What's the big deal about Bayes' Theorem?

In daily life, the basic move is to ask,

• What are the straightforward and the alternative explanations (hypotheses) for what I'm seeing?
• How much more likely is one compared to the other a priori (when ignoring what I'm seeing)?
• What probabilities do they assign to what I'm seeing?

and get the ratio of a posteriori probabilities of the hypotheses (a posteriori odds) from that (by multiplying the a priori odds by the likelihood ratio). Odds measure the relative strengths of hypotheses, so the move is to obtain relative strengths of a pair of hypotheses after observing a piece of data, with the choice of hypotheses inspired by the same piece of data. This is a very easy computation that becomes a habit and routinely fixes intuitive misestimates. Usually it's about explanation of data/claim as correct vs. constructed by a specific sloppy process that doesn't ensure correctness.

That is, for $D$ the data/claim you are observing, and $x$ and $y$ hypotheses chosen as possible explanations for $D$, $$\frac{P\left(x|D\right)}{P\left(y|D\right)}=\frac{P\left(x\right)}{P\left(y\right)}\times \frac{P\left(D|x\right)}{P\left(D|y\right)}.$$ This holds for any choice of two hypotheses $x$ and $y$, which don't have to be mutually exclusive or exhaust all possibilities, and there may be many other plausible hypotheses.

As you've noted, Bayes' Theorem is just a straight forward result of probability calculus. In that light, it is entirely uncontroversial. 

What people really seem to get excited about is Bayesianism, which is something more than just the application of Bayes' Theorem. 

To understand people's interest in Bayesianism, I think you then need to distinguish its use in two types of applications: how we use probabilities to deal with uncertainty when drawing inferences from data generated by scientific studies (i.e. statistical inference); and whether humans reason/learn, or should reason/learn, in a Bayesian manner. 

The latter would be well outside my own expertise, but I once got a fair number of interesting responses to this question from people that would know better than I. Regarding its use in statistical inference, Bayesianism is similarly controversial, and the many controversies are the subject of hundreds and thousands of papers and books. 

The link between probability theory and logic is most opportune. There is a whole ---albeit not very popular--- branch of probability theory and statistics that goes by various names (e.g., logical inference) that considers probability an extension of logic in the sense that it helps us come up with adequate solutions to problems whose premises are not stated in terms of true or false but are assigned some credibility. Names associated with this school of though are Keynes (because of his 1921 book on probability theory), Carnap and, more recently, Ian Hacking.

In this program, of course, Bayes Theorem plays an important role. If you consider logic more important than probability is, think again how many times do you work with premises that are 100% true or false and whether it would make sense to extend it to allow for plausibilities rather than certainties. 

For your first half-question, "A Technical Explanation of Technical Explanation" [? · GW] [Edit: added link] sums up the big deal; Bayes Theorem is part of what actually underpins how to make a map reflect the territory (given infinite compute) - it is a necessary component. In comparison with other necessary components required to do this (i.e. logic, math, other basic probability) I would conjecture that Bayes is only special in that it is 'often' the last piece of the puzzle that is assembled in someone's mind, and thus takes on psychological significance. 

For the second half-question, I think using Bayes in life is more about understanding just how much priors matter than about actually crunching the numbers.

As an example, commenters on LessWrong will sometimes refer to 'Outside View' vs 'Inside View'. 

A quick example to summarise roughly what these mean. If predicting how long 'Project X' will take, an Outside View goes 'well, project A took 3 weeks, and project B took 4, and project C took 3, and this project is similar, so 3-4 weeks', whereas the Inside View goes 'well to do project X, I need to do subprojects Y1, Y2, Y3 and Y4, and these should take me 3, 4, 5 and 4 days respectively, so 16 days = 2 and a bit weeks'. The Inside View is susceptible to the planning fallacy, etc etc. 

General perspective: Outside View Good, Inside View Can Go Very Wrong.

You've probable guessed the punchline; this is really about Bayes Theorem. The Outside View goes "I'm going to use previous projects as my prior (technically this forms a prior distribution of estimated project lengths), and then just go with that and try to avoid updating very much, because I have a second prior that says 'all the details of similar projects didn't matter in the past, so I shouldn't pay too much attention to them now", whereas the Inside View Going Very Wrong is what happens when you throw out your priors; you can end up badly calibrated very quickly.

I think you're coming at this question from a very different perspective than I do, but for me, it isn't about a mathematical result, which, as you and others noted, is very simple and entirely uncontroversial. I also expect it to be uncontroversial for say that, for most people, most of the time, it is not a useful idea to try to explicitly calculate conditional probabilities for making predictions and decisions. Instead, for me what matters is that some of the very basic, mathematically obvious, and uncontroversial implications of conditional probability calculations have no bearing at all on how most of the people in our world make decisions, not just personal decisions but ones with enormous consequences for our society and our collective futures.

In my experience most people, when trying to reason logically, think by default in binary categories and have no idea how to deal with uncertainty. I work in a job where my goal is to advise others on making various decisions based on my interpretation of often ambiguous technical, industry, and other data, and to help train others to do the same. I have never once made an explicit calculation using Bayes' theorem, but I consistently observe that a surprisingly (to me) large fraction of people start out extremely resistant to making reasonable assumptions based on prior knowledge and seeing where they lead, and instead insist that if they don't have "certainty" they can't say anything useful at all. Those people much more often fail to make good predictions, miss out on large opportunities, and waste exorbitant amounts of time and effort chasing down details that do not meaningfully affect the final output of their prediction and decision making processes. They're also more likely to miss huge, gaping holes in their models of the world that make all their careful attempts to be certain completely meaningless. 

In other words, for me the point is to develop a better intuitive understanding of how to make reasoned decisions that are likely to lead to desired outcomes - aka to become less wrong, and wrong less often. It's partly because Bayes' theorem is so simple that, for a certain type of person, it serves as a useful entry point on the path to that goal.