#expressive-power

Question 1: Single choice

hardAI-checkedProbabilistic & Graphical Models

Two independent inputs $x_1, x_2$ each drive one of two interacting outputs $y_1, y_2$ : $p = p(x_1)\,p(x_2)\,p(y_1 \mid x_1)\,p(y_2 \mid x_2)\,\phi(y_1, y_2)\,n(x_1, x_2)$ , where $n(x_1, x_2)$ normalises $p(y_1, y_2 \mid x_1, x_2)$ . The independencies of $p$ are $x_1 \perp x_2$ , $x_1 \perp y_2 \mid \{y_1, x_2\}$ and $x_2 \perp y_1 \mid \{y_2, x_1\}$ . Which statement is correct?

Neither an undirected graph nor a DAG is a perfect map for these independencies1
The undirected graph $x_1 - y_1 - y_2 - x_2$ is a perfect map2
The DAG $x_1 \to y_1 \to y_2 \leftarrow x_2$ is a perfect map3
The undirected four-cycle $x_1 - y_1 - y_2 - x_2 - x_1$ is a perfect map4

Timed test