Exercise 10.4 - Hidden variables in DGMs

For question (a), $p(X_i)$ where $i=1,2,3$ include three parameters. The term $p(H=h|X_{1:3})$ include $2^3=8$ free parameters. Each $p(X_i|H=h)$, where $i=4,5,6$ includes two parameters. Thus we have:

free parameters.

For question (b), $p(X_i)$ where $i=1,2,3$ include three parameters. $p(X_4|X_{1:3})$ contains $2^3$ free parameters. $p(X_5|X_{1:4})$ contains $2^4$ free parameters. $p(X_6|X_{1:4})$ contains $2^5$ free parameters. Thus we have:

free parameters.

For question (c), the second model is easier since we only have to estimate six terms independently by counting.