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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.


enter image description here

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The author here. You are right and the denominator on the first screenshot indeed misses a term. This was fixed several weeks ago and the book on Amazon and Leanpub now contains the fixed content.

As for your second screenshot, again in the updated version of the book, "maximum likelihood" was replaced by "maximum a posteriori".

In case of doubt, please refer to the online version of the book available at http://themlbook.com/wiki/doku.php as it is updated more regularly compared to the printed edition.

Sorry for the inconvenience.

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