Markov chain Monte Carlo methods without likelihoods

In Bayesian statistics, researchers often need to determine the posterior distribution of model parameters given observed data. Traditionally, this requires calculating a "likelihood function," which quantifies how well a specific set of parameters explains the observed data. However, for many complex scientific models—such as those used in population genetics—the likelihood function is either mathematically impossible to derive or computationally too expensive to evaluate. This paper addresses this challenge by introducing a method to perform Bayesian inference without ever calculating the likelihood.

The authors propose a Markov chain Monte Carlo (MCMC) method that avoids likelihoods by relying on stochastic simulation. The core idea is to simulate data from a model using proposed parameters and compare the simulated output to the actual observed data.

When the full data are too complex to compare directly, the authors utilize "summary statistics"—lower-dimensional representations of the data that capture the most relevant information. The algorithm accepts or rejects proposed parameters based on whether the distance between the summary statistics of the simulated data and the observed data falls within a predefined tolerance threshold ($\epsilon$). By iteratively proposing and accepting parameters, the algorithm generates a sequence of samples that approximates the true posterior distribution.

The researchers demonstrate their method using ancestral inference in population genetics, specifically analyzing mitochondrial DNA sequences. They show that their likelihood-free MCMC approach produces results comparable to traditional methods that rely on full likelihoods.

The study highlights a critical trade-off: while using summary statistics makes complex problems tractable, the accuracy of the posterior depends heavily on choosing statistics that are "sufficient" (i.e., they capture all the information relevant to the parameters). The authors conclude that this approach is a powerful tool for fields where simulation is easy but mathematical analysis is intractable, though they emphasize that further research is needed to automate the selection of optimal summary statistics.