= Solution
Within the positive component, <normal-normal conjugacy> gives
$$
p(\theta_i\mid y_i,+)\propto
\exp\left\{-\frac12(y_i-\theta_i)^2-\frac{\theta_i^2}{2V}\right\}.
$$
Completing the square yields
$$
\theta_i\mid y_i,+\sim N\left(\frac{V}{1+V}y_i,\frac{V}{1+V}\right),
\qquad\boxed{\mathbb E[\theta_i\mid y_i,+]=\frac{V}{1+V}y_i.}
$$
Here positive denotes membership in the variable-effect component, not the condition $\theta_i>0$; its normal prior permits effects of either sign.
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