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Is the cardinality finite (of the set of probability distributions on a finite set $A$)? Countable? Uncountable?

I think finite gets rule out because the probability distribution on a 2 element set can assign any probability in $[0,1]$ to the second element, so there should be an uncountable number of probability distributions?

This answer Say a probability distribution over reals has cardinality $2^{\mathbb{N}_0}$. So it is indeed uncountable?

user106860
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    Yes, it is uncountable for the very reason you give. – lulu Apr 30 '18 at 00:05
  • And adding (finitely many) more points to the finite set won't change the cardinality since $|\Bbb{R}\times\Bbb{R|$=$|\Bbb{R}|$. – C Monsour Apr 30 '18 at 00:20
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    Your linked answer shows uncountability in (1) for a distribution concentrated on a single point, so broadening the possibilities has at least as many – Henry Apr 30 '18 at 00:22
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    @Henry What you say (if i understood correctly the linked answer) is not correct. The linked answer shows uncountability of the set of Dirac (point mass) distributions, indeed, since you have uncountably many possible points in $\mathbb{R}$... but not on a discrete set. So that doesn't answer the OP's question for the cardinal of the set of probability distributions over a discrete set. – Clement C. Apr 30 '18 at 00:44
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    Now, as the OP (and @lulu) point out, a different but very simple argument shows uncountability of the set of distributions over two points (say $x,y$). Namely, because there is a bijection from $[0,1]$ to this set ($t\in[0,1]\mapsto p=(p_x,p_y)=(t,1-t)$). And so, immediately we get that the set of distributions over any discrete set of cardinality at least $2$ is also uncountable. – Clement C. Apr 30 '18 at 00:46
  • @ClementC. Is there such a thing as a distribution over a set with cardinality $1$? I guess the only distribution is a point mass? – user106860 Apr 30 '18 at 00:52
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    @user106860 Yes. It's not a very interesting case. – Clement C. Apr 30 '18 at 00:54
  • @ClementC. I would say that a Dirac distribution had the support of a discrete set (namely a single point), and that there were an uncountable number of possibilities for that set. But if you are saying that the discrete set must be fixed in advanced, then the Bernoulli distributions essentially suggested in this question would be almost as simple an uncountable example – Henry Apr 30 '18 at 07:44

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(Edited in response to comment) Answer is: Uncountable if the set has cardinality of at least two, as shown in the example in the question. If the set has cardinality 1 then there is only one possible probability distribution.

user106860
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  • As mentioned in a comment above, the linked answer does not provide an answer here, since the OP specifically asks about distributions over a discrete set (where the argument from the link answer does not go through). – Clement C. Apr 30 '18 at 00:56