Exercise 21.5 - Derivation of $\mathbb{E}[\log \pi_{k}]$ under a Dirichlet distribution

Answers

Dirichlet distribution is an exponential family distribution with:

ϕ (π) = {(\log π_{1}, \log π_{2}, . . . \log π_{K})}^{T},

𝜃 = α .

Its cumulant function is:

A (α) = \log B (α) = \sum_{i = 1}^{K} \log Γ (α_{i}) - \log Γ (\sum_{i = 1}^{K} α_{i}) .

Therefore we have:

𝔼 [\log π_{k}] = \frac{∂𝐴 (α)}{\partial α_{k}} = \frac{Γ^{'} (α_{k})}{Γ (α_{k})} - \frac{Γ^{'} (\sum_{i = 1}^{K} α_{k})}{Γ (\sum_{i = 1}^{K} α_{k})} = ψ (α_{k}) - ψ (\sum_{i = 1}^{K} α_{i}) .

Take exponential on both sides yields the desired form:

\exp (𝔼 [\log π_{k}]) = \exp (ψ (α_{k}) - ψ (\sum_{i = 1}^{K} α_{k})) = \frac{\exp (ψ (α_{k}))}{\exp (ψ (\sum_{i = 1}^{K} α_{i}))} .

solour_lfq

2021-03-24 13:42