  In counterfactual learning to rank (CLTR) user interactions are used as a
source of supervision. Since user interactions come with bias, an important
focus of research in this field lies in developing methods to correct for the
bias of interactions. Inverse propensity scoring (IPS) is a popular method
suitable for correcting position bias. Affine correction (AC) is a
generalization of IPS that corrects for position bias and trust bias. IPS and
AC provably remove bias, conditioned on an accurate estimation of the bias
parameters. Estimating the bias parameters, in turn, requires an accurate
estimation of the relevance probabilities. This cyclic dependency introduces
practical limitations in terms of sensitivity, convergence and efficiency.
  We propose a new correction method for position and trust bias in CLTR in
which, unlike the existing methods, the correction does not rely on relevance
estimation. Our proposed method, mixture-based correction (MBC), is based on
the assumption that the distribution of the CTRs over the items being ranked is
a mixture of two distributions: the distribution of CTRs for relevant items and
the distribution of CTRs for non-relevant items. We prove that our method is
unbiased. The validity of our proof is not conditioned on accurate bias
parameter estimation. Our experiments show that MBC, when used in different
bias settings and accompanied by different LTR algorithms, outperforms AC, the
state-of-the-art method for correcting position and trust bias, in some
settings, while performing on par in other settings. Furthermore, MBC is orders
of magnitude more efficient than AC in terms of the training time.
