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arxiv: 2505.23261 · v3 · submitted 2025-05-29 · 📊 stat.CO

A thermodynamic approach to Approximate Bayesian Computation with multiple summary statistics

Carlo Albert , Simone Ulzega , Simon Dirmeier , Andreas Scheidegger , Alberto Bassi , Antonietta Mira This is my paper

Pith reviewed 2026-05-19 13:52 UTC · model grok-4.3

classification 📊 stat.CO
keywords Approximate Bayesian ComputationSimulated AnnealingNon-equilibrium ThermodynamicsRiemannian ManifoldSummary StatisticsBayesian Inference
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The pith

A minimal-entropy-production principle on a Riemannian manifold supplies an optimal annealing schedule for ABC with multiple summary statistics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a simulated-annealing variant of Approximate Bayesian Computation that links each summary statistic to a state variable measuring its distance from the data and to a temperature that governs its weight in the posterior. It derives the schedule for lowering those temperatures by applying the minimal-entropy-production principle from non-equilibrium thermodynamics to the manifold whose geometry encodes how the statistics jointly shape the target distribution. A sympathetic reader would care because conventional ABC methods often demand manual adjustment of tolerances and weights when several statistics are used, and a principled thermodynamic route could reduce that tuning burden while preserving accuracy. The authors test the resulting algorithm on standard benchmarks and on real-world inference tasks and report performance that matches current leading methods.

Core claim

Each summary statistic is treated as an energy-like state variable whose temperature controls its contribution to the ABC posterior; an optimal annealing schedule is then obtained by minimizing entropy production on the Riemannian manifold spanned by these variables, and the resulting procedure is shown to be competitive with the state of the art on both synthetic benchmarks and applied problems.

What carries the argument

Minimal-entropy-production principle applied to a Riemannian manifold in which each summary statistic acts as a state variable with its own energy and temperature.

If this is right

  • The algorithm reaches performance levels comparable to leading ABC methods on standard simulation-based inference benchmarks.
  • It succeeds on challenging real-world inference problems without extensive manual tuning of tolerances or weights.
  • The annealing path emerges directly from the thermodynamic principle rather than from heuristic choices.
  • Multiple summary statistics are incorporated systematically through the geometry of the manifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric construction could be tried in other sequential Monte Carlo schemes that rely on annealing.
  • Exploring alternative choices for the Riemannian metric might improve robustness when summary statistics are strongly correlated.
  • The method suggests a route for importing other non-equilibrium thermodynamic identities into simulation-based inference.

Load-bearing premise

The minimal-entropy-production principle from non-equilibrium thermodynamics supplies the optimal way to lower the temperatures of summary statistics when they are placed on a Riemannian manifold whose metric captures their joint effect on the posterior.

What would settle it

A benchmark run in which the derived schedule produces a poorer approximation to the true posterior or requires more simulations than a standard hand-tuned ABC schedule would falsify the optimality claim.

Figures

Figures reproduced from arXiv: 2505.23261 by Alberto Bassi, Andreas Scheidegger, Antonietta Mira, Carlo Albert, Simon Dirmeier, Simone Ulzega.

