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# prob-models - A Little Probability Coding and Information...

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A Little Probability . . . . . . Coding and Information Theory Fall, 2004 Tom Carter http://astarte.csustan.edu/˜ tom/ October, 2004 1

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Some probability background There are two notions of the probability of an event happening. The two general notions are: 1. A frequentist version of probability: In this version, we assume we have a set of possible events, each of which we assume occurs some number of times. Thus, if there are N distinct possible events ( x 1 , x 2 , . . . , x N ) , no two of which can occur simultaneously, and the events occur with frequencies ( n 1 , n 2 , . . . , n N ), we say that the probability of event x i is given by P ( x i ) = n i N j =1 n j This definition has the nice property that N X i =1 P ( x i ) = 1 2
2. An observer relative version of probability: In this version, we take a statement of probability to be an assertion about the belief that a specific observer has of the occurrence of a specific event. Note that in this version of probability , it is possible that two different observers may assign different probabilities to the same event. Furthermore, the probability of an event, for me, is likely to change as I learn more about the event, or the context of the event. 3

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3. In some (possibly many) cases, we may be able to find a reasonable correspondence between these two views of probability. In particular, we may sometimes be able to understand the observer relative version of the probability of an event to be an approximation to the frequentist version, and to view new knowledge as providing us a better estimate of the relative frequencies. 4
I won’t go through much, but some probability basics, where a and b are events: P ( not a ) = 1 - P ( a ) . P ( a or b ) = P ( a ) + P ( b ) - P ( a and b ) . We will often denote P ( a and b ) by P ( a, b ). If P ( a, b ) = 0, we say a and b are mutually exclusive. Conditional probability: P ( a | b ) is the probability of a , given that we know b . The joint probability of both a and b is given by: P ( a, b ) = P ( a | b ) P ( b ) .

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