1. 5 points. In lecture we mentioned that the directionality of edges in Bayesian networks do...

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1. 5 points. In lecture we mentioned that the directionality of edges in Bayesian networks do not necessarily reflect causality, but having causal edges often lead to simpler network structures (and hence greater inference efficiency). We will explore this concept further using the burglar alarm example, shown above. (a) Write out all the independence / conditional independence assumptions implied by the graphical structure of the original network. (Use B, E, A, J, M as the random variables.) Hint: There should be 21 in total. Recall that Al B|C is equivalent to BlLA|C (so they only count as one conditional independence), but AL B|C is different from AlILB|C, D. (b) Now consider networks (a) and (b) that involve the same variables. In cach of these cases, we consider variables in a different order, and when introducing the variable, we introduced all the edges necessary to ensure the original probability distribution can still be represented. For example, in network (a), when A was introduced, both arrows leading from M and J were necessary, otherwise conditional independence assumptions not present in the original network would have been introduced (e.g., if M A was not present, then MLA|J would be introduced). Write out all the independence/ conditional independence assumptions implied by the graphical structure of network (a). MaryCalls MaryCalls P(B) P(E) Burglary Earthquake 001 002 JohnCalls JohnCalls E P(A) Earthquake 95 Alarm Alarm 94 29 001 Burglary Bur glary A PU) A PM) Earthquake Alarm JohnCalls I| 90 05 MaryCalls I70 (a) (b) Figure 1: From left to right, we will refer to the above Bayesian networks as the original network, network (a), and network (b). 1. 5 points. In lecture we mentioned that the directionality of edges in Bayesian networks do not necessarily reflect causality, but having causal edges often lead to simpler network structures (and hence greater inference efficiency). We will explore this concept further using the burglar alarm example, shown above. (a) Write out all the independence / conditional independence assumptions implied by the graphical structure of the original network. (Use B, E, A, J, M as the random variables.) Hint: There should be 21 in total. Recall that Al B|C is equivalent to BlLA|C (so they only count as one conditional independence), but AL B|C is different from AlILB|C, D. (b) Now consider networks (a) and (b) that involve the same variables. In cach of these cases, we consider variables in a different order, and when introducing the variable, we introduced all the edges necessary to ensure the original probability distribution can still be represented. For example, in network (a), when A was introduced, both arrows leading from M and J were necessary, otherwise conditional independence assumptions not present in the original network would have been introduced (e.g., if M A was not present, then MLA|J would be introduced). Write out all the independence/ conditional independence assumptions implied by the graphical structure of network (a). MaryCalls MaryCalls P(B) P(E) Burglary Earthquake 001 002 JohnCalls JohnCalls E P(A) Earthquake 95 Alarm Alarm 94 29 001 Burglary Bur glary A PU) A PM) Earthquake Alarm JohnCalls I| 90 05 MaryCalls I70 (a) (b) Figure 1: From left to right, we will refer to the above Bayesian networks as the original network, network (a), and network (b).