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But what if a sequence one actually observes has 9 black squares out of 10? Even though this is not the most likely thing to see, one certainly cannot conclude from seeing it that the model is wrong. For the model does not say that such sequences are impossible—it merely says that they should occur only about 1% of the time.

And indeed there is no meaningful way without more information to deduce any kind of absolute probability for the model to be correct. So in practice what almost universally ends up being done is to consider not just an individual model, but rather a whole class of models, and then to try to identify which model from this class is the best one—as measured, say, by the criterion that its likelihood of generating the observed data is as large as possible.

For sequences of black and white squares a simple class of models to consider are those in which each square is taken to be black with some fixed independent probability p. Given a set of raw data the procedure for finding which model in this class is best—according, say, to the criterion of maximum likelihood—is extremely straightforward: all one does is to compute what fraction of squares in the data are black, and this value then immediately gives the value of p for the best model.

So what about more complicated models? Instead of taking each square to have a color that is chosen completely independently, one can for example take blocks of squares of some given length to have their colors chosen together. And in this case the best model is again straightforward to find: it simply takes the probabilities for different blocks to be equal to the frequencies with which these blocks occur in the data.

If one does not decide in advance how long the blocks are going to be, however, then things can become more complicated. For in such a case one can always just make up an extreme model in which only one very long block is allowed, with this block being precisely the sequence that is observed in the data.

Needless to say, such a model would for most purposes not be considered particularly useful—and certainly it does not succeed in providing any kind of short summary of the data. But to exclude models like this in a systematic way requires going beyond criteria such as

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From Stephen Wolfram: A New Kind of Science [citation]