Bayesian inference plays an important role in phylogenetics, evolutionary biology, and in many other branches of science. It provides a principled framework for dealing with uncertainty and quantifying how it changes in the light of new evidence. For many complex models and inference problems, however, only approximate quantitative answers are obtainable. Approximate Bayesian computation (ABC) refers to a family of algorithms for approximate inference that makes a minimal set of assumptions by only requiring that sampling from a model is possible. We explain here the fundamentals of ABC, review the classical algorithms, and highlight recent developments. [ABC; approximate Bayesian computation; Bayesian inference; likelihood-free inference; phylogenetics; simulator-based models; stochastic simulation models; tree-based models.]

Understanding the transmission dynamics of infectious diseases is important for both biological research and public health applications. It has been widely demonstrated that statistical modeling provides a firm basis for inferring relevant epidemiological quantities from incidence and molecular data. However, the complexity of transmission dynamic models presents two challenges: (1) the likelihood function of the models is generally not computable, and computationally intensive simulation-based inference methods need to be employed, and (2) the model may not be fully identifiable from the available data. While the first difficulty can be tackled by computational and algorithmic advances, the second obstacle is more fundamental. Identifiability issues may lead to inferences that are driven more by prior assumptions than by the data themselves. We consider a popular and relatively simple yet analytically intractable model for the spread of tuberculosis based on classical IS6110 fingerprinting data. We report on the identifiability of the model, also presenting some methodological advances regarding the inference. Using likelihood approximations, we show that the reproductive value cannot be identified from the data available and that the posterior distributions obtained in previous work have likely been substantially dominated by the assumed prior distribution. Further, we show that the inferences are influenced by the assumed infectious population size, which generally has been kept fixed in previous work. We demonstrate that the infectious population size can be inferred if the remaining epidemiological parameters are already known with sufficient precision.KEYWORDS approximate Bayesian computation; identifiability; intractable likelihood; transmission dynamic models; tuberculosis S TATISTICAL models for transmission dynamics are widely employed to answer fundamental questions about the infectivity of bacteria and viruses and to make predictions for intervention policies such as vaccines, decolonization, and case containment. For some types of infectious diseases, the complexity of the transmission process and the corresponding model, combined with the characteristics of the available data, makes the inference an intricate task. A particular difficulty arises from the need to use computationally intensive methods. Examples include the work by Tanaka et al. (2006), Sisson et al. (2007), Blum (2010), Stadler (2011), Fearnhead and Prangle (2012), Del Moral et al. (2012, Baragatti et al. (2013), andAlbert et al. (2015), who considered the transmission dynamics of Mycobacterium tuberculosis based on IS6110 fingerprinting data from tuberculosis (M. tuberculosis) cases in San Francisco reported earlier by Small et al. (1994). Except for Stadler (2011), who proposed an inference scheme based on likelihood and Markov chain Monte Carlo approximations, the abovementioned studies employed and improved an approximate inference technique known as approximate Bayesian computation (ABC), which was originally introduced...

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