Problem $1(20$ points$)$

Given the probability tables and independence assumptions, write an expression for the following questions. You can write "not possible" if you cannot come up with an expression by given probability tables and independence assumptions. You are required to write your work in details for full credit.

a$.$ Given the probability tables $P(x|Y,Z),P(Z|x,Y),P(Z),P(x,Y)$ and no conditional independence assumptions, write an expression for the CPT of $P(x,Y,Z).$

b$.$ Given the probability tables $P(x,Y,Z),P(x),P(Y,Z|x),P(Y|Z)$ and no conditional independence assumptions, write an expression for the CPT of $P(x|Y,Z).$

c$.$ Given the probability tables $P(Y|x,Z),P(x,Z),P(Y),P(Y|Z),P(x|Z),P(Z)$ and conditional independence assumptions $xY|Z||,$ write an expression for the marginal probability table of $P(x|Y).$

d$.$ Given the probability tables $P(Z),P(Y|Z),P(x),P(x|Y)$ and conditional independence assumptions $xY|Z||$ and $xZ,$ write an expression for the marginal probability table of $P(x,Y,Z).$

Problem $2(20$ points$)$

Select the minimal set of conditional independence assumptions required for each of the following equation. Explain and show the derivation for full credit.

a$.P(x,Z)=P(x|Y)P(Z)$

i$)xY$

ii$)YZ$

iii$)xY|Z||$

iv$)YZ|x||$

v$)xz$

vi$)xZ|Y||$

vii$)$ No independence assumptions needed.

Explanation:

b$.,P(x|Y,Z)=\frac{P(x)P(Y|x)P(Z|x)}{P(Y|Z)P(Z)}$

i$)xY$

ii$)YZ$

iii$)xY|Z||$

iv$)YZ|x||$

v$)xz$

vi$)xZ|Y||$

vii$)$ No independence assumptions needed.

Explanation:

c$.,P(x,Y)={\displaystyle \underset{Z}{\overset{?}{}}}P(x|Y,Z)P(Y|z)P(z)$

i$)xY$

ii$)YZ$

iii$)xY|Z||$

iv$)YZ|x||$

v$)xz$

vi$)xZ|Y||$

vii$)$ No independence assumptions needed.

Explanation:

d$.,P(x,Y|Z,W)=P(x|Z,W)P(Y|x,Z,W)$

i$)xY$

ii$)ZW|x||$

iii$)xY|Z||$

iv$)ZW|Y||$

v$)xY|W||$

vi$)ZW$

vii$)$ No independence assumptions needed.

Explanation: