  We develop a novel identification strategy as well as a new estimator for
context-dependent causal inference in non-parametric triangular models with
non-separable disturbances. Departing from the common practice, our analysis
does not rely on the strict monotonicity assumption. Our key contribution lies
in leveraging on diffusion models to formulate the structural equations as a
system evolving from noise accumulation to account for the influence of the
latent context (confounder) on the outcome. Our identifiability strategy
involves a system of Fredholm integral equations expressing the distributional
relationship between a latent context variable and a vector of observables.
These integral equations involve an unknown kernel and are governed by a set of
structural form functions, inducing a non-monotonic inverse problem. We prove
that if the kernel density can be represented as an infinite mixture of
Gaussians, then there exists a unique solution for the unknown function. This
is a significant result, as it shows that it is possible to solve a
non-monotonic inverse problem even when the kernel is unknown. On the
methodological front we leverage on a novel and enriched Contaminated
Generative Adversarial (Neural) Networks (CONGAN) which we provide as a
solution to the non-monotonic inverse problem.
