{"id":"72b0e5ae-b0b3-4a9d-a18b-ed3cc157d837","arxiv_id":"2608.00841","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Using the expected posterior as a common design measure for both optimization and uncertainty reduction yields robust finite-budget posterior approximations.","lead":"SALE is a new Gaussian process active learning method that uses the surrogate's expected posterior to choose where to evaluate expensive likelihoods, combining optimization and calibration in one loop. If it works as claimed, it lowers the cost of Bayesian inference in settings where each likelihood evaluation is costly.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 17's TV rate is proved for exact EP refresh samples (Assumption 44), but SALE uses ssMCMC with L=10; Online Appendix H Remark 51 disclaims the resulting errors, so the implemented-proxy guarantee is not established.","rationale":"The paper's central contribution has two parts: a robust empirical design procedure and a theoretical guarantee for the implemented UR proxy. The theory for the proxy, Theorem 17, is the piece that would make the strongest claim rigorous. That theorem rests on Assumption 44, which requires exact draws from the expected posterior at refresh times. The implementation deliberately uses an approximate ssMCMC sample, and the paper itself, in Online Appendix H Remark 51, disclaims control of the finite-inner-chain error. This is precisely the gap the reader identified. I do not see a more load-bearing flaw: the perturbation/stability results for AO, the allocation-proxy arguments, and the empirical comparisons are internally coherent, and the empirical evidence is substantial. However, because the headline claim explicitly mentions an expected-TV rate for the implemented proxy, the gap between Assumption 44 and the L=10 ssMCMC implementation is a genuine correctness risk, not merely a stylistic caveat. The empirical TV numbers are also computed on π_{μ_T} rather than π̄_T, which weakens the direct link between the experiments and the theorem. These are fixable in revision by adding a perturbation result that bounds the effect of finite-L approximate EP samples, or by restating the theoretical claim as applying to an idealized SALE variant. Since the reader already renders a CONDITIONAL verdict with the same concern, my read does not move the verdict: UNCHANGED is appropriate.","tokens_in":57938,"tokens_out":5234,"duration_ms":68195,"concrete_test":"Run a small benchmark (e.g., 4D Quartic or 6D Funnel) with fixed budgets and compare SALE using the default L=10 inner MH steps against variants with L=100 and L=1000 (and, if feasible, a high-accuracy reference refresh sample obtained from long runs of many independent GP-path chains), holding all other settings fixed. If final dTV(π_{μ_T}, π_X), query selections, and PSRF diagnostics change materially with L, the finite-chain approximation is consequential and Theorem 17's assumption is not benign in practice. If all L variants are statistically indistinguishable, the practical gap is minor, though the theory would still need an explicit perturbation statement. To test the theorem more directly, implement the exact-refresh version on a small 1D/2D target where perfect sampling from π̄_{r_i} is feasible and compare the estimated E[TV(π̄_T, π_X)] trajectories with the predicted polynomia","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical support for the implemented UR proxy is Theorem 17, whose proof assumes Assumption 44: at each refresh time r_i, the sample points satisfy θ_{r_i,s} | H_{r_i} ∼ π̄_{r_i}. In the implementation, these samples are produced by Algorithm 1 with S=10,000 outer draws and only L=10 inner Metropolis–Hastings steps per surrogate path (Appendix A.1), and Algorithm 3 line 5 explicitly says the sample 'approximately target[s] π̄_0'. Online Appendix H, Remark 51, states that the practical finite inner chain and noncompact latent state 'introduce errors not controlled by this result.' Thus the actual SALE run in all experiments does not satisfy the hypothesis of Theorem 17. If the L=10 inner chain has not mixed, the candidate set U_{r(t)} used in the UR rule (Eq. 8) and the state-tempered weights in Eq. (7) are drawn from the wrong measure, so the claimed budget-dependent expected-TV rate for the implemented proxy need not hold for the real algorithm. This does not disprove the empirical robustness claim, but it does mean that 'establishes an expected total-variation rate for the implemented proxy' overstates what is proved. Moreover, the empirical comparisons report dTV(π_{μ_T}, π_X), not TV(π̄_T, π_X), so they do not directly validate the theorem's output.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes SALE, an active-learning framework for Bayesian inference when log-posterior evaluations are expensive. A GP surrogate is built sequentially, and the expected posterior (EP) induced by normalised GP sample paths is used as the common design measure. SALE alternates between Bayesian optimisation via an annealed objective (AO) and calibration via EP-weighted uncertainty reduction (UR), with a state-dependent allocation rule. The theoretical core consists of: a perturbation-stability bound for AO and a Bayesian regret bound (Theorem 10); an explicit finite-budget expected-TV bound for an ideal EP-weighted UR rule (Theorem 14); and an expected-TV rate for the implemented UR proxy (Theorem 17). Numerical experiments on analytic benchmarks, simulated likelihoods, and two applications report that SALE reduces TV relative to baselines across settings and avoids several severe failures. Code, data, and an R package are archived.","tokens_in":58304,"tokens_out":5420,"duration_ms":62264,"significance":"If the theoretical results and numerics hold, the paper makes a substantial contribution: it is, to the authors' knowledge, the first to give an explicit budget-dependent expected-TV bound for a sequential posterior-weighted UR design, and it introduces the EP as a common design measure for both localisation and calibration. The empirical protocol is careful: paired initial designs, many replicates, failure counts, ablations, and archived code strengthen the robustness claim. The paper is also honest about many limitations. However, the central theoretical support for the implemented UR proxy is not matched to the implementation: Theorem 17 requires exact EP samples at refresh times, while Algorithm 1 provides only approximate samples with L=10 inner MH steps, and Online Appendix H explicitly says the resulting errors are not controlled. This