Dirichlet distribution - moments

  • Thread starter Boot20
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  • #1
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For a Dirichlet variable, I know the means and covariances, that is,

[itex] E[X_i] = \alpha_i/\alpha_0 [/itex]
[itex] Cov[X_i,X_j] = \frac{ \alpha_i (\alpha_0I[i=j] - \alpha_j)}{\alpha_0^2(\alpha_0 + 1)}[/itex]

But how can I prove these facts?
 
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  • #2
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[itex] p(x_1,...,x_{k-1}) = \frac{1}{B(\alpha) } \left[ \prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] [/itex]

[itex] E[X_1] = \frac{1}{B(\alpha) } \int^1_0 ...\int^{1 - \sum^{K}_{i=2}x_i}_0 x_1 \left[\prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] d x_1 ... d x_{k} [/itex]
 
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  • #3
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[itex] p(x_1,...,x_{k-1}) = \frac{1}{B(\alpha) } \left[ \prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] [/itex]

[itex] E[X_1] = \frac{1}{B(\alpha) } \int^1_0 ...\int^{1 - \sum^{K}_{i=2}x_i}_0 x_1 \left[\prod^{K}_{i=1} x_i^{\alpha_i - 1} \right] d x_1 ... d x_{k} [/itex]
Hint: the expectation integrals closely resemble the definition of the normalizing constant B (which itself can be expressed in terms of gamma functions).
 

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