[CrazyStat]

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).

[gweinberg]

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%.

[CrazyStat]

Srsly? 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 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.