I read the page on Lindsey's paradox, and it's astonishing bullshit. It's well known that with sufficiently insane priors you can come up with stupid conclusions. The page asserts that a Bayesian would accept as reasonable priors that it's equally likely that the probability of child being born male is precisely 0.5 as it is that it has some other value, and also that if it has some other value that all values in the interval from zero to one are equally likely. But nobody on God's green earth would accept those as reasonable values, least of all a Bayesian. A Bayesian would say there's zero chance of it being precisely 0.5, but it is almost certainly really close to 0.5, just like a normal human being would.
A few points because I actually think Lindley’s paradox is really important and underappreciated.
(1) You can get the same effect with a prior distribution concentrated around a point instead of a point prior. The null hypothesis prior being a point prior is not what causes Lindley’s paradox.
(2) Point priors aren’t intrinsically nonsensical. I suspect that you might accept a point prior for an ESP effect, for example (maybe not—I know one prominent statistician who believes ESP is real).
(3) The prior probability assigned to each of the two models also doesn’t really matter, Lindley’s
paradox arises from the marginal likelihoods (which depend on the priors for parameters within each model but not the prior probability of each model).
Are you seriously saying that, because a point distribution may well make sense if the point in question is zero (or 1) other points are plausible also? Srsly?
The nonsense isn't just that they're assuming a point probability, it's that, conditional on that point probability not being true, there's only a 2% chance that theta is .5 += .01. Whereas the actual a priori probability is more like 99.99%.
> The nonsense isn't just that they're assuming a point probability, it's that, conditional on that point probability not being true, there's only a 2% chance that theta is .5 += .01. Whereas the actual a priori probability is more like 99.99%.
The birth sex ratio in humans is about 51.5% male and 48.5% female, well outside of your 99.99% interval. That’s embarrassing.
You are extremely overconfident in the ratio because you have a lot of prior information (but not enough, clearly, to justify your extreme overconfidence). In many problems you don’t have that much prior information. Vague priors are often reasonable.
Indeed, Bayesian approaches need effort to correct bad priors, and indeed the original hypothesis was bad.
That said. First, in defense of the prior, it is infinitely more likely that the probability is exactly 0.5 than it is some individual uniformly chosen number to each side. There are causal mechanisms that can explain exactly even splits. I agree that it's much safer to use simpler priors that can at least approximate any precise simple prior, and will learn any 'close enough' match, but some privileged probability on 0.5 is not crazy, and can even be nice as a reference to help you check the power of your data.
One really should separate out the update part of Bayes from the prior part of Bayes. The data fits differently under a lot of hypotheses. Like, it's good to check expected log odds against actual log odds, but Bayes updates are almost never going to tell you that a hypothesis is "true", because whether your log loss is good is relative to the baselines you're comparing it against. Someone might come up with a prior on the basis that particular ratios are evolutionarily selected for. Someone might come up with a model that predicts births sequentially using a genomics-over-time model and get a loss far better than any of the independent random variable hypotheses. The important part is the log-odds of hypotheses under observations, not the posterior.
This Veritasium video does a great job at explaining how such skewed priors can easily appear in our current academic system and the paradox in general: https://youtu.be/42QuXLucH3Q?si=c56F7Y3RB5SBeL4m