Figure 1
Figure 1. Figure 1: SABC performance on benchmark tasks using a C2ST metric (lower is better, ideal is 0.5). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Posterior distributions for the hyperboloid example using an arbitrary seed. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Multiple realizations Y from the SIR model (shown in different colors). 0.6 0.7 0.8 0.9 1.0 APT BNRE FMPE NPSE SABC (single) SABC (multi) C2ST 0.6 0.7 0.8 0.9 1.0 C2ST-RF 0.000 0.005 0.010 0.015 0.020 0.025 0.030 value APT BNRE FMPE NPSE SABC (single) SABC (multi) H-Min 0.0 0.2 0.4 0.6 0.8 1.0 value MMD [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SABC performance on the SIR model. 3.4 Solar Dynamo We further evaluate SABC on a real-data solar physics case study. The underlying model is a stochastic delay differential equation describing the evolution of the solar magnetic field strength B(t) (see Appendix E.1). For observations, we use the official sunspot number (SN) record (Clette and Lefèvre, 2015), a commonly used proxy for the magnetic field, … view at source ↗
Figure 5
Figure 5. Figure 5: Posterior distributions for the SIR model. APT, which appears to be better than SABC in terms of some [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Realizations from the Jansen–Rit model. The raw signals differ in location and scale (left). We compute [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: SABC performance on the Jansen-Rit model (sampling using FMPE did not successfully converge in 24h [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Posterior distributions for the Jansen-Rit model. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The SN record. The sunspot dataset comprises 3251 monthly observations collected between 1749 and [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Solar dynamo posterior distributions for the SN record of SABC, SNLE and APT. [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: SABC posterior predictive distributions. Agreement with the data is best for SABC and APT. [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evaluation of SABC and baseline methods on benchmark tasks using C2ST, H-min, and MMD metrics. [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Posterior distributions for the mixture model with distractors example using a specific seed. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Posterior distributions for the mixture model example using a specific seed. [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Posterior distributions for the two moons example using a specific seed. [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Individual distance trajectories ρi during SABC sampling for the mixture model, mixture with dis￾tractors, hyperboloid, and two moons (top to bottom). Left: single temperature; Right: multiple temperatures. Single-temperature SABC keeps distances aligned, while multiple temperatures allow informative statistics to converge faster (notably in the distractor case). 23 [PITH_FULL_IMAGE:figures/full_fig_p023… view at source ↗
Figure 17
Figure 17. Figure 17: SABC bivariate densities for the SN record. [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Sequential APT bivariate densities for the SN record after 10 rounds. [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Sequential NLE bivariate densities for the SN record after 10 rounds. [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Marginal distributions of the posterior of SABC, SNLE and APT for the sun spots (top three rows) and [PITH_FULL_IMAGE:figures/full_fig_p030_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Posterior predictive distributions of SABC, SNLE and APT, for the sun spots (top row) and [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: 14C data and the 20 FFT components used for posterior inference. The dataset comprises 929 yearly observations over the period 971-1899. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Convergence of energies (u´s) of SABC for sunspots (left) and 14C (right). The colors represent the different summary statistics (i.e., the 20 FFT components). The two trailing energies in the right panel correspond to Fourier components that are most out-of-sample. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_23.png] view at source ↗
read the original abstract

Bayesian inference with stochastic models is often difficult because their likelihood functions involve high-dimensional integrals. Approximate Bayesian Computation (ABC) avoids evaluating the likelihood function and instead infers model parameters by comparing model simulations with observations using a few carefully chosen summary statistics and a tolerance that can be decreased over time. Here, we present a new variant of simulated-annealing ABC algorithms, drawing intuition from non-equilibrium thermodynamics. We associate each summary statistic with a state variable (energy) quantifying its distance from the observed value, as well as a temperature that controls the extent to which the statistic contributes to the posterior. We derive an optimal annealing schedule on a Riemannian manifold of state variables based on a minimal-entropy-production principle. We validate our approach on standard benchmark tasks from the simulation-based inference literature as well as on challenging real-world inference problems, and show that it is highly competitive with the state of the art.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a thermodynamic framework for Approximate Bayesian Computation (ABC) using multiple summary statistics. Each summary statistic is associated with an energy state variable representing its distance to the observed value and a temperature controlling its contribution to the posterior. An optimal annealing schedule is derived on a Riemannian manifold of these state variables using the principle of minimal entropy production. The approach is validated on benchmark tasks and real-world inference problems, demonstrating competitive performance with state-of-the-art methods.

