Psychologists would like to say that a probability distribution on {0,1}~n is d-dimensional if (1) the distribution can be represented by some smooth d-dimensional latent variable model and (2) the distribution cannot be represented by any smooth d-1 dimensional model. This does not work out because for d > 1, every distribution that can be represented by a smooth d-dimensional model can also be represented by a smooth one-dimensional model. A proof and discussion of some implications of this mathematical result follow.
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