[1] It forms the basis for a recommendation that a deck of cards should be riffled seven times in order to thoroughly randomize it.
[2] It is named after the work of Edgar Gilbert, Claude Shannon, and J. Reeds, reported in a 1955 technical report by Gilbert[3] and in a 1981 unpublished manuscript of Reeds.
In this way, it describes the probability of obtaining each permutation, when a shuffle is performed at random.
The model may be defined in several equivalent ways, describing alternative ways of performing this random shuffle: Among all of the possible riffle shuffle permutations of a card deck, the Gilbert–Shannon–Reeds model gives almost all riffles equal probability,
[5] The inverse permutation of a random riffle may be generated directly.
Bayer and Diaconis reported that, for decks of n cards shuffled
times, where θ is an arbitrary constant, the total variation distance is close to one when θ is significantly less than zero, and close to zero when θ is significantly greater than zero, independently of n. In particular their calculations showed that for n = 52, five riffles produce a distribution whose total variation distance from uniform is still close to one, while seven riffles give total variation distance 0.334.
This result was widely reported as implying that card decks should be riffled seven times in order to thoroughly randomize them.
[6][7][8] Similar analyses have been performed using the Kullback–Leibler divergence, a distance between two probability distributions defined in terms of entropy; the divergence of a distribution from uniform can be interpreted as the number of bits of information that can still be recovered about the initial state of the card deck.
(at which point the number of remaining bits of information is linear, smaller by a logarithmic factor than its initial value) and then decreasing exponentially until, after