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corrected some pages
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Several small mistakes/errors were corrected in several proofs/definitions.
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JoramSoch committed Jan 5, 2022
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6 changes: 3 additions & 3 deletions P/dir-ep.md
Original file line number Diff line number Diff line change
Expand Up @@ -170,7 +170,7 @@ $$ \label{eq:Gam-EP2}
p(\forall_{j \neq i} \left[ q_i > q_j \right]|q_i) = \prod_{j \neq i} p(q_i > q_j|q_i) = \prod_{j \neq i} \frac{\gamma(\alpha_j,q_i)}{\Gamma(\alpha_j)} \; .
$$

In order to obtain the exceedance probability $\varphi_i$, the dependency on $q_i$ in this probability still has to be removed. From equations (\ref{eq:Dir-EP-def}) and (\ref{eq:Gam-Dir-eq}), it follows that
In order to obtain the exceedance probability $\varphi_i$, the dependency on $q_i$ in this probability still has to be removed. From equations \eqref{eq:Dir-EP-def} and \eqref{eq:Gam-Dir-eq}, it follows that

$$ \label{eq:Dir-EP2a}
\varphi_i = p(\forall_{j \neq i} \left[ r_i > r_j \right]) = p(\forall_{j \neq i} \left[ q_i > q_j \right]) \; .
Expand All @@ -182,13 +182,13 @@ $$ \label{eq:Dir-EP2b}
\varphi_i = \int_0^\infty p(\forall_{j \neq i} \left[ q_i > q_j \right]|q_i) \, p(q_i) \, \mathrm{d}q_i \; .
$$

With (\ref{eq:Gam-EP2}) and (\ref{eq:Gam-Dir-A}), this becomes
With \eqref{eq:Gam-EP2} and \eqref{eq:Gam-Dir-A}, this becomes

$$ \label{eq:Dir-EP2c}
\varphi_i = \int_0^\infty \prod_{j \neq i} \left( p(q_i > q_j|q_i) \right) \cdot \mathrm{Gam}(q_i;\alpha_i,1) \, \mathrm{d}q_i \; .
$$

And with (\ref{eq:Gam-EP1}) and (\ref{eq:Gam-pdf}), it becomes
And with \eqref{eq:Gam-EP1} and \eqref{eq:Gam-pdf}, it becomes

$$ \label{eq:Dir-EP-qed}
\varphi_i = \int_0^\infty \prod_{j \neq i} \left( \frac{\gamma(\alpha_j,q_i)}{\Gamma(\alpha_j)} \right) \cdot \frac{q_i^{\alpha_i-1} \exp[-q_i]}{\Gamma(\alpha_i)} \, \mathrm{d}q_i \; .
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