Dimension in latent variable models

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 - I 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. (C) 2003 Elsevier Inc. All rights reserved.