it seems like a weird response to say “oh, well who cares about explicit assumptions anyways?”
Yeah, sorry. I was getting a little off topic there. It’s just that in your post, you were able to connect the explicit assumptions being true to some kind of performance guarantee. Here I was musing on the fact that I couldn’t. It was meant to undermine my point, not to support it.
What does it mean to “assume that the prior satisfies these constraints”?
?? The answer to this is so obvious that I think I’ve misunderstood you. In my example, the constraints are on moments of the prior density. In many other cases, the constraints are symmetry constraints, which are also easy to express mathematically.
But then you bring up concrete statements about the world? Are you asking how you get from your prior knowledge about the world to constraints on the prior distribution?
EDIT: you don’t “assume a constraint”, a constraint follows from an assumption. Can you re-ask the question?
Yeah, sorry. I was getting a little off topic there. It’s just that in your post, you were able to connect the explicit assumptions being true to some kind of performance guarantee. Here I was musing on the fact that I couldn’t. It was meant to undermine my point, not to support it.
Ah my bad! Now I feel silly :).
But then you bring up concrete statements about the world? Are you asking how you get from your prior knowledge about the world to constraints on the prior distribution?
So the prior is this thing you start with, and then you get a bunch of data and update it and get a posterior. In general it’s pretty unclear what constraints on the prior will translate to in terms of the posterior. Or at least, I spent a while musing about this and wasn’t able to make much progress. And furthermore, when I look back, even in retrospect it’s pretty unclear how I would ever test if my “assumption” held if it was a constraint on the prior. I mean sure, if there’s actually some random process generating my data, then I might be able to say something, but that seems like a pretty rare case… sorry if I’m being unclear, hopefully that was at least somewhat more clear than before. Or it’s possible that I’m just nitpicking pointlessly.
Hmm. Considering that I was trying to come up with an example to illustrate how explicit the assumptions are, the assumptions aren’t that explicit in my example are they?
Prior knowledge about the world --> mathematical constraints --> prior probability distribution
The assumptions I used to get the constraints are that the best estimate of your next measurement is the average of your previous ones, and that the best estimate of its squared deviation from that average is some number s^2, maybe the variance of your previous observations. But those aren’t states of the world, those are assumptions about your inference behavior.
Then I added later that the real assumptions are that you’re making unbiased measurements of some unchanging quantity mu, and that the mechanism of your instrument is unchanging. These are facts about the world. But these are not the assumptions that I used to derive the constraints, and I don’t show how they lead to the former assumptions. In fact, I don’t think they do.
Well. Let me assure you that the assumptions that lead to the constraints are supposed to be facts about the world. But I don’t see how that’s supposed to work.
Yeah, sorry. I was getting a little off topic there. It’s just that in your post, you were able to connect the explicit assumptions being true to some kind of performance guarantee. Here I was musing on the fact that I couldn’t. It was meant to undermine my point, not to support it.
?? The answer to this is so obvious that I think I’ve misunderstood you. In my example, the constraints are on moments of the prior density. In many other cases, the constraints are symmetry constraints, which are also easy to express mathematically.
But then you bring up concrete statements about the world? Are you asking how you get from your prior knowledge about the world to constraints on the prior distribution?
EDIT: you don’t “assume a constraint”, a constraint follows from an assumption. Can you re-ask the question?
Ah my bad! Now I feel silly :).
So the prior is this thing you start with, and then you get a bunch of data and update it and get a posterior. In general it’s pretty unclear what constraints on the prior will translate to in terms of the posterior. Or at least, I spent a while musing about this and wasn’t able to make much progress. And furthermore, when I look back, even in retrospect it’s pretty unclear how I would ever test if my “assumption” held if it was a constraint on the prior. I mean sure, if there’s actually some random process generating my data, then I might be able to say something, but that seems like a pretty rare case… sorry if I’m being unclear, hopefully that was at least somewhat more clear than before. Or it’s possible that I’m just nitpicking pointlessly.
Hmm. Considering that I was trying to come up with an example to illustrate how explicit the assumptions are, the assumptions aren’t that explicit in my example are they?
Prior knowledge about the world --> mathematical constraints --> prior probability distribution
The assumptions I used to get the constraints are that the best estimate of your next measurement is the average of your previous ones, and that the best estimate of its squared deviation from that average is some number s^2, maybe the variance of your previous observations. But those aren’t states of the world, those are assumptions about your inference behavior.
Then I added later that the real assumptions are that you’re making unbiased measurements of some unchanging quantity mu, and that the mechanism of your instrument is unchanging. These are facts about the world. But these are not the assumptions that I used to derive the constraints, and I don’t show how they lead to the former assumptions. In fact, I don’t think they do.
Well. Let me assure you that the assumptions that lead to the constraints are supposed to be facts about the world. But I don’t see how that’s supposed to work.