Causal inference is a challenging problem with observational data alone. The
task becomes easier when having access to data from perturbing the underlying
system, even when happening in a non-randomized way: this is the setting we
consider, encompassing also latent confounding variables. To identify causal
relations among a collections of covariates and a response variable, existing
procedures rely on at least one of the following assumptions: i) the response
variable remains unperturbed, ii) the latent variables remain unperturbed, and
iii) the latent effects are dense. In this paper, we examine a perturbation
model for interventional data, which can be viewed as a mixed-effects linear
structural causal model, over a collection of Gaussian variables that does not
satisfy any of these conditions. We propose a maximum-likelihood estimator --
dubbed DirectLikelihood -- that exploits system-wide invariances to uniquely
identify the population causal structure from unspecific perturbation data, and
our results carry over to linear structural causal models without requiring
Gaussianity. We illustrate the utility of our framework on synthetic data as
well as real data involving California reservoirs and protein expressions