Fig. 2

A VI-VS overview. VI-VS identifies conditional dependencies between molecular features X, e.g., genes, and a response variable Y of cell properties, e.g., protein expression levels, in single-cell genomics. (1) We first randomly split the observed data into a validation and a development set. On the development set, we fit a generative model \(p_\theta\) and importance score \(T_\phi\), which is a scalar-valued function taking Y and X as inputs on the development set. In a simple case, \(T_\phi\) may correspond to the prediction error of the ordinary least squares of Y on X. Here, \(\theta\) and \(\phi\) denote the parameters of these models, learned on the development set. (2) We compute the importance score of the observed data on the validation set. (3) In parallel, we sample K synthetic feature samples for gene g using the trained generative model. We then compute synthetic importance scores, computed based on Y and on the modified feature matrix X where the gth column was replaced by the synthetic samples. (4) We compare the observed importance score to the distribution of synthetic importance scores to compute a p-value. B Power limitation of conditional approaches. Features \(X_1, X_2, X_3, X_4\) are mildly correlated and form a first cluster. Features \(X_1', X_2', X_3', X_4'\) are strongly correlated and form a second cluster. If the target response causally depends on \(X_1\) and \(X_1'\), then a conditional independence test may fail to detect \(X'_1\) due to its strong correlations with features of the same cluster. C Illustrative example of multi-resolution testing on B in the case where conditional dependencies at assessed at three resolutions (res. 3 being the finest at the feature level). VI-VS can detect groups of features that are conditionally dependent on the response variable, even if no individual gene can be identified as conditionally dependent, as well as individual features, if the sample size allows. For instance, VI-VS could detect, in additional to individual feature \(X_1\), that group \(\{X'_1, X'_2\}\) (marked as a star in the figure) is conditionally dependent on the response without being able which of the two features is responsible for the association