  Measurement error in the observed values of the variables can greatly change
the output of various causal discovery methods. This problem has received much
attention in multiple fields, but it is not clear to what extent the causal
model for the measurement-error-free variables can be identified in the
presence of measurement error with unknown variance. In this paper, we study
precise sufficient identifiability conditions for the measurement-error-free
causal model and show what information of the causal model can be recovered
from observed data. In particular, we present two different sets of
identifiability conditions, based on the second-order statistics and
higher-order statistics of the data, respectively. The former was inspired by
the relationship between the generating model of the
measurement-error-contaminated data and the factor analysis model, and the
latter makes use of the identifiability result of the over-complete independent
component analysis problem.
