Revisiting concentration results for approximate Bayesian computation
As the tolerance ε → 0, the ABC posterior concentrates, but it can only concentrate on the equivalence set Θ₀ of parameters that generate the same data law: everything a discrepancy cannot distinguish stays indistinguishable. This demo draws the prior (dotted) and the paper's limiting ABC posterior πε(θ) ∝ π(θ) w(|D∞(θ₀,θ)|, ε) in closed form over a fine grid, for the three normal models of Section 6: identifiable, finitely unidentifiable, and infinitely unidentifiable. Sweep ε and watch the mass collapse, onto a point, onto two symmetric points, or onto a one-dimensional L1 diamond.
The unknown is θ with prior π(θ) = N(0, 1) (in 2D, N(0, I₂)). Two independent normal samples are compared through the mean distance D(Xn, Yn) = |X̄n − Ȳn|, which converges almost surely to D∞(θ₀, θ). The ABC posterior weights the prior by the kernel w(D, ε), and as n → ∞ it converges to the paper's limiting density πε(θ) ∝ π(θ) w(|D∞(θ₀, θ)|, ε). Here we plot exactly that limit (with a finite-n Gaussian blur of width √(2/n) on the summary, so the n slider visibly sharpens it). Identifiable (D∞ = |θ₀ − θ|): Θ₀ = {θ₀}, the mass collapses to a single point. Finitely unidentifiable (D∞ = |θ₀² − θ²|): Θ₀ = {−θ₀, +θ₀}, two symmetric spikes that never merge, no matter how small ε gets. Infinitely unidentifiable (D∞ = | ‖θ₀‖₁ − ‖θ‖₁ |): Θ₀ is the whole L1 diamond ‖θ‖₁ = ‖θ₀‖₁, so the posterior collapses onto a one-dimensional curve, never a point. The right panel tracks the effective support |Θε| shrinking as ε → 0: concentration on sets of arbitrarily small measure that still contain Θ₀. (This is a faithful 1D/2D toy of the paper's Section 6 illustrations, computed in closed form over a grid: the real paper proves general almost-sure convergence and total-variation concentration for a broad class of kernelized ABC algorithms, without summary statistics, and applies it to a sound-source-localization problem. The "triweight" here is the paper's smooth kernel, standing in for the generic Gaussian-like kernel.)
Run the experiments
Every animation runs live in your browser. Click a button to run that experiment on the demo (it scrolls up and starts); drag any control to take over. Nothing is downloaded, it is generated on the fly.
Sweep the tolerance ε → 0
The star control. As the ABC tolerance shrinks toward zero, the limiting posterior concentrates on the equivalence set: a single spike when identifiable, two spikes that never merge when finitely unidentifiable, or a thinning L1 diamond when infinitely unidentifiable.
Move the true parameter θ₀
Slide the data-generating parameter and watch the equivalence set Θ₀ (the point, the symmetric pair ±θ₀, or the L1 radius) move with it while the posterior tracks it.
Sharpen with more data n
The discrepancy is a sample mean, so it converges almost surely to its limit D∞ only as n grows. Increasing n removes the √(2/n) blur and sharpens the posterior toward the paper's closed-form limiting density.
Tour the three identifiability regimes
Cycle identifiable (collapse to a point), finitely unidentifiable (two symmetric spikes that never merge), and infinitely unidentifiable (mass collapses onto a 1-D L1 diamond, never a point).
Swap the ABC kernel
Compare the accept/reject (uniform) kernel with the paper's smooth triweight kernel: the accepted-region shape softens but the support width, and the equivalence set it concentrates on, is unchanged.
The idea in three steps
ABC accepts parameter proposals whose simulated data land within ε of the observation. What does the accepted cloud look like when the tolerance shrinks and the model is not identifiable?
The posterior collapses
As ε → 0 the ABC posterior concentrates on smaller and smaller sets. With enough data it converges almost surely to a limiting density that is just the prior reweighted by the kernel of the discrepancy.
Onto what it can resolve
It concentrates on the equivalence set Θ₀: all parameters that produce the same data law. Identifiable models give a single point; a square summary gives two symmetric points; an L1 summary gives an entire diamond.
Concentration is not resolution
ABC concentrates, but it cannot resolve the unidentifiability that the discrepancy discards. Tightening ε sharpens the modes; it never merges the two spikes or shrinks the diamond to a point.
For the almost-sure and total-variation convergence of the pseudo-posterior and coarsened posterior, the concentration-of-mass theorems without identifiability, and the assumptions on the kernel and prior, see Revisiting concentration results for approximate Bayesian computation (H. D. Nguyen, TrungTin Nguyen, Arbel & Forbes, Bayesian Analysis 2025).