Predicting When Cheap Simulators Break Neural Inference of Bursty Gene Expression

Simulation-based inference (SBI) makes stochastic models of gene expression amenable to amortized Bayesian parameter estimation, but practitioners routinely replace the exact chemical master equation (CME) in the training loop with a cheaper approximate simulator such as the chemical Langevin equation (CLE). We ask how this substitution biases inference, and whether we can predict the resulting failure a priori rather than merely detect it after the fact. Using the two-state (telegraph) model, whose exact stationary distribution and moments have closed form, we show that the CLE misspecifies the model directionally in the bursty, low-copy regime: it overestimates mean expression, suppresses the Fano factor, and erases the zero-inflation that signals bursty transcription. We train neural posterior estimators on the cheap simulators and on the exact CME and evaluate all against exact CME data. The CLE-trained estimator is biased and overconfident across the whole burstiness range, and a second, structurally different Gaussian surrogate, the linear noise approximation, also loses calibration as burstiness grows, precisely the regime of biological interest. An ablation that restricts inference to low-order moments does not rescue it, because enforcing non-negative counts distorts even those moments where bursts dominate; the culprit is the nability of a Gaussian to represent bursty count distributions, not any single simulator. A matched non Gaussian surrogate that carries the same moments is instead calibrated, which pins the cause on Gaussianity and points to a cheap fix. The CME trained control stays calibrated throughout. Because the telegraph model admits closed-form moments, we anticipate the onset of this failure from the analytic Fano factor, which converts a post hoc diagnosis into a simple budget rule for when a practitioner can trust the cheap simulator. We frame the work as a controlled-sandbox characterization rather than a new inference method; its value lies in a transferable diagnostic for surrogate-simulator misspecification in SBI.