# A Novel Test for Independence Derived from an Exact Distribution of *i - Figure 2 *th Nearest Neighbours

**A**: Conditional distribution for , (*top*) and the entropy (*bottom*). The probability of observing large is zero for distances larger than when . The lower triangle is empty because and the entropy is constantly decreasing for increasing values of because the possible decrease towards . **B**: Marginal distribution for (*top*) and entropy (*bottom*). With increasing , the distribution becomes narrower and the entropy tends towards 0, as the number of possible distances to the th nearest neighbor decrease. The non-monotonic behavior of the entropy for large values of is due to downstream constraints imposed by the maximal distance . For testing independence, we advise using all until the value of where the entropy starts increasing again ( in this example).