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Is it possible to have a function which maps the set of all distributions to the real number line?

I am asking this because I am trying to think of a Bayesian prior distribution which concerns all possible distributions.

If this is possible how would one then update the prior in a meaningful way?

Lorenzo
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  • The set of distributions has the same cardinality as the set of reals, so yes there exists a bijective function. https://math.stackexchange.com/questions/1945297/is-the-set-of-probability-distributions-infinite-in-length – 79037662 Oct 01 '19 at 13:21
  • @79037662 my question is more if a bijection can be defined rather than its existence and if there is a "sensible" map of distributions to reals so that a distribution over all distributions "makes sense". (I am aware that this is an extremely vague question so I don't need a precise answer) – Lorenzo Oct 01 '19 at 14:34

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