Also also in case anybody's forgotten, the reason they needed this sum was to renormalize the relative probabilities into absolute probabilities.  That's another step along the direct path to deriving a prediction about the next ball observed.

After seeing N balls go LEFT, the posterior distribution ends up proportional to a distribution where each hypothesis p has weight p^N.

But if you forget to normalize that prior-times-likelihood distribution, and ask what happens if every part of the distribution then contributes p likelihood to the next ball going LEFT, you'll end up adding one extra factor of p everywhere, and calculating:

1/INF * [ (0/INF)^(N+1)  +  (1/INF)^(N+1)  +  ...  +  (INF/INF)^(N+1) ] = 1/(N+2)

After seeing 100 balls go LEFT, you shouldn't calculate a 1/102 chance of the next ball going LEFT - you should think it's more likely than not for the next ball to go LEFT - so something has gone wrong here; and what's wrong is that we didn't normalize the prior-times-likelihood distribution.

The pre-normalized distribution sums to 1/(N+1); the normalized distribution should sum to 1 -

Though, Keltham quickly remarks, that's just it summing to 1 within that whole metahypothesis.  If you see balls going LEFT RIGHT LEFT RIGHT repeating, you might form, after a few steps, the hypothesis that the ball goes LEFT RIGHT repeating, which can't be expressed within that whole metahypothesis.  All of this reasoning is taking place inside the metahypothesis about the balls having some stable but otherwise not-further-scrutable fractional propensity to go LEFT or RIGHT, with all such propensities between 0 and 1 being equally likely on priors.  This whole metahypothesis then may do better or worse compared to some other hypotheses.

He's not saying that the metahypothesis absolutely has to be true, by saying that the sum of all the hypotheses inside it should be 1.  If you started believing that, you'd never in principle be able to notice any other pattern like LEFT RIGHT repeating.

Rather, this whole metahypothesis is itself being entertained as one of many in-principle possible metahypotheses about how things could be, and the probabilities of all the hypotheses inside it summing to 1, are to be understood as being what we get when we entertain the possibility, condition on, reason inside, the metahypothesis being true.

After seeing N balls go left, the whole metahypothesis has racked up a joint probability of 1/(N+1) and scored log(1/(N+1)) in terms of how well it's doing.  If you previously started out with any prior credence on an incompatible metahypothesis, like "Actually maybe the ball just always goes LEFT or RIGHT, equally likely a priori," that other metahypothesis may be doing better - that particular hypothesis has now racked up probability 1/2 and a score of log(1/2) = -1, which is probably quite a bit better than 1/(N+1) if you've seen a lot of balls go LEFT.  You'd probably be believing in that metahypothesis a bit more, now, even if it started out less meta-credible than the other.

Also also in case anybody's forgotten, the reason they needed this sum was to renormalize the relative probabilities into absolute probabilities.  That's another step along the direct path to deriving a prediction about the next ball observed.

After seeing N balls go LEFT, the posterior distribution ends up proportional to a distribution where each hypothesis p has weight p^N.

But if you forget to normalize that prior-times-likelihood distribution, and ask what happens if every part of the distribution then contributes p likelihood to the next ball going LEFT, you'll end up adding one extra factor of p everywhere, and calculating:

1/INF * [ (0/INF)^(N+1)  +  (1/INF)^(N+1)  +  ...  +  (INF/INF)^(N+1) ] = 1/(N+2)

After seeing 100 balls go LEFT, you shouldn't calculate a 1/102 chance of the next ball going LEFT - you should think it's more likely than not for the next ball to go LEFT - so something has gone wrong here; and what's wrong is that we didn't normalize the prior-times-likelihood distribution.

The pre-normalized distribution sums to 1/(N+1); the normalized distribution should sum to 1 -

Though, Keltham quickly remarks, that's just it summing to 1 within that whole metahypothesis.  If you see balls going LEFT RIGHT LEFT RIGHT repeating, you might form, after a few steps, the hypothesis that the ball goes LEFT RIGHT repeating, which can't be expressed within that whole metahypothesis.  All of this reasoning is taking place inside the metahypothesis about the balls having some stable but otherwise not-further-scrutable fractional propensity to go LEFT or RIGHT, with all such propensities between 0 and 1 being equally likely on priors.  This whole metahypothesis then may do better or worse compared to some other hypotheses.

He's not saying that the metahypothesis absolutely has to be true, by saying that the sum of all the hypotheses inside it should be 1.  If you started believing that, you'd never in principle be able to notice any other pattern like LEFT RIGHT repeating.

Rather, this whole metahypothesis is itself being entertained as one of many in-principle possible metahypotheses about how things could be, and the probabilities of all the hypotheses inside it summing to 1, are to be understood as being what we get when we entertain the possibility, condition on, reason inside, the metahypothesis being true.

After seeing N balls go left, the whole metahypothesis has racked up a joint probability of 1/(N+1) and scored log(1/(N+1)) in terms of how well it's doing.  If you previously started out with any prior credence on an incompatible metahypothesis, like 'Actually maybe the ball just always goes LEFT or RIGHT, both equally credible on priors,' that other metahypothesis may be doing better.  In fact, the always-left-or-always-right metahypothesis has now racked up probability 1/2 and a score of log(1/2) = -1, which is probably quite a bit better than 1/(N+1) if you've seen a lot of balls go LEFT.  You'd probably be believing in that metahypothesis a bit more, now, even if it started out less meta-credible than the other.

Also also in case anybody's forgotten, the reason they needed this sum was to renormalize the relative probabilities into absolute probabilities.  That's another step along the direct path to deriving a prediction about the next ball observed.

After seeing N balls go LEFT, the posterior distribution ends up proportional to a distribution where each hypothesis p has weight p^N.

