ai-bayes - Probability Review and Intro to Bayes Nets...

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Probability Review and Intro to Bayes Nets
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Probability The world is a very uncertain place . As of this point, we’ve basically danced around that  fact.  We’ve assumed that what we see in the world  is really there, what we do in the world has  predictable outcomes, etc. 
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Some limitations we’ve encountered so far . .. A A B B C C move(A,up) = B move(A,down) = C In the search algorithms we’ve explored so far, we’ve assumed a deterministic relationship between moves and successors
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A A B B C C move(A,up) = B 50% of the time move(A,up) = C 50% of the time Lots of problems aren’t this way! 0.5 0.5 Some limitations we’ve encountered so far . ..
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A A B B C C Moreover, lots of times we don’t know exactly where we are in our search . .. Based on what we see, there’s a 30% chance we’re in A, 30% in B and 40% in C . ... Some limitations we’ve encountered so far . ..
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How to cope? We have to incorporate probability into our graphs, to help us reason and make good decisions. This requires a review of probability basics.
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Boolean Random Variable A boolean random variable is a variable that can be true or false with some probability. A = The next president is a liberal. A = You wake up tomorrow with a headache. A = You have the flu.
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Visualizing P(A) Call P(A) as “the fraction of possible worlds in which A is true”.  Let’s visualize  this:
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The axioms of probability 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B) We will visualize each of these axioms in turn.
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Visualizing the axioms
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Visualizing the axioms
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Visualizing the axioms
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Visualizing the axioms
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Theorems from the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A and B)  From these we can prove: P(not A) = P(~A) = 1-P(A) .
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Conditional Probability
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Conditional Probability
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Reasoning with Conditional Probability One day you wake up with a headache. You think: “Drat!  50% of flu cases are associated with headaches so I must have a  50-50 chance of coming down with flu”. Is that good reasoning?
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What we just did, more formally.
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Using Bayes Rule to gamble Trivial question : Someone picks an envelope and random and asks you to bet as to whether or not it holds a dollar. What are your odds?
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Using Bayes Rule to gamble Not trivial question : Someone lets you take a bead out of the envelope before you bet. If it is black, what are your odds? If it is red, what are your odds?
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Joint Distributions A joint distribution records the probabilities that multiple variables will hold particular values. They can be represented much like truth tables. They can be populated using
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This note was uploaded on 01/20/2011 for the course CS 6810 taught by Professor Hecker during the Spring '10 term at CSU East Bay.

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ai-bayes - Probability Review and Intro to Bayes Nets...

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