Typo in Deepleariningbook.org or am I misunderstanding Bayesian stats?

Question

This is on page 133 of the book: https://www.deeplearningbook.org/contents/ml.html#pf10

In the above, it says that

the data set is directly observed and so is not random

If that data we observe is not random, wouldn't the probability of the data occurring equal to 1. And thus, the denominator of Bayes' theorem also be 1? That should be incorrect as we wouldn't be able to normalize the numerator (a likelihood) if we divide it by 1...

In Bayesian stats, are both the data AND parameters of the data generating process both considered random? The book seems to imply that the data is fixed. In Frequentist, seems like only the data is viewed to be random.

Answer 1

This quote is a bit of an ambiguous statement if not a typo per se. Bayesian statistics fully exploits the fact that the data $x$ is the realisation of a random variable, $X=x$, with $$X\sim f(x|\theta)$$by actualising the prior distribution $\pi(\theta)$ into the posterior $$\pi(\theta|x) = \dfrac{\pi(\theta)f(x|\theta)}{\int_\Theta\pi(\theta)f(x|\theta)\,\text{d}\theta}$$ It would thus have been clearer to avoid such a statement and replace it with

the data set is directly observed and so is conditioned upon