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I am reading the paper available at this link:

I am having trouble understanding section 5.1 (page 2741).

Essentially it says the following:

$\theta_{ABi} \sim \mathrm{N}(\mu_{AB}, \tau^2)$

$\theta_{ACi} \sim \mathrm{N}(\mu_{AC}, \tau^2)$

$\theta_{BCi} \sim \mathrm{N}(\mu_{BC}, \tau^2)$

$\mu_{BC} = \mu_{AC}-\mu_{AB}$


$ \begin{pmatrix}\theta_{ABi} \\ \theta_{ACi}\end{pmatrix} \sim \mathrm{N} \left(\begin{pmatrix}\mu_{ABi} \\ \mu_{ACi}\end{pmatrix}, \begin{pmatrix}\tau^2 & \tau^2/2 \\ \tau^2 /2 & \tau^2\end{pmatrix} \right)$

I do not understand how $\mathrm{Cov} \left[ \theta_{ABi}, \theta_{ACi} \right] = \tau^2 /2$ ? Could someone please explain this?

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up vote 2 down vote accepted

Variable $\theta_{BC}$ is defined as $\theta_{AC} - \theta_{AB}$ as can be seen from the definition of $E(\theta_{BC}) = \mu_{BC}$. That being said: $$ \begin{align} \textrm{Var}(\theta_{BC})&=\textrm{Var}(\theta_{AC} - \theta_{AB})\\ \textrm{Var}(\theta_{BC})&=\textrm{Var}(\theta_{AC}) + (-1^2)\cdot\textrm{Var}(\theta_{AB}) + 2(1)(-1)\textrm{Cov}(\theta_{AC}, \theta_{AB})\\ \end{align} $$ Substituting, we get: $$ \begin{align} \tau^2 &= \tau^2 + \tau^2 - 2\textrm{Cov}(\theta_{AC}, \theta_{AB})\\ 2\textrm{Cov}(\theta_{AC}, \theta_{AB}) &= \tau^2\\ \textrm{Cov}(\theta_{AC}, \theta_{AB}) &= \frac{\tau^2}{2} \end{align} $$

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