I've been working through the paper Bell nonlocality by Brunner et al. after seeing it in user glS' answer here. Early on in the paper, the standard Bell experimental setup is defined:
Where $x, y \in \{0,1\}$, $a, b \in \{-1, 1\}$, and the two people (Alice & Bob) measure a shared quantum system generated by $S$ according to their indepedent inputs $x$ and $y$, outputting the results as $a$ and $b$.
The paper then has the following equation:
$P(ab|xy) \ne P(a|x)P(b|y)$
And claims the fact this is an inequality means the two sides are not statistically independent. It's been a long time since I took probability & statistics in university, so I'm interested in this equation, what it means, and why it is a test for statistical independence. Why is this equation used, and what is the intuitive meaning of each side? I have basic knowledge of conditional probability and Bayes' theorem.