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cs-401r:assignment-2 [2014/09/12 14:51]
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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-Bayes Nets =+Probability Theory ​=
  
 == Objectives == == Objectives ==
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 == Exercises == == Exercises ==
  
-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.
-# [10 points] Does your pseudo-code scale to a distribution over ten thousand values? ​ If not, rewrite it.  If so, say why. +#* You may assume that you have access to a function that samples a uniform random variable with support [0,1]. 
-# [20 points] Implement your pseudo-code,​ run it 100 times, ​and give the results as a vector of counts over the four colors.+# [10 points] Does your pseudo-code scale to run efficiently on a distribution over ten thousand values? ​ If not, rewrite it.  If so, say why. 
 +# [20 points] Implement your pseudo-code, ​choose values for the four probabilities on the spinner as parameters to your procedure, and run it 100 times, ​keeping track of how many times each color shows up.  Give the results as a vector of counts over the four colors.
 # [10 points] Normalize your count vector by 100.  How does the result compare with your chosen parameters? # [10 points] Normalize your count vector by 100.  How does the result compare with your chosen parameters?
-# [10 points] Compute the mean and variance of the estimated multinomial distribution you just discovered. 
  
  
cs-401r/assignment-2.1410533490.txt.gz · Last modified: 2014/09/12 14:51 by ringger
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