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Is there a formal treatment of the role/power of latent/hidden variables in graphical models and other machine learning models (e.g., structural equation models)?

For example, the Restricted Boltzman Machine is a graphical model which is a compact representation of a probability distribution of $x$, but it is modeled as a joint distribution with a hidden variable $h$ (i.e., we model $p(x,h)$ instead of $p(x)$). Why is $p(x,h)$ any more powerful than a suitable representation for $p(x)$?

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Hidden variables simplify the description of certain models. They do not give any more modelling power. For any model with hidden variables, there is an equivalent model without them.

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  • $\begingroup$ Thanks. I figured out and also saw the same in this webpage: cs.ubc.ca/~murphyk/Bayes/bnintro.html (under the title "Inventing new hidden nodes"). Now I understand hidden variables lower model complexity. $\endgroup$
    – Vimal
    Sep 22, 2014 at 0:38

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