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But priors are not just numbers between 0 and 1, as GP claimed. It is true that in your example, the outcome is binary: there is only a single proposition (t

But priors are not *just* numbers between 0 and 1, as GP claimed. It is true that in your example, the outcome is binary: there is only a single proposition (the coin landed heads). And so the prior distribution is a Bernoulli distribution, but I would agree you could think of this prior as a single number between 0 and 1. But in such a scenario, the only possibilities for updating beliefs based on evidence is to observe heads, then update the posterior to P(Heads)=100%, or observe tails, then update posterior to P(Heads)=0%. I might say that this is updating your beliefs based on evidence, but only trivially so. If you observe heads and form a posterior of P(Heads)=99.999%, then it means that your prior probability distribution was more complex: it included the possibility that "the coin landed heads but I mistakenly observed tails". In this case, your priors include a family of propositions, not a single number. In GPs example, the prior included the possibility that the coin was magnetic, etc. So it could not be represented by a single number.

lisper 4y ago
> the prior distribution is a Bernoulli distribution No, it isn't. The prior on the outcome of multiple tosses is a Bernoulli distribution, but the prior on the outcome of a single toss is a number. And for a fair coin, that number is 0.5 before you see the result, and it is either 0 or 1 after. Note that "This coin will land/landed heads" is a different proposition than "this coin is a fair coin." The latter is much more problematic because it is hard to define what a "fair coin" is. > your prior probability distribution was more complex No. You are confusing my prior on a single proposition (a number) with the set of priors on all propositions which I have considered in my lifetime. The latter is obviously a set of numbers, one for each proposition. > "the coin landed heads but I mistakenly observed tails" Yes, that is two separate propositions: 1. The coin landed heads. 2. My senses are reliable. All evidence must be assessed relative to some "background model", a set of propositions whose priors are very nearly 1. And it is also possible to model an infinite family of propositions about continuous properties, e.g. The length of that rod is between X and Y for any two real numbers X and Y. But none of this changes the fact that for any single proposition the prior is a number. > it could not be represented by a single number That depends on what you mean by "it". The probability that the coin landed heads certainly can be represented by a single number, as can the probability that the coin is magnetic. Two different propositions, hence two different priors.

But priors are not *just* numbers between 0 and 1, as GP claimed. It is true that in your example, the outcome is binary: there is only a single proposition (t

lisper 4y ago

> the prior distribution is a Bernoulli distribution No, it isn't. The prior on the outcome of multiple tosses is a Bernoulli distribution, but the prior on the outcome of a single toss is a number. And for a fair coin, that number is 0.5 before you see the result, and it is either 0 or 1 after. Note that "This coin will land/landed heads" is a different proposition than "this coin is a fair coin." The latter is much more problematic because it is hard to define what a "fair coin" is. > your prior probability distribution was more complex No. You are confusing my prior on a single proposition (a number) with the set of priors on all propositions which I have considered in my lifetime. The latter is obviously a set of numbers, one for each proposition. > "the coin landed heads but I mistakenly observed tails" Yes, that is two separate propositions: 1. The coin landed heads. 2. My senses are reliable. All evidence must be assessed relative to some "background model", a set of propositions whose priors are very nearly 1. And it is also possible to model an infinite family of propositions about continuous properties, e.g. The length of that rod is between X and Y for any two real numbers X and Y. But none of this changes the fact that for any single proposition the prior is a number. > it could not be represented by a single number That depends on what you mean by "it". The probability that the coin landed heads certainly can be represented by a single number, as can the probability that the coin is magnetic. Two different propositions, hence two different priors.