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Say I'm doing Bayesian inference on a Dirichlet-Multinomial model:

$$ x \in [1,2,3]; \\ x \sim Multinomial(p_1, p_2, p_3); \\ p_1, p_2, p_3 \sim Dirichlet(\alpha_1, \alpha_2, \alpha_3); \\ \alpha_n = \alpha_{prior} + \sum (x=n) $$

I'm interested in whether $p_1 > p_2$. I can sample from the posterior for this parameter directly (in R):

alphas = c(10, 12, 5)
samp.from.dirichlet = MCMCpack::rdirichlet(100000, alphas)
delta.dirichlet = samp.from.dirichlet[,2] - samp.from.dirichlet[,1]
# Prob. that p2 > p1
mean(delta.dirichlet > 0) # > 0.66683

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Is there a closed-form way to calculate this probability?

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