Probability in Decline

• Activity Number: 119
• Type: Topic Contributed
• Date/Time: Monday, August 2, 2010 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Education
• Author(s): Dean Michael Brooks*+
• Companies: Ekaros Analytical Inc.
• Keywords: maximum entropy ; decline ; epidemiology ; Web traffic ; Spencer-Brown ; randomization

Abstract

Spencer-Brown advanced a provocative thesis in "Probability and Scientific Inference" (1957). From experiments with early "chance machines," he argued that a fundamental flaw exists in our view of randomness. Long sequences of random digits generated by a variety of methods show long-term declines in repetition of rare items or sequences.

I revisit this neglected topic and show that a variety of modern random number generators (including pseudo-random algorithms) exhibit this same property. For simple schemes the decline is short and the system soon lapses into classic equipartition. For sufficiently complex schemes the decline continues indefinitely. Standard tests like DIEHARD do not detect this pattern.

I suggest the principle of maximum entropy as the underlying cause. Similar decline patterns show up empirically in epidemiology, Web traffic, and other probabilistic settings.