ai-bayes

# 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.
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 . ..
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.
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:
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
Visualizing the axioms

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Visualizing the axioms
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) .
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?
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?
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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