GLLiM-ABC: Likelihood-Free Inference with Surrogate Posteriors
When the likelihood is intractable but you can simulate data, approximate Bayesian computation (ABC) accepts parameter proposals whose simulated data look like the observation. The hard part is how to compare: which summary statistics, and which distance. This method learns a surrogate posterior once (a Gaussian mixture from an inverse-regression fit), then uses the whole distribution as a functional summary and compares surrogates with an L2 or optimal-transport distance. This is the paper's flagship multiple-hyperboloid experiment, a source-localization problem whose posterior is genuinely multimodal. Click the map and compare the four ABC discrepancies.
The unknown is a source location θ = (θ₁, θ₂) in [−2, 2]². Two anchor pairs are fixed (blue at (±0.5, 0), purple at (0, ±0.5)); one pair is chosen at random and the observation is R = 10 noisy Student-t (df = 3) measurements of F(θ) = |d₁ − d₂|, the absolute difference of the distances to that pair's anchors. Since |d₁ − d₂| is constant along a hyperbola and the generating pair is unknown, many source locations explain the data equally well: the true posterior (the heat map) concentrates on hyperbola arcs and is genuinely multimodal. This is exactly the paper's non-identifiable flagship (its ITD / interaural-time-difference model). A GLLiM model, fitted once by inverse regression (the GLLiM-iid variant, since the 10 measurements are exchangeable), turns any observation into a full surrogate posterior pG(θ | y) = ∑k ηk(y) N(θ; μk(y), Σ), a Gaussian mixture (blue discs) that traces the arcs. Rejection ABC keeps the closest 2% of simulated sources under a chosen discrepancy. The baseline, Semi-automatic ABC (Fearnhead-Prangle), summarizes each observation by its estimated posterior mean E[θ | y]; because the arcs are symmetric, that mean (amber □) lands in a low-density valley between the modes, so it is nearly uninformative. The paper's GLLiM methods instead use the surrogate posterior: GLLiM-EV adds the log-variance, while GLLiM-L2 (closed-form L2 between the mixtures) and GLLiM-MW2 (Mixture-Wasserstein optimal transport) compare the whole distribution. The right panel scores each method by how much of its accepted sample falls in high-density regions of the true posterior: the semi-automatic mean summary is weakest, and the functional distances L2 / MW2 clearly outperform it, recovering the multimodal structure most faithfully. Toggle SMC-ABC to switch from one-shot rejection to the paper's sequential variant, which perturbs and re-simulates the accepted particles under a shrinking threshold to refine them onto the arcs. Move the source by clicking, draw a new observation, and switch discrepancies. (The GLLiM fit φ* is a real GLLiM-iid estimate (30 components) precomputed offline; conditioning uses the exact GLLiM-iid formula, L2 is exact, and MW2 is solved by exact integer optimal transport. The paper applies the method to real audio-localization and planetary-science data.)
Run the experiments
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The four discrepancies
Tour the semi-automatic point summary, the mean+variance, the closed-form L2 and the exact Wasserstein (MW2) discrepancy; the functional ones recover the posterior arcs the point summary misses.
Acceptance rate
Tighten the ABC acceptance threshold and watch the accepted cloud concentrate onto the true posterior.
New observations
Draw fresh observations and see the posterior arcs move with the data.
The idea in three steps
ABC needs a way to say how far a simulation is from the observation. Reducing data to a handful of summary statistics can throw away exactly what distinguishes competing explanations. This method keeps a whole distribution as the summary.
A surrogate posterior
One inverse-regression (GLLiM) fit turns any observation into a full posterior over the parameters, a Gaussian mixture available in closed form, at no per-observation cost.
Distances between distributions
Instead of comparing summary numbers, compare the surrogate posteriors themselves with an L2 or a Mixture-Wasserstein distance, so the full shape, including multiple modes, has to match.
Where the mean fails
When several parameter values explain the data equally well, their average is meaningless. The functional distances keep the modes separate; the posterior-mean summary cannot.
For the surrogate-posterior construction, the L2 and Mixture-Wasserstein distances, the convergence of the ABC quasi-posterior, and the SMC-ABC and real-data experiments, see Summary statistics and discrepancy measures for approximate Bayesian computation via surrogate posteriors (Forbes, H. D. Nguyen, TrungTin Nguyen & Arbel, Statistics and Computing 2022).