|y) \propto P(y|\theta)P(\theta)$. Why is it like this? I don’t get why $P(y)$ is ignored.

$Pr(y)$, the marginal probability of $y$, is not “ignored.” It is simply constant. Dividing by $Pr(y)$ has the effect of “rescaling” the $Pr(y|\theta)P(\theta)$ computations to be measured as proper probabilities, i.e. on a $[0,1]$ interval. Without this scaling, they are still perfectly valid relative measures, but are not restricted to the $[0,1]$ interval.
$Pr(y)$ is often “left out” because $Pr(y)=\int Pr(y|\theta)Pr(\theta)d\theta$ is often difficult to evaluate, and it is usually convenient enough to indirectly perform the integration via simulation.