ABC-$D^2$: Gradient-Informed Inference for Discrete Stochastic Simulators

Sequential Monte Carlo approximate Bayesian computation (SMC-ABC) enables likelihood-free Bayesian inference with expensive simulators, by distributing simulators across parallel workers. However, its random-walk moves can become ineffective as the parameter dimension grows and when the particle population is small. Gradient-informed proposals offer a potential solution, but stochastic simulators used in scientific contexts such as population genetics, epidemiology and chemical physics often use discrete distributions, which prevent conventional pathwise differentiation. We introduce ABC-$D^2$, a Metropolis-adjusted Langevin method that uses discrete stochastic automatic differentiation ($D^2$) to guide particle mutation in SMC-ABC, with unbiased and low-variance gradient estimates of an expected log-kernel ABC objective from a single simulation and without resorting to relaxations or learned surrogates. We prove that drift along an unbiased gradient estimate improves the expected objective at the proposal over a random walk with the same covariance, to leading order, which a biased drift does not guarantee, and that fixing the drift's random numbers within each sweep keeps the Metropolis-Hastings correction exact. On Bernoulli generalised linear models, Gaussian mixture models and a multi-population SIR epidemic model, ABC-$D^2$ reduces posterior-mean error of random-walk SMC-ABC by up to two-thirds and matches the random walk's accuracy with at least an order of magnitude fewer particles. These results demonstrate the potential of discrete stochastic differentiation to improve the accuracy of SMC-ABC for discrete simulators in common scientific applications, where conventional gradient-based proposals cannot be deployed.