Conditional Mean Flows: Few-Step Densities for Simulation-Based Inference

Sampling from the posterior is the central task of computational Bayesian inverse problems. The standard workhorse in Bayesian inference - Markov chain Monte Carlo (MCMC) - is sequential, yields correlated samples, and must be rerun for each observation. Conditional flow matching offers an amortized alternative: a transport map, trained once on joint samples of parameter and data, that yields independent approximate posterior samples for any observation at negligible online cost, without new likelihood evaluations. We give a careful, MCMC-literate assessment of flow matching for PDE-based inverse problems with function-valued parameters. Exploiting the flow's tractable density, we derive computable accuracy estimates of the underlying approximate posterior in total-variation distance and Kullback-Leibler divergence and, moreover, propose a hybrid sampler that is asymptotically exact by Metropolization. We validate the accuracy estimates and demonstrate the amortization in several numerical examples, including electrical impedance tomography and a likelihood-free Lotka-Volterra model.

Simulation-based inference (SBI) methods typically amortize a single query, usually the posterior, so each further conditional or marginal needs its own model. Existing generalists amortize many queries in one transformer with one token per variable, so their attention cost grows quadratically with the number of variables. We introduce OneFlowSBI, a masked flow matching framework that trains one network on the joint distribution of parameters and observations. A ternary mask marks each coordinate as observed, generated or absent, so any conditional or marginal is sampled by changing the mask. The mask is an input, rather than a tokenization, so the backbone can follow the data's structure and scale to many variables. On high-dimensional image denoising and shallow water inference, OneFlowSBI achieves lower posterior error than the baselines, samples far faster than a transformer-based generalist, and handles missing regions and partial observation windows without retraining. On most of the ten SBIBM tasks at the largest simulation budget, it matches or exceeds neural posterior estimation and is comparable to transformer-based generalists, while also providing the likelihood and marginals.

The cost of simulator evaluations is a key practical bottleneck for Simulation Based Inference (SBI). In hierarchical settings with shared global parameters and exchangeable site-level parameters and observations, this structure can be exploited to improve simulation efficiency. Existing hierarchical SBI approaches factorise the posterior yet still simulate across multiple sites per training sample; We instead explore likelihood factorisation (LF) to train from single-site simulations. In LF sampling we learn a per-site neural surrogate of the simulator and then assemble synthetic multi-site observations to amortise inference for the full hierarchical posterior. Building on this, we propose Tokenised Flow Matching for Posterior Estimation (TFMPE), a tokenised flow matching approach that supports function-valued observations through likelihood factorisation. To enable systematic evaluation, we introduce a benchmark for hierarchical SBI. We validate TFMPE on this benchmark and on realistic infectious disease and computational fluid dynamics models, finding well-calibrated posteriors while reducing computational cost.