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Abstract The Li-Stephens (LS) haplotype copying model forms the basis of a number of important statistical inference procedures in genetics. LS is a probabilistic generative model which supposes that a sampled chromosome is an imperfect mosaic of other chromosomes found in a population. In the frequentist setting which is the focus of this paper, the output of LS is a “copying path” through chromosome space. The behavior of LS depends crucially on two user-specified parameters,$$\theta $$ $\theta$and$$\rho $$ $\rho$, which are respectively interpreted as the rates of mutation and recombination. However, because LS is not based on a realistic model of ancestry, the precise connection between these parameters and the biological phenomena they represent is unclear. Here, we offer an alternative perspective, which considers$$\theta $$ $\theta$and$$\rho $$ $\rho$as tuning parameters, and seeks to understand their impact on the LS output. We derive an algorithm which, for a given dataset, efficiently partitions the$$(\theta ,\rho )$$ $(\theta ,\rho )$plane into regions where the output of the algorithm is constant, thereby enumerating all possible solutions to the LS model in one go. We extend this approach to the “diploid LS” model commonly used for phasing. We demonstrate the usefulness of our method by studying the effects of changing$$\theta $$ $\theta$and$$\rho $$ $\rho$when using LS for common bioinformatic tasks. Our findings indicate that using the conventional (i.e., population-scaled) values for$$\theta $$ $\theta$and$$\rho $$ $\rho$produces near optimal results for imputation, but may systematically inflate switch error in the case of phasing diploid genotypes.