  Methods that bypass analytical evaluations of the likelihood function have
become an indispensable tool for statistical inference in many fields of
science. These so-called likelihood-free methods rely on accepting and
rejecting simulations based on summary statistics, which limits them to low
dimensional models for which the absolute likelihood is large enough to result
in manageable acceptance rates. To get around these issues, we introduce a
novel, likelihood-free Markov-Chain Monte Carlo (MCMC) method combining two key
innovations: updating only one parameter per iteration and accepting or
rejecting this update based on subsets of statistics sufficient for this
parameter. This increases acceptance rates dramatically, rendering this
approach suitable even for models of very high dimensionality. We further
derive that for linear models, a one dimensional combination of statistics per
parameter is sufficient and can be found empirically with simulations. Finally,
we demonstrate that our method readily scales to models of very high
dimensionality using both toy models as well as by jointly inferring the
effective population size, the distribution of fitness effects of new mutations
(DFE) and selection coefficients for each locus from data of a recent
experiment on the evolution of drug-resistance in Influenza.
