tag:blogger.com,1999:blog-3891434218564545511.post1872133785069684651..comments2024-03-28T13:23:50.623-05:00Comments on Alexander Pruss's Blog: Product spaces for hyperreal and full conditional probabilitiesAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-3891434218564545511.post-88312830826432907332021-01-31T20:11:09.779-06:002021-01-31T20:11:09.779-06:00Here is an example of different joint hyperreal pr...Here is an example of different joint hyperreal probabilities with the same (strictly, isomorphic) pairs of independent marginal probabilities. Usual warning: amateur at work.<br /><br />The example uses hyperreal probabilities constructed by the sequence-and-ultrafilter method. To form a product of such probabilities, we need a cross product ultrafilter. The usual product W of an ultrafilter U on the powerset of A and V on the powerset of B can be described like this: a subset C of the powerset of AxB is in the product ultrafilter W iff the set of ‘a’ in A such that (the set of ‘b’ in B such that (‘a’ x ‘b’ is in C) is in V) is in U.<br /><br />Note (this is the key point) that this product treats the factors differently. We would get a different product ultrafilter if we said ‘… iff the set of ‘b’ in B such that (the set of ‘a’ in A such that (‘a’ x ‘b’ is in C) is in U) is in V.’<br /><br />Here is the construction. Define a probability P on the rationals in [0, 1) as follows. Define a sequence of sets Sn (n = 1, 2, 3 …) by Sn = { o/n!, 1/n!, …, (n!-1)/n!}. For any set A of rationals in [0, 1) define Pn(A) = (size of A intersect Sn)/n!. Use the Pn and a (free) ultrafilter U on n to define a hyperreal probability P in the usual way. P is finitely additive, regular and invariant under (folded) rational rotations.<br /><br />Define a second probability Q, again on the rationals in [0, 1), in a similar way, with sets Tm, probabilities Qm, and a possibly different ultrafilter V on m.<br /><br />Define a joint probability R on the rationals in [0, 1) x [0, 1) like this: for each pair n, m define Rn,m(A) as (size of A intersect (Sn x Tm))/(n!*m!). Use the Rn,m and the product ultrafilter W of U and V as described above to define a hyperreal probability R. R is finitely additive, regular and invariant under (folded) rational rotations on both coordinates.<br /><br />The marginal distribution R((.) x ([0, 1)) is isomorphic to P. (Because Rnm(A x [0, 1)) = Pn(A) for all A, n, and m, and any set u in U corresponds to a set u x {1, 2, 3, …} in W.) Similarly, the marginal R([0, 1) x (.)) is isomorphic to Q. Also, R makes its marginals independent. (Because Rn,m(A x B) = Rn,m(A x [0, 1)) * Rn,m([0, 1) x B) for all n and m.)<br /><br />Here is the interesting bit. Associate the random variable X with the first coordinate, Y with the second. Then R(Y=0) is strictly less than R(X=0). In fact, R(Y=0)/R(X=0) is infinitesimal. To see this, note that, for any natural k, the set of all pairs (n, m) with n = 1, 2, 3 … and m strictly greater than n+k is in the product ultrafilter W. (Because all free ultrafilters on the naturals contain the Frechet filter. Note also, this is where the asymmetry of the product ultrafilter comes in: a similar set with n and m swapped would <i>not</i> be in W.) On this set, Rn,m(Y=0)/Rn,m(X=0) = n!/m!, which is less than 1/k!. Then take k arbitrarily large. <br /><br />We can construct another joint probability R´ similarly, but using the alternate product ultrafilter mentioned above. R´ will have all the properties mentioned above, except that R´(Y=0) will be <i>greater</i> than R´(X=0) and R´(Y=0)/R´(X=0) will be 1/infinitesimal. So R´ will be genuinely different from R.<br /><br />So we have two different joint distributions, both with the same (strictly, isomorphic) independent marginal distributions. Neither is more obviously natural. As the post suggests, independence alone is not enough to force a unique joint distribution.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-29813300318692225542020-08-26T11:18:47.421-05:002020-08-26T11:18:47.421-05:00OK, thanks.OK, thanks.Andrew Dabrowskihttps://www.blogger.com/profile/14194210589133048249noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-30432015114977675422020-08-26T11:12:28.891-05:002020-08-26T11:12:28.891-05:00Maybe. Mine are constructed via an ultraproduct of...Maybe. Mine are constructed via an ultraproduct of the reals like his. But from what I recall in Robinson's textbook, his were constructed via a different ultrafilter, namely one over the naturals.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-68175415899318231762020-08-26T10:11:32.961-05:002020-08-26T10:11:32.961-05:00Are your "hyperreals" synonymous with Ro...Are your "hyperreals" synonymous with Robinson's nonstandard reals?Andrew Dabrowskihttps://www.blogger.com/profile/14194210589133048249noreply@blogger.com