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cs-401r:assignment-2 [2014/09/12 14:27]
ringger
cs-401r:assignment-2 [2014/09/18 04:34]
ringger [Exercises]
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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.'''​
- +
-=== Bayes Net === +
- +
-[[http://​nlp.cs.byu.edu/​~plf1/​cs401rpml/​GraphicalModel.png|Model (open in a new tab)]]+
  
 # [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 (a) or (b) (since they are equivalent),​ and apply the same [[Proofs|standard of proof]] as in assignment 1:
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 #* (b) $P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)$ if and only if $P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)$ #* (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).
-Consider the graphical model (shown above) over five binary random variables:​ +# [20 points: ​10 points ​each] (based on exercise 2.in Koller ​and Friedman) Independence:​ 
-# [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. +#* 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
-# [20 points] List the independence assumptions captured in the model (in the above figure) between the following pairs of random variablesBe sure to consider all of the cases in which the other variables in the model have known values ​and when they do not. +#* Give a counterexample for nonbinary ​variables. 
-#* $L$ and $R$ +# [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. 
-#* $T$ and $R$ +#* You may assume that you have access to a function that samples a uniform random variable with support [0,1]
-# [20 points] More independence+# [10 points] ​Does your pseudo-code scale to run efficiently on distribution over ten thousand values? ​ If notrewrite it.  ​If so, say why
-#* Assuming that the value of $Lis known, list ''​all'' ​independence ​relations between ​$Tand 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+# [20 points] ​Implement your pseudo-codechoose values for the four probabilities on the spinner as parameters to your procedureand 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
-#* 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] ​Normalize your count vector by 100 How does the result compare with your chosen parameters?​ 
-# [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 $P(T=1R=0, L=0, M=0, S=1)$ in terms of the probabilities given in the model (use the symbolic forms).  ​(b) Then compute the actual probability+
-# [10 points] ​Compute $P(T=1R=0L=0)$Show your work+
-# [10 points] ​Compute $P(T=1 | R=0, L=0)$Show your work.+
  
 == Report == == Report ==
cs-401r/assignment-2.txt · Last modified: 2014/09/24 21:22 by cs401rPML
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