Exercise 14.19 [bn-complexity-exercise]

Investigate the complexity of exact inference in general Bayesian networks:

1. Prove that any 3-SAT problem can be reduced to exact inference in a Bayesian network constructed to represent the particular problem and hence that exact inference is NP-hard. (Hint: Consider a network with one variable for each proposition symbol, one for each clause, and one for the conjunction of clauses.)

2. The problem of counting the number of satisfying assignments for a 3-SAT problem is #P-complete. Show that exact inference is at least as hard as this.