Semi-supervised interpolation in an anticausal learning scenario

According to a recently stated 'independence postulate', the distribution Pcause contains no information about the conditional Peffect|cause while Peffect may contain information about Pcause|effect. Since semi-supervised learning (SSL) attempts to exploit information from PX to assist in predicting Y from X, it should only work in anticausal direction, i.e., when Y is the cause and X is the effect. In causal direction, when X is the cause and Y the effect, unlabelled x-values should be useless. To shed light on this asymmetry, we study a deterministic causal relation Y = f(X) as recently assayed in Information-Geometric Causal Inference (IGCI). Within this model, we discuss two options to formalize the independence of PX and f as an orthogonality of vectors in appropriate inner product spaces. We prove that unlabelled data help for the problem of interpolating a monotonically increasing function if and only if the orthogonality conditions are violated - which we only expect for the anticausal direction. Here, performance of SSL and its supervised baseline analogue is measured in terms of two different loss functions: first, the mean squared error and second the surprise in a Bayesian prediction scenario.