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lec5 - C260A Lecture 5 Probabilistic Modeling Christopher...

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C260A Lecture 5: Probabilistic Modeling Christopher Lee October 7, 2009
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Defining Events vs. Variables event : a subset of our total probability space S . p ( e ) = a number. variable : some slicing of S into disjoint subsets, each labeled with a distinct value . Now p ( X ) = f ( X ) 1
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Multiple Variables = Multiple Slicings Each variable just represents another way of slicing up S . Different slicings could be very similar, or very different. Each time we draw one item from S , it is “labeled” with a value for each of our different variables. 2
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Independence? Are events A , B statistically independent? 3
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Independence is About Variables Independence is a statement about the function p ( X , Y ) = p ( X ) p ( Y ) over the entire space, i.e. true x , y ! Don’t confuse independence with simple set intersection. 4
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Event Independence? Create variables X : { A , ¬ A } , Y : { B , ¬ B } . p ( A , B ) = p ( A ) p ( B ) ⇐⇒ p ( X , Y ) = p ( X ) p ( Y ) 5
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Unconditional Sampling “Throw a dart at the dartboard”, get one data point ( x , y , z ) yielding a value for each of the variables defined in our information graph. Each time we sample, we get a new value for every vari- able, including our hidden variables. By contrast, in in- ference we typically draw multiple samples of the observ- ables from one specific inference problem (i.e. one value of θ ).
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