This point can't be completed, though, without explaining what happens to the 'any fraction between 0 and 1 is equally likely' metahypothesis when it updates.

Let's say that we see 10 balls go LEFT.

After seeing 10 balls go LEFT, it has a cumulative score of 1/(N+1) = 1/11, or equivalently, if you asked this metahypothesis at the dawn of time what probability it assigned to the first 10 balls going left, it would answer "1/11".  Equivalently, isomorphically, these are stating the same thing, if you multiply out the prior distribution (1/INF everywhere) and the likelihood distribution (p^10 everywhere) you'd get a new distribution p^10/INF which, if you summed over the infinitely many terms inside it, would sum to 1/11.

Continuing to reason within this metahypothesis, one of the internal subhypotheses must by assumption be true, and so within the metahypothesis the posterior probabilities of the (infinite collection of) subhypotheses must sum to 1.

Which you'd obviously do by dividing all the terms through by 1/11, that is, multiplying them by 11.

The distribution...

(0/INF)^10     (1/INF)^10             (INF/INF)^10
----------  +  ----------  +  ...  +  ------------
INF * 1/11     INF * 1/11              INF * 1/11

...will properly sum to 1.

To be clear, by having the distribution sum to 1, we are assuming-for-the-sake-of-extrapolation that some hypothesis in the collection must be true.  But no matter how far the posterior updates on finite data, every individual hypothesis in the collection will still have posterior 0.

After rolling 1000 balls and seeing 985 go left, there's finite probability between 0.98 and 0.99, but 0 probability on the fraction being exactly 0.985.

That's just how this metahypothesis rolls.  If you wanted a metahypothesis that could put little finite dots of probability mass on all the big round numbers like 0.985, 1/2pi, and so on, that would be a much more complicated ask.

This point can't be completed, though, without explaining what happens to the 'any fraction between 0 and 1 is equally likely' metahypothesis when it updates.

Let's say that we see 10 balls go LEFT.

After seeing 10 balls go LEFT, it has a cumulative score of 1/(N+1) = 1/11, or equivalently, if you asked this metahypothesis at the dawn of time what probability it assigned to the first 10 balls going left, it would answer "1/11".  Equivalently, isomorphically, these are stating the same thing, if you multiply out the prior distribution (1/INF everywhere) and the likelihood distribution (p^10 everywhere) you'd get a new distribution p^10/INF which, if you summed over the infinitely many terms inside it, would sum to 1/11.

Continuing to reason within this metahypothesis, one of the internal subhypotheses must by assumption be true, and so within the metahypothesis the posterior probabilities of the (infinite collection of) subhypotheses must sum to 1.

Which you'd obviously do by dividing all the terms through by 1/11, that is, multiplying them by 11.

The distribution...

(0/INF)^10     (1/INF)^10             (INF/INF)^10
----------  +  ----------  +  ...  +  ------------
INF * 1/11     INF * 1/11              INF * 1/11

...will properly sum to 1.

To be clear, by having the distribution sum to 1, we are assuming-for-the-sake-of-extrapolation that some hypothesis in the collection must be true.  But no matter how far the posterior updates on finite data, every individual hypothesis in the collection will still have posterior 0.

After rolling 1000 balls and seeing 985 go left, there's finite probability between 0.98 and 0.99, but 0 probability on the fraction being exactly 0.985.

That's just how this metahypothesis rolls.  If you wanted a metahypothesis that could put little finite dots of probability mass on all the big round numbers like 0.985, 1/2pi, and so on, that would be a much more complicated ask.

Keltham will now zoom in closer on what happens to the 'any fraction between 0 and 1 is equally likely' metahypothesis when it updates.

Let's say that we see 10 balls go LEFT.

After seeing 10 balls go LEFT, the metahypothesis as a whole has a cumulative score of 1/(N+1) = 1/11, or equivalently, if you asked this metahypothesis at the dawn of time what probability it assigned to the first 10 balls going left, it would answer "1/11".  Equivalently, isomorphically, these are stating the same thing, if you multiply out the prior distribution (1/INF everywhere) and the likelihood distribution (p^10 everywhere) you'd get a new distribution p^10/INF which, if you summed over the infinitely many terms inside it, would sum to 1/11.

Continuing to reason within this metahypothesis, one of the internal subhypotheses must by assumption be true, and so within the metahypothesis the posterior probabilities of the (infinite collection of) subhypotheses must sum to 1.

Which you'd obviously do by dividing all the terms through by 1/11, that is, multiplying them by 11.

The distribution...

(0/INF)^10     (1/INF)^10             (INF/INF)^10
----------  +  ----------  +  ...  +  ------------
INF * 1/11     INF * 1/11              INF * 1/11

...will, indeed, sum to 1.

To be clear, by having the distribution sum to 1, we are assuming-for-the-sake-of-extrapolation that some hypothesis in the collection must be true.  But no matter how far the posterior updates on finite data, every individual hypothesis in the collection will still have posterior 0.

After rolling 1000 balls and seeing 985 go left, there's finite probability between 0.98 and 0.99, but 0 probability on the fraction being exactly 0.985.

That's just how this metahypothesis rolls.  If you wanted a metahypothesis that could put little finite dots of probability mass on all the big round numbers like 0.985, 1/2pi, and so on, that would be a much more complicated ask.