## Semi-Supervised Interpolation in an Anticausal Learning Scenario

*Dominik Janzing, Bernhard Schölkopf*; 16(Sep):1923−1948, 2015.

### Abstract

According to a recently stated 'independence postulate', the
distribution $P_{\rm cause}$ contains no information about the
conditional $P_{\rm effect | cause}$ while $P_{\rm effect}$ may
contain information about $P_{\rm cause | effect}$. Since semi-
supervised learning (SSL) attempts to exploit information from
$P_X$ 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 $P_X$ 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.

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