  Unbiased learning to rank has been proposed to alleviate the biases in the
search ranking, making it possible to train ranking models with user
interaction data. In real applications, search engines are designed to display
only the most relevant k documents from the retrieved candidate set. The rest
candidates are discarded. As a consequence, position bias and sample selection
bias usually occur simultaneously. Existing unbiased learning to rank
approaches either focus on one type of bias (e.g., position bias) or mitigate
the position bias and sample selection bias with separate components,
overlooking their associations. In this study, we first analyze the mechanisms
and associations of position bias and sample selection bias from the viewpoint
of a causal graph. Based on the analysis, we propose Causal Likelihood
Decomposition (CLD), a unified approach to simultaneously mitigating these two
biases in top-k learning to rank. By decomposing the log-likelihood of the
biased data as an unbiased term that only related to relevance, plus other
terms related to biases, CLD successfully detaches the relevance from position
bias and sample selection bias. An unbiased ranking model can be obtained from
the unbiased term, via maximizing the whole likelihood. An extension to the
pairwise neural ranking is also developed. Advantages of CLD include
theoretical soundness and a unified framework for pointwise and pairwise
unbiased top-k learning to rank. Extensive experimental results verified that
CLD, including its pairwise neural extension, outperformed the baselines by
mitigating both the position bias and the sample selection bias. Empirical
studies also showed that CLD is robust to the variation of bias severity and
the click noise.
