A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approximations to the posterior have been suggested, and these are reviewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors.In addition, taking inspiration from the Physics literature, we study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher-Bingham distributions on the d-Sphere and a large-scale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.

The persistence of cancer cells resistant to therapy remains a major clinical challenge. In triple-negative breast cancer, resistance to chemotherapy results in the highest recurrence risk among breast cancer subtypes. The drug-tolerant state seems largely defined by non-genetic features, but the underlying mechanisms are poorly understood. Here, by monitoring epigenomes, transcriptomes and lineages with single-cell resolution, we show that the repressive histone mark H3K27me3 regulates cell fate at the onset of chemotherapy. We report that a persister expression program is primed with both H3K4me3 and H3K27me3 in unchallenged cells, H3K27me3 being the lock to its transcriptional activation. We further demonstrate that depleting H3K27me3 enhances the potential of cancer cells to tolerate chemotherapy. Conversely, preventing H3K27me3 demethylation simultaneously to chemotherapy inhibits the transition to a drug-tolerant state, and delays tumor recurrence in vivo . Our results highlight how chromatin landscapes shape the potential of cancer cells to respond to initial therapy.