  An old problem in multivariate statistics is that linear Gaussian models are
often unidentifiable, i.e. some parameters cannot be uniquely estimated. In
factor (component) analysis, an orthogonal rotation of the factors is
unidentifiable, while in linear regression, the direction of effect cannot be
identified. For such linear models, non-Gaussianity of the (latent) variables
has been shown to provide identifiability. In the case of factor analysis, this
leads to independent component analysis, while in the case of the direction of
effect, non-Gaussian versions of structural equation modelling solve the
problem. More recently, we have shown how even general nonparametric nonlinear
versions of such models can be estimated. Non-Gaussianity is not enough in this
case, but assuming we have time series, or that the distributions are suitably
modulated by some observed auxiliary variables, the models are identifiable.
This paper reviews the identifiability theory for the linear and nonlinear
cases, considering both factor analytic models and structural equation models.