But if you forget to normalize that prior-times-likelihood distribution, and ask what happens if every part of the distribution then contributes p likelihood to the next ball going LEFT, you'll end up adding one extra factor of p everywhere, and calculating:

1/INF * [ (0/INF)^(N+1)  +  (1/INF)^(N+1)  +  ...  +  (INF/INF)^(N+1) ] = 1/(N+2)

After seeing 100 balls go LEFT, you shouldn't calculate a 1/102 chance of the next ball going LEFT - you should think it's more likely than not for the next ball to go LEFT - so something has gone wrong here; and what's wrong is that we didn't normalize the prior-times-likelihood distribution.

The pre-normalized distribution sums to 1/(N+1); the normalized distribution should sum to 1.  So we divide everything through by 1/(N+1), meaning we multiply by (N+1), to make it sum to 1.  For this example, that means multiplying everything uniformly by 101 to normalize it.  Which yields a 101/102 chance of the next ball going LEFT, which is what it should be.

Though, Keltham quickly remarks, the principle of making the posterior sum to 1, is something that only applies within that whole metahypothesis.  If you see balls going LEFT RIGHT LEFT RIGHT repeating, you might form, after a few steps, the hypothesis that the ball goes LEFT RIGHT repeating, which can't be expressed within that whole metahypothesis.  All of this reasoning is taking place inside the metahypothesis about the balls having some stable but otherwise not-further-scrutable fractional propensity to go LEFT or RIGHT, with all such propensities between 0 and 1 being equally likely on priors.  This whole metahypothesis then may do better or worse compared to some other hypotheses.

He's not saying that the metahypothesis absolutely has to be true, by saying that the sum of all the hypotheses inside it should be 1.  If you started believing that, you'd never in principle be able to notice any other pattern like LEFT RIGHT repeating.

Rather, this whole metahypothesis is itself being entertained as one of many in-principle possible metahypotheses about how things could be, and the probabilities of all the hypotheses inside it summing to 1, are to be understood as being what we get when we entertain the possibility, condition on, reason inside, the metahypothesis being true.

After seeing N balls go left, the whole metahypothesis has racked up a joint probability of 1/(N+1) and scored log(1/(N+1)) in terms of how well it's doing.  If you previously started out with any prior credence on an incompatible metahypothesis, like 'Actually maybe the ball just always goes LEFT or RIGHT, both equally credible on priors,' that other metahypothesis may be doing better.  In fact, the always-left-or-always-right metahypothesis has now racked up probability 1/2 and a score of log(1/2) = -1, which is probably quite a bit better than 1/(N+1) if you've seen a lot of balls go LEFT.  You'd probably be believing in that metahypothesis a bit more, now, even if it started out less meta-credible than the other.

Also also in case anybody's forgotten, the reason they needed this sum was to renormalize the relative probabilities into absolute probabilities.  That's another step along the direct path to deriving a prediction about the next ball observed.

After seeing N balls go LEFT, the posterior distribution ends up proportional to a distribution where each hypothesis p has weight p^N.

But if you forget to normalize that prior-times-likelihood distribution, and ask what happens if every part of the distribution then contributes p likelihood to the next ball going LEFT, you'd first add one extra factor of p everywhere, and then calculate:

1/INF * [ (0/INF)^(N+1)  +  (1/INF)^(N+1)  +  ...  +  (INF/INF)^(N+1) ] = 1/(N+2)

After seeing 100 balls go LEFT, you shouldn't calculate a 1/102 chance of the next ball going LEFT - you should think it's more likely than not for the next ball to go LEFT - so something has gone wrong here; and what's wrong is that we didn't normalize the prior-times-likelihood distribution.

The pre-normalized distribution sums to 1/(N+1); the normalized distribution should sum to 1.  So we divide everything through by 1/(N+1), meaning we multiply by (N+1), to make it sum to 1.  For this example, that means multiplying everything uniformly by 101 to normalize it.  Which yields a 101/102 chance of the next ball going LEFT, which is what it should be.

Though, Keltham quickly remarks, the principle of making the posterior sum to 1, is something that only applies within that whole metahypothesis.  If you see balls going LEFT RIGHT LEFT RIGHT repeating, you might form, after a few steps, the hypothesis that the ball goes LEFT RIGHT repeating, which can't be expressed within that whole metahypothesis.  All of this reasoning is taking place inside the metahypothesis about the balls having some stable but otherwise not-further-scrutable fractional propensity to go LEFT or RIGHT, with all such propensities between 0 and 1 being equally likely on priors.  This whole metahypothesis then may do better or worse compared to some other hypotheses.

He's not saying that the metahypothesis absolutely has to be true, by saying that the sum of all the hypotheses inside it should be 1.  If you started believing that, you'd never in principle be able to notice any other pattern like LEFT RIGHT repeating.

Rather, this whole metahypothesis is itself being entertained as one of many in-principle possible metahypotheses about how things could be, and the probabilities of all the hypotheses inside it summing to 1, are to be understood as being what we get when we entertain the possibility, condition on, reason inside, the metahypothesis being true.

After seeing N balls go left, the whole metahypothesis has racked up a joint probability of 1/(N+1) and scored log(1/(N+1)) in terms of how well it's doing.  If you previously started out with any prior credence on an incompatible metahypothesis, like 'Actually maybe the ball just always goes LEFT or RIGHT, both equally credible on priors,' that other metahypothesis may be doing better.  In fact, the always-left-or-always-right metahypothesis has now racked up probability 1/2 and a score of log(1/2) = -1, which is probably quite a bit better than 1/(N+1) if you've seen a lot of balls go LEFT.  You'd probably be believing in that metahypothesis a bit more, now, even if it started out less meta-credible than the other.