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 cs-401r:homework-0.2 [2014/09/03 00:00]ringger created cs-401r:homework-0.2 [2014/12/31 16:19] (current)ringger 2014/12/31 16:19 ringger 2014/09/03 00:00 ringger created 2014/12/31 16:19 ringger 2014/09/03 00:00 ringger created Line 1: Line 1: - <​big>'''​Bayes Nets'''​ + = Bayes Nets = == Objectives == == Objectives == Line 24: Line 24: #* Assuming that the value of S is known, list ''​all''​ independence relations between T and other variables. ​ Be sure to consider all of the cases in which the other variables in the model have known values and when they do not. #* Assuming that the value of S is known, list ''​all''​ independence relations between T and other variables. ​ Be sure to consider all of the cases in which the other variables in the model have known values and when they do not. # [10 points] Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model. # [10 points] Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model. - # [10 points] Write the four entries and their values in the conditional distribution for <​math>​P(L=0 | M=m,S=s) ​(for m in {0,1} and s in {0,1}). + # [10 points] Write the four entries and their values in the conditional distribution for $P(L=0 | M=m,S=s)$ (for m in {0,1} and s in {0,1}). - # [10 points] (a) Write an expression for the joint probability ​<​math>​P(T=1, R=0, L=0, M=0, S=1) ​in terms of the probabilities given in the model (use the symbolic forms). ​ (b) Then compute the actual probability. + # [10 points] (a) Write an expression for the joint probability ​$P(T=1, R=0, L=0, M=0, S=1)$ in terms of the probabilities given in the model (use the symbolic forms). ​ (b) Then compute the actual probability. - # [10 points] Compute ​<​math>​P(T=1, R=0, L=0)​. Show your work. + # [10 points] Compute ​$P(T=1, R=0, L=0)$. Show your work. - # [10 points] Compute ​<​math>​P(T=1 | R=0, L=0)​. Show your work. + # [10 points] Compute ​$P(T=1 | R=0, L=0)$. Show your work. # [10 points] Prove that the relationship we call conditional independence is symmetric. ​ Apply the same [[Proofs|standard of proof]] as in homework 0.1. # [10 points] Prove that the relationship we call conditional independence is symmetric. ​ Apply the same [[Proofs|standard of proof]] as in homework 0.1. - #* In other words, show that (a) <​math>​P(X | Y, Z) = P(X | Z) ​if and only if <​math>​P(Y | X, Z) = P(Y | Z)​ + #* In other words, show that (a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$ - #* Equivalently,​ show that (b) <​math>​P(X, Y | Z) = P(X | Z) \cdot P(Y|Z) ​if and only if <​math>​P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)​ + #* Equivalently,​ show that (b) $P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)$ if and only if $P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)$ - #* (in other words, the "​given ​<​math>​Z​" stays the same, while <​math>​X ​and <​math>​Y ​trade places). + #* (in other words, the "​given ​$Z$" stays the same, while $X$ and $Y$ trade places). #* To be clear, prove (a) or (b). #* To be clear, prove (a) or (b).