[extropy-chat] what is probability?
Benjamin Goertzel
ben at goertzel.org
Tue Jan 16 21:12:10 UTC 2007
> I think that these results
>
> http://citeseer.ist.psu.edu/calude94borel.html
>
> imply that Chaitin randomness implies exchangeability for infinite
> sequences. Also, they show that for finite sequences, almost all
> sequences are exchangeable (so in that sense, a 'randomly chosen'
> sequence is very likely to be exchangeable).
Sorry: I meant in the last sentence that almost all RANDOM (i.e.
incompressible) finite sequences are exchangeable [for any specific
observer, and as sequence length gets long enough...]
I don't know whether anyone has proved that, conversely, almost all
finite exchangeable sequences are Chaitin-random. But I would bet
that they are.
For instance, tossing a coin according to the bits of the binary
Champernowne sequence [scroll down in
http://en.wikipedia.org/wiki/Champernowne_constant
] may lead to an exchangeable but not Chaitin random series. But,
this sort of case is probably a rare one statistically...
-- Ben G
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