Cursed Bayesianism

Take this system of updating your beliefs in the face of evidence, but beware it will make you a stranger to the mainstream ways of working with statistical evidence in your culture.

That’s bad

But the system results in you having beliefs that are locally optimal: any small departure from them would leave you open to exploitation by a clever packaging of bets. But the system will save you from that, and in fact is the only thing that can!

That’s good!

But, the system requires having a complete enumeration of every hypothesis that could explain the evidence. And then it asks you to go through every element of this infinitude, and say precisely and numerically with what probability the hypothesis predicts the data. And then it asks you to tot up that infinitude of data so you can use it as a normalising constant.

That’s bad!

But! As long as you’re just comparing hypotheses to each other, as long as you’re just asking “does this evidence favour hypothesis A or hypothesis B more?”, as long as you do not want an actual probability of how likely some specific hypothesis is, then you only need the ratio of how likely each hypothesis makes each piece of evidence.

Because, remember that big old totality of every which way your evidence could come about that you had to consider? Well that plays the same role in the calculation regardless of which hypothesis you’re looking at: it’s something you divide by, something that normalises. So because you’re comparing a ratio of two things, each feature the same divisor, you can throw that divisor away. That horrible tricky thing isn’t needed.

That’s good!

But you do actually have to have the likelihoods. Suppose you want to know how temperature impacts frogurt sales. You come up with some models that take in a temperature, and output a probability distribution over frogurt sales.

There are multiple messy steps in this process, for example, the model you make might not end up being the One Ultimate Model of how temperature affects frogurt sales, but end up encoding a bunch of substantive assumptions you didn’t mean it to.

Or, if you avoid that fate, and you make a great model that reflects the hypothesis you’re trying to investigate, you might find that even pulling the likelihood out of just that one hypothesis is basically intractable, and that you can’t do it.

That’s bad.

But nowadays we can use gradient-based methods to approximate the likelihood for a wide class of flexibly-specified models. You can more or less just code up the model that you’re trying to consider, and powerful, general tools like STAN will use MCMC and a bunch of fancy tricks to spit out an approximate answer. You barely have to write any special custom code, and you don’t need to learn any nasty formulae.

That’s good!

But yeah sometimes it’s unbearably slow, and it doesn’t even get substantially different inferences than hacky fast solutions.

…

That’s bad.