  Latent variable models are used to estimate variables of interest quantities
which are observable only up to some measurement error. In many studies, such
variables are known but not precisely quantifiable (such as "job satisfaction"
in social sciences and marketing, "analytical ability" in educational testing,
or "inflation" in economics). This leads to the development of measurement
instruments to record noisy indirect evidence for such unobserved variables
such as surveys, tests and price indexes. In such problems, there are
postulated latent variables and a given measurement model. At the same time,
other unantecipated latent variables can add further unmeasured confounding to
the observed variables. The problem is how to deal with unantecipated latents
variables. In this paper, we provide a method loosely inspired by canonical
correlation that makes use of background information concerning the "known"
latent variables. Given a partially specified structure, it provides a
structure learning approach to detect "unknown unknowns," the confounding
effect of potentially infinitely many other latent variables. This is done
without explicitly modeling such extra latent factors. Because of the special
structure of the problem, we are able to exploit a new variation of composite
likelihood fitting to efficiently learn this structure. Validation is provided
with experiments in synthetic data and the analysis of a large survey done with
a sample of over 100,000 staff members of the National Health Service of the
United Kingdom.
