If parameter p ~ Beta(1,1), this would reflect we know nothing about parameter $p$. Generalizing to the multivariate case, how would the same be said about a vector $P$ of probability parameters $p_i$?
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1"this would reflect we know nothing about parameter $p$" — it is a uniform prior, but it's not an uninformative prior. – Neil G Dec 05 '13 at 08:42
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but would this not mean $p$ could be anything with equal probability? – user2896468 Dec 05 '13 at 19:31
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Yes, but what if you had used odds $o \in [0, \infty)$ instead of probability $p$. Would you not want to choose a uniform prior for that? – Neil G Dec 05 '13 at 21:52
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A Beta distribution is just a special case of the Dirichlet distribution, that is, a Beta distribution is a Dirichlet distribution with two parameters, alpha and beta.
Dirichlet is the multidimensional generalisation (of Beta) with 'n' parameters instead of two. The parameters of Dirichlet are denoted by alpha with an index as a subscript. Setting all the alphas of a Dirichlet to 1 (no matter how many dimensions we are taking about) will give us the 'n' equivalents of Beta(1,1).
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