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Take some independent random variables. Put them on a list, in a catalog, in a box, etc., and now they are still independent. That each random variable is abo

Take some independent random variables. Put them on a list, in a catalog, in a box, etc., and now they are still independent. That each random variable is about a galaxy and that they are all independent but now know that they are all visible only from the southern hemisphere of the earth, they are still independent. For such things will want to learn about the careful definition of independence and, for more, about conditional independence and the Radon-Nikodym theorem. There is a nice proof of this result by von Neumann in, e.g., the W. Rudin Real and Complex Analysis. > But these are _not_ independent datum You have not shown that. > ... unequal footing Your "unequal footing" is not part of probability theory. Pick a galaxy X. Note what you know about it. Now someone tells you it is in the catalog and also lets you see the other galaxies in the catalog. Now what more do you know about X than you did before???? Can read more about independence in Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/}

vlovich123 4y ago
I think what I implied (and others are trying to communicate across) is how do we know the galaxies are independent variables as you claim? After all, there are galactic forces at work here (gravity, initial conditions etc). How do we know that everything that’s observable to us isn’t biased? Ie the galaxies that are asymmetric are more easily observed for some secondary reason (closer to us, brighter to us, etc)? Even if the galaxies in our observable universe are independent variables, how do we know that our observable universe is an independent variable of all observable universes? Eg what if the structure of the Big Bang sent different asymmetries in every direction and thus each observable universe will always have some asymmetry from the perspective of any single observer but overall there’s no asymmetry. Or heck. The ages of the galaxies are presumably hard to account for individually in this kind of combinatorial analysis. How do we know that the asymmetry isn’t because the galaxies are all in different stages of development / the light from them took variable amounts of time to reach us?
- graycat 4y ago
  > I think what I implied (and others are trying to communicate across) is how do we know the galaxies are independent variables as you claim? Simple. I never claimed that the random variables were independent. Instead I just observed that if one made an ASSUMPTION of independence (and the rest of i.i.d.) then could use the theorem the law of large numbers and get a result without being concerned with what fraction the sample size was of the whole population. I didn't even try to argue that there was independence or even approximate independence. The independence assumption is really common in applications of probability and statistics. Maybe then people comfort themselves by believing that independence holds approximately and close enough. As I recall, there has been at least one attempt to investigate approximate independence, but that work is likely not well known. Questioning independence in this discussion of cosmology is appropriate. Again, in practice, an independence assumption is common, and with all of "i.i.d." could apply the strong law of large numbers and get results about a large population from a comparably small sample thus responding to the stated concern I was responding to: > ... a frighteningly small sample to draw conclusions from?
  - vlovich123 4y ago
    Fair point. Thanks for explaining.

vlovich123 4y ago

I think what I implied (and others are trying to communicate across) is how do we know the galaxies are independent variables as you claim? After all, there are galactic forces at work here (gravity, initial conditions etc). How do we know that everything that’s observable to us isn’t biased? Ie the galaxies that are asymmetric are more easily observed for some secondary reason (closer to us, brighter to us, etc)? Even if the galaxies in our observable universe are independent variables, how do we know that our observable universe is an independent variable of all observable universes? Eg what if the structure of the Big Bang sent different asymmetries in every direction and thus each observable universe will always have some asymmetry from the perspective of any single observer but overall there’s no asymmetry. Or heck. The ages of the galaxies are presumably hard to account for individually in this kind of combinatorial analysis. How do we know that the asymmetry isn’t because the galaxies are all in different stages of development / the light from them took variable amounts of time to reach us?

graycat 4y ago
> I think what I implied (and others are trying to communicate across) is how do we know the galaxies are independent variables as you claim? Simple. I never claimed that the random variables were independent. Instead I just observed that if one made an ASSUMPTION of independence (and the rest of i.i.d.) then could use the theorem the law of large numbers and get a result without being concerned with what fraction the sample size was of the whole population. I didn't even try to argue that there was independence or even approximate independence. The independence assumption is really common in applications of probability and statistics. Maybe then people comfort themselves by believing that independence holds approximately and close enough. As I recall, there has been at least one attempt to investigate approximate independence, but that work is likely not well known. Questioning independence in this discussion of cosmology is appropriate. Again, in practice, an independence assumption is common, and with all of "i.i.d." could apply the strong law of large numbers and get results about a large population from a comparably small sample thus responding to the stated concern I was responding to: > ... a frighteningly small sample to draw conclusions from?
- vlovich123 4y ago
  Fair point. Thanks for explaining.