This paper introduces a Markov chain Monte Carlo (MCMC) method for generating observations from a posterior distribution without the use of likelihoods. The method is applicable in both Bayesian and frequentist contexts, including maximum-likelihood estimation. The approach is illustrated using an example of ancestral inference in population genetics. The paper discusses the challenges of computing posterior distributions when likelihoods are intractable or computationally prohibitive, and presents an MCMC algorithm that does not require likelihood evaluation. The algorithm is based on simulating data from the model and comparing it to the observed data, using a distance metric to determine acceptance. The method is shown to be effective in scenarios where likelihoods are difficult to compute, such as in population genetics. The paper also compares the performance of different approaches, including rejection sampling, estimated likelihood methods, and likelihood-free MCMC, using a dataset of mitochondrial DNA sequences. The results show that the likelihood-free MCMC approach provides a good approximation to the true posterior distribution, even when the data are summarized using a small set of statistics. The paper concludes that further research is needed to develop more efficient methods for combining summary statistics to improve posterior estimates.This paper introduces a Markov chain Monte Carlo (MCMC) method for generating observations from a posterior distribution without the use of likelihoods. The method is applicable in both Bayesian and frequentist contexts, including maximum-likelihood estimation. The approach is illustrated using an example of ancestral inference in population genetics. The paper discusses the challenges of computing posterior distributions when likelihoods are intractable or computationally prohibitive, and presents an MCMC algorithm that does not require likelihood evaluation. The algorithm is based on simulating data from the model and comparing it to the observed data, using a distance metric to determine acceptance. The method is shown to be effective in scenarios where likelihoods are difficult to compute, such as in population genetics. The paper also compares the performance of different approaches, including rejection sampling, estimated likelihood methods, and likelihood-free MCMC, using a dataset of mitochondrial DNA sequences. The results show that the likelihood-free MCMC approach provides a good approximation to the true posterior distribution, even when the data are summarized using a small set of statistics. The paper concludes that further research is needed to develop more efficient methods for combining summary statistics to improve posterior estimates.