tell, the only kinds of correlations that are ultimately important to our auditory system are those that lead to some form of repetition.

So in the end, any features of the behavior of a system that go beyond pure repetition will tend to seem to our ears essentially random.

## Statistical Analysis

When it comes to studying large volumes of data the method almost exclusively used in present-day science is statistical analysis. So what kinds of processes does such analysis involve? What is typically done in practice is to compute from raw data various fairly simple quantities whose values can then be used to assess models which could provide summaries of the data.

Most kinds of statistical analysis are fundamentally based on the assumption that such models must be probabilistic, in the sense that they give only probabilities for behavior, and do not specifically say what the behavior will be. In different situations the reasons for using such probabilistic models have been somewhat different, but before the discoveries in this book one of the key points was that it seemed inconceivable that there could be deterministic models that would reproduce the kinds of complexity and apparent randomness that were so often seen in practice.

If one has a deterministic model then it is at least in principle quite straightforward to find out whether the model is correct: for all one has to do is to compare whatever specific behavior the model predicts with behavior that one observes. But if one has a probabilistic model then it is a much more difficult matter to assess its validity—and indeed much of the technical development of the field of statistics, as well as many of its well-publicized problems, can be traced to this issue.

As one simple example, consider a model in which all possible sequences of black and white squares are supposed to occur with equal probability. By effectively enumerating all such sequences, it is easy to see that such a model predicts that in any particular sequence the fraction of black squares is most likely to be 1/2.