A typical example is improving ranking in (custom/site specific) search. E.g. a client of a company I used to work at spent a huge amount with them having developers manually tweak search results. To add to it, they had multiple different stakeholders (e.g. sales team, executive team, their customers) who had different ideas about what should rank highly for different queries.
But they had a very simple metric right in front of their nose:
Their "product" was bookings at their customers restaurants. They were paid per cover.
One trivial improvement over the constant manual tweaks would be to simply record the probability of a user choosing to book at a given restaurant if that restaurant is present in the search result for a given term or set of terms.
Once you have that data, you can use Bayes theorem to take a set of tokens for a search and produce a list of probabilities that a given restaurant will be a good match, and rank accordingly. And you don't even need to understand Bayes theorem (though as someone who usually don't like maths papers, Bayes paper is remarkably understandable with very little mathematical background) - you just need to be aware of it and be able to use a simple library.
[you'll want to do some tweaks to account for the difference that position in the result makes]
There'll be an endless stream of improvements and more advanced methods you can use, but the beauty of applying Bayes theorem to data like this is that it is very simple, and you can explain to people why it ends up ranking the way it does very easily (the action you record, such as the decision to make a booking, serves as an "upvote" effectively).
There are lots of more advanced approaches you can take if you have the time and the skill and sufficient data, but the above works reasonably well with very little effort and is very often a substantial improvement over whatever manual attempts people make at guessing how the ranking ought to be. And it can be applied not just to search results, but to pretty much anything where you can derive an input set of terms (be it e.g. categories, or a paragraph from the text on the page) that has some likelihood of having some relation to how people will like another item, and where you can record an action against a related item as a "vote" for that connection.