# Clarification regarding Parameter Estimation (Andriy Burkov's book)

So I recently decided to read Andriy Burkov's "The 100-Page Machine Learning Book" and got confused in Chapter Two (Page 11) where he discusses Parameter Estimation techniques. A picture of the relevant section has been attached in the end and the problematic parts have been highlighted.

$$(1)\$$To me it seems that applying Bayes' Theorem in this particular context would yield:$$Pr(\theta = \hat \theta |X=x)=\frac{Pr(X=x|\theta=\hat \theta)\ Pr(\theta = \hat \theta)}{Pr(X=x)}=\frac{Pr(X=x|\theta=\hat \theta)\ Pr(\theta = \hat \theta)}{\sum_{\tilde\theta}Pr(X=x|\theta=\tilde\theta)\ \color{red}{Pr(\theta=\tilde\theta)}}$$ but the book doesn't have the same denominator in the last fraction and I wonder why that is the case.

$$(2)$$ As per my understanding of Maximum Likelihood Estimate, the objective is to find that value of $$\theta$$ which maximizes the Likelihood function given by:$$L(\theta)=\prod_{i=1}^n f(x_i|\theta)$$ but the book seems to have used a different expression for the Likelihood function, the origins of which remain unknown to me.

If someone could shed some light here, that'd be really helpful.