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cs-401r:assignment-2 [2014/09/18 04:34] ringger [Exercises] |
cs-401r:assignment-2 [2014/09/24 21:22] (current) cs401rPML Updated labeling of sub-problems and/or problem options. |
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Show your work. Be clear and concise. '''This assignment must be typed.''' | Show your work. Be clear and concise. '''This assignment must be typed.''' | ||

- | # [10 points] Prove that the relationship we call ''conditional independence'' is symmetric. In other words, Prove either (a) or (b) (since they are equivalent), and apply the same [[Proofs|standard of proof]] as in assignment 1: | + | # [10 points] Prove that the relationship we call ''conditional independence'' is symmetric. In other words, Prove either (option a) or (option b) (since they are equivalent), and apply the same [[Proofs|standard of proof]] as in assignment 1: |

- | #* (a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$ | + | #* (option a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$ |

- | #* (b) $P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)$ if and only if $P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)$ | + | #* (option 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 $Z$" stays the same, while $X$ and $Y$ trade places). | #** (in other words, the "given $Z$" stays the same, while $X$ and $Y$ trade places). | ||

# [20 points: 10 points each] (based on exercise 2.2 in Koller and Friedman) Independence: | # [20 points: 10 points each] (based on exercise 2.2 in Koller and Friedman) Independence: | ||

- | #* Prove that for binary random variables $X$ and $Y$, the event-level independence $(x^0 \bot y^0)$ implies random-variable independence $(X \bot Y)$. Use the usual standard of proof. | + | #* (2.1) Prove that for binary random variables $X$ and $Y$, the event-level independence $(x^0 \bot y^0)$ implies random-variable independence $(X \bot Y)$. Use the usual standard of proof. |

- | #* Give a counterexample for nonbinary variables. | + | #* (2.2) Give a counterexample for nonbinary variables. |

# [20 points] Consider how to sample from a categorical distribution over four colors. Think of a spinner with four regions having probabilities $p_{red}$, $p_{green}$, $p_{yellow}$, and $p_{blue}$. Write pseudo-code for choosing a sample from this distribution. | # [20 points] Consider how to sample from a categorical distribution over four colors. Think of a spinner with four regions having probabilities $p_{red}$, $p_{green}$, $p_{yellow}$, and $p_{blue}$. Write pseudo-code for choosing a sample from this distribution. | ||

#* You may assume that you have access to a function that samples a uniform random variable with support [0,1]. | #* You may assume that you have access to a function that samples a uniform random variable with support [0,1]. |