So I don't have a rigorous background in probability theory but I am working my way through the Beta-Binomial model for Bayesian Data Analysis. The author says that given prior distribution $P(\theta)$ is a uniform distribution on an interval $[0,1]$ using Baye's rule we get that $$P(\theta|y) = \frac{P(\theta|y) P(\theta)}{p(y)}$$ Where likelihood = $P(y|\theta) = {n\choose y}\theta^n(1-\theta)^{n-y}$ Hence, $$P(\theta|y) \propto P(y|\theta) P(\theta)$$ $$P(\theta|y) \propto \theta^{y}(1-\theta)^{n-y}$$ I do understand why we can drop $n \choose y$ and the denominator(from Bayes rule) but I dont understand why the effect of multiplying with uniform distribution ($p(\theta)$) is like multiplying with identity.
I basically don't understand how multiplication with a uniform distribution takes place?