2\theta + V_1 \quad \text{and} \quad Y_2 = V_2,$$ where $V_1,V_2$ are independent ${{\sf N}}(0,2)$. This completes the A-step. Following the basic procedure described in Section \[S:ims\], for the P-step, we should predict the pair $(V_1,V_2)$ with a predictive random set ${\mathcal{S}}$. A simple $L_\infty$ generalization of the default predictive random set to the case of a two-dimensional auxiliary variable is a random square: $${\mathcal{S}}= \{(v_1,v_2): \max(|v_1|,|v_2|) \leq \max(|V_1|,|V_2|)\}, \quad V_1,V_2 {\overset{\text{\tiny iid}}{\,\sim\,}}{{\sf N}}(0,2).$$ For a singleton assertion $\{\theta\}$, the C-step gives plausibility function $${\mathsf{pl}}_y(\theta) = \frac{1 - G( 2^{-1/2}\max\{|y_1-2\theta|,|y_2|\} )^2}{1 - G( 2^{-1/2}|y_2| )^2},$$ where $G(z)=1-2(1-\Phi(z))$ is the $|{{\sf N}}(0,1)|$ distribution function. The unusual form here is due to the conditioning in to remove conflict cases where $\Theta_y({\mathcal{S}}) = \varnothing$.

As an alternative approach, note that the value of $V_2$ is known once $Y_2$ is observed. So rather than trying to predict this component, as in the approach just described, we might condition on this observed value, to sharpen our uncertainty for predicting $V_1$. Since $V_1$ and $V_2$ are actually independent, it suffices to work with the marginal distribution, $V_1 \sim {{\sf N}}(0,2)$. For the A-step, we get $Y_1 = 2\theta + V_1$ and, for the P-step, we use 