Should you believe that this coin is fair?

Faced with a sequence of N binary events, such as coin flips (or Ising spins), it is natural to ask whether these events reflect some underlying dynamic signals or are just random. Plausible models for the dynamics of hidden biases lead to surprisingly high probabilities of misidentifying random sequences as biased. In particular, this probability decays as N −1/4 , so that no reasonable amount of data would be sufficient to induce the concept of a fair coin with high probability. I suggest that these theoretical results may be relevant to understanding experiments on the apparent misperception of random sequences by human observers. There is a large literature testifying to the errors that humans make in reasoning about probability [1, 2]. Perhaps most fundamental is the claim that people routinely detect order and hidden causes in genuinely random sequences [3]. These apparent limitations on human ratio-nality have broad implications, not least for economics, and have attracted considerable attention in the popular press. In contrast with these results, a number of experiments indicate that humans and other animals can change their behavior in response to changes in the probabilities of stimuli and rewards, sometimes making optimal use of the available data [4, 5, 6, 7]. Similarly, many perceptual discriminations approach the limits to reliability set by noise near the sensory input [8, 9, 10], and related ideas of statistical optimization have emerged in recent work on motor control [11, 12, 13]. There is even the suggestion that if the detection of order vs. random-ness is cast in the standard two–alternative format for perceptual discrimination experiments, then people can learn to perform with close to the statistically maximum reliability [14]. While nearly perfect neural processing of statistical data under some conditions could coexist with qualitative failures at similar problems under different conditions, it would be attractive to have a more unified view of the brain as an engine for probabilistic inference. The problem of identifying genuinely random sequences has a number of subtleties that seem not to have been emphasized in the previous literature. In particular , from a Bayesian point of view our confidence that a given sequence really was generated at random depends entirely on the universe of alternative models that we are willing to consider. Here I consider a family of models that involve time–dependent biases of unknown magnitude , similar in spirit to the models that would allow …