BN-Assignment-2

Consider the following Bayesian networks with the following nodes

• R (Rain): it is raining or not, with binary values $r$: it is raining and $r^c$: it is not raining. $val(R)=\{r,r^c\}$
• L (Leaves): there are juicy leaves or not, $val(L)=\{l,l^c\}$
• Q (Quokkas Happiness): the quokkas are happy or unhappy, $val(Q)=\{q,q^c\}$
• T (Tourists): there are lots of tourists or not many, $val(T)=\{t,t^c\}$
• S (Selfie Quokka): people are taking lots of quokka selfies, or not, $val(S)=\{s,s^c\}$

Question 1.

Write the chain rule for the joint distribution $P(R,L,Q,T,S)$ Note: You must use VE (variable elimination) method to solve these questions below!

$$P(R,L,Q,T,S)=P(R)P(L|R)P(Q|R,L)P(T|R,L,Q)P(S|R,L,Q,T)$$$$P(R,L,Q,T,S)=P(R)P(L|R)P(Q|L)P(T|R,Q)P(S|Q,T)$$

Question 2.

What is the probability that there are many tourists?
Given joint distribution:

$$P(R,L,Q,T,S)={\varphi }_R(R){\varphi }_L(L,R){\varphi }_Q(Q,L){\varphi }_T(T,R,Q){\varphi }_S(S,Q,T)$$

We want:

$$P(T)\text{or specifically}P(T=t)$$

using Variable Elimination (VE).

We will keep $T$ (the query variable) and eliminate the others in the order:

$$\rho =\{S,L,R,Q\}$$

• Find all factors containing $S$: ${\varphi }_S(S,Q,T)$
• Eliminate $S$
  • Only factor with $S$: ${\varphi }_S(S\mid Q,T)$.

$${\tau }_1(Q,T)=\sum _S{\varphi }_S(S\mid Q,T)=1(\text{constant, drop it})$$

• Remaining: $\{{\varphi }_R(R),{\varphi }_L(L\mid R),{\varphi }_Q(Q\mid L),{\varphi }_T(T\mid Q,R)\}$.

• Eliminate $L$

  • Multiply the factors with $L$ and sum out $L$:

$$g_2(L,R,Q)={\varphi }_L(L\mid R){\varphi }_Q(Q\mid L),{\tau }_2(R,Q)=\sum _L{g}_2(L,R,Q).$$$$\begin{array}{rl}{\tau }_2(R=r,Q=q)& =0.8\cdot 0.9+0.2\cdot 0.7=0.86,\\ {\tau }_2(R=r^c,Q=q)& =0.3\cdot 0.9+0.7\cdot 0.7=0.76,\\ {\tau }_2(R=r,Q=q^c)& =0.8\cdot 0.1+0.2\cdot 0.3=0.14,\\ {\tau }_2(R=r^c,Q=q^c)& =0.3\cdot 0.1+0.7\cdot 0.3=0.24.\end{array}$$

• Eliminate $R$
  • Combine factors with $R$ and sum out $R$:

$$g_3(R,Q,T)={\varphi }_R(R){\tau }_2(R,Q){\varphi }_T(T\mid Q,R),{\tau }_3(Q,T)=\sum _R{g}_3(R,Q,T).$$

• With $P(R=r)=0.2,P(R=r^c)=0.8$:

$$\begin{array}{rl}{\tau }_3(Q=q,t)& =0.2\cdot 0.86\cdot 0.7+0.8\cdot 0.76\cdot 0.9=0.1204+0.5472=0.6676,\\ {\tau }_3(Q=q^c,t)& =0.2\cdot 0.14\cdot 0.2+0.8\cdot 0.24\cdot 0.3=0.0056+0.0576=0.0632.\end{array}$$

• Eliminate $Q$

$${\tau }_4(T)=\sum _Q{\tau }_3(Q,T),P(T=t)={\tau }_4(t)=0.6676+0.0632=0.7308.$$

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Question 3.

What is the probability that the quokkas are happy, given there are lots of quokka selfies being taken and it is raining.

$$P(Q=q|S=s,R=r)$$$$P(Q=q\mid S=s,R=r)$$

• Factorization & evidence

$$P(R,L,Q,T,S)={\varphi }_R(R){\varphi }_L(L\mid R){\varphi }_Q(Q\mid L){\varphi }_T(T\mid Q,R){\varphi }_S(S\mid Q,T)$$

• Evidence: $R=r,S=s$. Substitute to get factors

$$\{{\varphi }_L(L\mid r),{\varphi }_Q(Q\mid L),{\varphi }_T(T\mid Q,r),{\varphi }_S(s\mid Q,T)\}$$

(${\varphi }_R(r)$ is a constant and cancels at normalization).

• Eliminate variables $\rho =\{L,T\}$ (keep $Q$).

• Eliminate $L$

$${\tau }_1(Q)=\sum _L{\varphi }_L(L\mid r){\varphi }_Q(Q\mid L)$$

• Numbers:

$${\tau }_1(q)=0.8\cdot 0.9+0.2\cdot 0.7=0.86,{\tau }_1(q^c)=0.8\cdot 0.1+0.2\cdot 0.3=0.14.$$

• Eliminate $T$

$${\tau }_2(Q)=\sum _T{\varphi }_T(T\mid Q,r){\varphi }_S(s\mid Q,T)$$

• Numbers:

$${\tau }_2(q)=0.7\cdot 0.9+0.3\cdot 0.6=0.81,{\tau }_2(q^c)=0.2\cdot 0.4+0.8\cdot 0.1=0.16.$$

• Combine & normalize

$$h(Q)={\tau }_1(Q){\tau }_2(Q)$$$$h(q)=0.86\cdot 0.81=0.6966,h(q^c)=0.14\cdot 0.16=0.0224.$$$$P(Q=q\mid s,r)=\frac{h(q)}{h(q)+h(q^c)}=\frac{0.6966}{0.6966+0.0224}\approx 0.9688.$$