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The second step p = Probability(X_i = heads) is technically wrong (although a common abuse of notation). It's strictly speaking p = Probability(X_i = heads | p)

The second step p = Probability(X_i = heads) is technically wrong (although a common abuse of notation). It's strictly speaking p = Probability(X_i = heads | p). So while it is meaningful to say: "p has this probability distribution." It is not meaningful to say: "The probability the coin comes up heads has this probability distribution." This is because p and the probability the coin comes up heads are not the same thing (this is even more apparent if you have a non-Bernoulli scenario) when p is no longer fixed. It's not really just a case of marginal vs conditional probabilities either. Even in the case of a conditional probability, it's only defined for a fixed condition. Even in the case of random measures, while you can reasonably talk about the probability of a probability distribution, it still doesn't really make sense to talk about the probability distribution of a probability.

conjectures 8y ago
> Even in the case of random measures, while you can reasonably talk about the probability of a probability distribution, it still doesn't really make sense to talk about the probability distribution of a probability. But that's exactly what it does mean. For the Dirichlet Process, the probability of hitting a set in the partition of a sample space is random. A DP is a couple of generalisations up from the beta distribution on p under discussion [1]. I.e. this specializes to the current situation as: * The sample space theta is the set {H, T}. * The only interesting partition to talk about is {A_1, A_2}={{H}, {T}}. * The Dirichlet distribution on (G(A_1), G(A_2)) is therefore 2 dimensional, which is a Beta. * G(A_1) is the probability measure of hitting heads. No specific parametric model involving univariate p needs to be involved. [1] c.f. eqn 2 of http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/dp.pdf http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/dp.pdf

conjectures 8y ago

> Even in the case of random measures, while you can reasonably talk about the probability of a probability distribution, it still doesn't really make sense to talk about the probability distribution of a probability. But that's exactly what it does mean. For the Dirichlet Process, the probability of hitting a set in the partition of a sample space is random. A DP is a couple of generalisations up from the beta distribution on p under discussion [1]. I.e. this specializes to the current situation as: * The sample space theta is the set {H, T}. * The only interesting partition to talk about is {A_1, A_2}={{H}, {T}}. * The Dirichlet distribution on (G(A_1), G(A_2)) is therefore 2 dimensional, which is a Beta. * G(A_1) is the probability measure of hitting heads. No specific parametric model involving univariate p needs to be involved. [1] c.f. eqn 2 of http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/dp.pdf http://www.gatsby.ucl.ac.uk/~ywteh/research/npbayes/dp.pdf