I’m not sure what you mean by proper or improper priors here. At first you seem to be talking about self-consistent but bad priors in your coinflip example, but then when you talk about proper allocation of mass you seem to be talking about self-consistency of priors. These are different issues.
This isn’t true with an improper prior. If I wanted to predict the value of a random real number, and used a normal distribution with a mean of zero and a standard deviation of one, I’d be pretty darn surprised if it doesn’t end up being pretty close to zero, but I’d be infinitely surprised if I used a uniform distribution.
There is no uniform distribution on the real line.
Your new first paragraph is not the definition. Partly it goes opposite the definition and partly it is orthogonal. It is so confused, I’m surprised that the other material is (or looked) correct. You should separate your consideration of continuous priors from improper priors. An example of an improper prior in a discrete setting is the uniform prior on positive integers. Another example is the prior p(n) = 1/n.
I’m not sure what you mean by proper or improper priors here. At first you seem to be talking about self-consistent but bad priors in your coinflip example, but then when you talk about proper allocation of mass you seem to be talking about self-consistency of priors. These are different issues.
There is no uniform distribution on the real line.
“Improper prior” is a technical term for using an infinite measure as a prior.
Ah, thanks. I was not aware of that term. Maybe linking or explaining that in the post might not be a bad idea.
Edited to add this.
Your new first paragraph is not the definition. Partly it goes opposite the definition and partly it is orthogonal. It is so confused, I’m surprised that the other material is (or looked) correct. You should separate your consideration of continuous priors from improper priors. An example of an improper prior in a discrete setting is the uniform prior on positive integers. Another example is the prior p(n) = 1/n.
I am also confused. More specifically, improper priors are priors that integrate to infinity and thus cannot be normalized.
That’s almost the definition, except that improper priors are not priors.
Is that your confusion?
No, I mean I share your confusion that the rest of the conversation appeared reasonable given the incorrect definition in the post.
Sorry. Probably part of the miscommunication is that I used “confused” to describe Daniel LC and “surprised” to describe myself.