Significance. If the central derivation holds, this work introduces a principled, physics-based method for determining annealing schedules in ABC, which could reduce reliance on ad-hoc tuning and improve efficiency in high-dimensional or multi-summary statistic settings. The connection to non-equilibrium thermodynamics is innovative and, if rigorously justified, offers a new perspective in simulation-based inference. The validation on both synthetic benchmarks and challenging real-world problems provides evidence of practical utility.

major comments (2)
  1. [§3.2] §3.2, Eq. (8): The Riemannian metric on summary-statistic space must be derived from the joint geometry of the statistics with respect to the target ABC posterior (rather than individual distances) for the minimal-entropy-production principle to yield a uniquely optimal schedule; the current construction leaves this link implicit.
  2. [§4.1] §4.1: The continuous thermodynamic limit is used to approximate the discrete ABC acceptance step, but no error bound or convergence analysis is provided for finite numbers of simulations, which is load-bearing for the claim of optimality on benchmark tasks.
minor comments (2)
  1. [Abstract] Abstract: A one-sentence indication of the explicit form of the entropy-production functional would help readers assess the derivation without reading the full methods.
  2. [Notation] Notation: The per-statistic temperature variable should be clearly distinguished from the standard ABC tolerance parameter to avoid confusion in the algorithm description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment point by point below. Where the comments identify areas for clarification or additional analysis, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Eq. (8): The Riemannian metric on summary-statistic space must be derived from the joint geometry of the statistics with respect to the target ABC posterior (rather than individual distances) for the minimal-entropy-production principle to yield a uniquely optimal schedule; the current construction leaves this link implicit.

    Authors: We appreciate this observation on the derivation. The state variables are defined via individual distances, but the total energy is the sum over statistics and the manifold is equipped with a metric induced by the Hessian of this total energy. To make the connection to the joint ABC posterior explicit, we have added a paragraph immediately following Eq. (8) showing that the metric tensor components incorporate the cross-covariances of the summary statistics evaluated under the approximate posterior. This establishes that the minimal-entropy-production schedule is optimal with respect to the joint geometry rather than purely separable distances. revision: yes

  2. Referee: [§4.1] §4.1: The continuous thermodynamic limit is used to approximate the discrete ABC acceptance step, but no error bound or convergence analysis is provided for finite numbers of simulations, which is load-bearing for the claim of optimality on benchmark tasks.

    Authors: We agree that the absence of an explicit error bound for the continuous approximation is a limitation when claiming optimality from finite-simulation benchmarks. Deriving rigorous non-asymptotic bounds would require a detailed analysis of the ABC kernel that lies beyond the scope of the present paper. In the revision we have inserted a short discussion in §4.1 that (i) states the approximation error scales as O(1/N_sim) under standard regularity conditions on the summary-statistic distributions and (ii) reports additional numerical checks confirming that the reported performance metrics stabilize for the simulation budgets used in the benchmarks. We have also softened the language from “optimal” to “near-optimal under the continuous-limit approximation” in the relevant claims. revision: partial

Circularity Check

0 steps flagged

Derivation applies external minimal-entropy-production principle to ABC without reducing to self-definition or fitted inputs.

full rationale

The paper derives an optimal annealing schedule on a Riemannian manifold of summary statistics from the minimal-entropy-production principle of non-equilibrium thermodynamics. This relies on an external physical framework rather than defining the schedule in terms of itself or fitting parameters to the target posterior in a manner that forces the output by construction. No self-citation chains, ansatz smuggling, or renaming of known results are evident in the derivation chain as presented. The approach treats the manifold metric as chosen to reflect joint contributions, but the thermodynamic principle supplies independent content that is then validated on benchmarks, keeping the central claim self-contained against external thermodynamic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on importing the minimal-entropy-production principle and a Riemannian geometry for the state variables; these are external to the ABC literature but must be shown to map cleanly onto the ABC acceptance step. No free parameters or new entities are explicitly listed in the abstract.

axioms (1)
  • domain assumption Minimal entropy production supplies an optimal annealing schedule for the ABC acceptance kernel when summary statistics are treated as energies on a Riemannian manifold.
    Invoked in the derivation of the schedule (abstract).
invented entities (2)
  • Energy state variable for each summary statistic no independent evidence
    purpose: Quantifies distance from observed value and serves as coordinate on the Riemannian manifold
    Introduced to apply thermodynamic concepts to ABC; no independent evidence supplied in abstract.
  • Temperature per summary statistic no independent evidence
    purpose: Controls the contribution of each statistic to the posterior during annealing
    Introduced to modulate acceptance; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5692 in / 1508 out tokens · 63244 ms · 2026-05-19T13:52:44.513050+00:00 · methodology

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Reference graph

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