Debiasing FracMinHash and deriving confidence intervals for mutation rates across a wide range of evolutionary distances

Sketching methods offer computational biologists scalable techniques to analyze data sets that continue to grow in size. MinHash is one such technique to estimate set similarity that has enjoyed recent broad application. However, traditional MinHash has previously been shown to perform poorly when applied to sets of very dissimilar sizes. FracMinHash was recently introduced as a modification of MinHash to compensate for this lack of performance when set sizes differ. This approach has been successfully applied to metagenomic taxonomic profiling in the widely used tool sourmash gather. While experimental evidence has been encouraging, FracMinHash has not yet been analyzed from a theoretical perspective. In this paper, we perform such an analysis to derive various statistics of FracMinHash, and prove that while FracMinHash is not unbiased (in the sense that its expected value is not equal to the quantity it attempts to estimate), this bias is easily corrected for both the containment and Jaccard index versions. Next, we show how FracMinHash can be used to compute point estimates as well as confidence intervals for evolutionary mutation distance between a pair of sequences by assuming a simple mutation model. We also investigate edge cases where these analyses may fail, to effectively warn the users of FracMinHash indicating the likelihood of such cases. Our analyses show that FracMinHash estimates the containment of a genome in a large metagenome more accurately and more precisely when compared to traditional MinHash, and the point estimates and confidence intervals perform significantly better in estimating mutation distances.