Thursday, July 2, 2015

Multinomial probability derivation

For an optimization problem related to media planning I needed to evaluate the probability that a multinomial variate assume the value zero. This can be written as:

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However it was suggested to use the following expression:

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I did not see immediately that these things are approximately the same. With a little bit of effort:

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where the last step is based on a Taylor approximation:

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To be precise:

image image

This approximation in evaluating the reach for some advertisement plans seem to work very well.