gap tempers the claim that the paper 'establishes an expected total-variation rate for the implemented proxy' and should be addressed before publication.","major_comments":[{"comment":"Theorem 17 relies on Assumption 44, under which every refresh-time sample point θ_{r_i,s} is an exact draw from π̄_{r_i}. The implementation generates these samples by ssMCMC with S=10,000 outer draws and only L=10 inner Metropolis–Hastings steps per surrogate path (Appendix A.1), and Algorithm 3 line 5 states the sample 'approximately target[s] π̄_0'. Online Appendix H, Remark 51, explicitly states that the practical finite inner chain and noncompact latent state 'introduce errors not controlled by this result.' Thus the implemented UR proxy does not satisfy the hypothesis of Theorem 17. The abstract and contribution list should not claim an established expected-TV rate for the implemented proxy unless a controlled approximate-sampling theorem is added, the claim is scaled back to the ideal proxy, or numerical evidence shows the L=10 approximation error is negligible.","section":"Section 4.4, Theorem 17; Algorithm 1, Algorithm 3, Appendix A.1, Online Appendix H Remark 51"},{"comment":"The empirical evaluation reports dTV(π_{μ_T}, π_X), i.e. total variation between the plug-in posterior based on the GP mean and the true posterior. The paper acknowledges this in §5.1.2. Theorems 14 and 17, however, bound TV(π̄_T, π_X), the expected posterior. The reported figures therefore validate SALE's posterior accuracy as a method but do not directly validate the theorem's output. The paper should separate these claims and, ideally, include a small-scale check of TV(π̄_T, π_X) at final budgets, or explicitly state that the empirical curves are only indirect support for the EP-based theory.","section":"Section 5.1.2, Figures 2–4 and 6"},{"comment":"Theorem 10 assumes exact AO sampling over the full domain Ω, while the SALE implementation uses ESA with a finite-dimensional RFF–Matheron representation and restricts BO to the current search region Ω_{r(t)}. The paper notes this after the theorem, but the same structure—an idealised result separated from an approximate implementation—also affects the AO contribution. A remark or supplementary result quantifying the restriction and the ESA approximation would make the claim 'characterise the annealed objective's stability–bias trade-off' more precise. This is less central than the Theorem 17 gap, but should be clarified.","section":"Section 4.3, Theorem 10 and Algorithm 2"}],"minor_comments":[{"comment":"The levelwise ESA convergence result assumes an exact exchange auxiliary draw, while Appendix A.3 uses L_u=20–30 MH updates for the auxiliary draw. The main text should state this discrepancy where Algorithm 2 is introduced.","section":"Section 4.3/Online Appendix H, Assumption 47(E3)"},{"comment":"The theorem states the Matérn rate for ν>1/2, but Assumption 1(A5) requires C^1 sample paths, which for Matérn kernels usually requires ν>1. Please harmonise the smoothness condition or explain the convention.","section":"Theorem 17"},{"comment":"For GP-UCB+, the text says one iteration comprises two evaluations. If the x-axis uses 'Evaluation budget T', this is fine; if it refers to iterations, please relabel to avoid ambiguity. The caption and text should be consistent.","section":"Section 5.1, Figure 2"},{"comment":"The notation π_{τ,t} is introduced as the AO law over Ω and then 'unless otherwise stated' is used for the restricted law. Please define both versions explicitly at first use to avoid confusion in the analysis and algorithms.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: This is a strong paper with a genuinely useful empirical framework, a careful benchmark protocol, and a serious theoretical attempt. The main issue is not circularity or lack of novelty; it is the mismatch between Theorem 17's assumptions and the implemented algorithm, which is acknowledged in Online Appendix H but not reflected in the abstract and contributions. If the authors either close this gap with a controlled approximate-sampling result or explicitly scale back the claim, the paper would be publishable. I do not see grounds for rejection, but the current overstatement of the implemented-proxy theorem needs major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this is the first framework I've seen that uses the expected posterior as the common design measure for both localisation and calibration, and that decision pays off in practice. The paper is genuinely new: EP-guided search region, state-dependent BO/UR allocation, annealed objective with an explicit stability–optimization trade-off, and finite-budget TV bounds for posterior-weighted UR. The empirical section is unusually thorough and honest—analytic benchmarks across d=4,6,8, two simulated likelihoods, two real expensive-likelihood applications, an ablation, archived code/data, and a clear statement that SALE is not best on every target. Its robustness claim—consistently lower TV and fewer catastrophic failures—holds up in the reported results. I would use this as a reference design for expensive posterior evaluation.\n\nNow the soft spot, and it is substantial though not fatal. The headline theoretical rates are for a cleaner process than the one actually run. Theorem 17's explicit expected-TV rate for the implemented proxy requires Assumption 44: refresh samples are exact draws from the expected posterior. The code generates them with ssMCMC using only L=10 inner MH steps, and Online Appendix H Remark 51 explicitly says the resulting errors are not controlled. So the sentence 'establishes an expected TV rate for the implemented proxy' overstates what is proved. The same pattern appears for the BO branch: Theorem 10 assumes exact AO sampling, while implementation uses RFF-ESA with MH-approximated auxiliary draws. The empirical comparisons use TV(π_μ,T,π_X), not TV(π̄_T,π_X), so they don't directly validate the theorem's output either. That doesn't kill the central contribution, but the authors need to either prove the rate under a verifiable mixing condition, or reword the claim and report a diagnostic on the refresh samples.\n\nWho gets value: method developers in surrogate-based Bayesian inference, especially those with expensive simulators; practitioners less so, because the full system is complex and many constants need care. The paper deserves a serious referee. The idea is clearly not incremental, the code and data are shipped, and the gap I describe is fixable in revision. I'd ask for a precise statement of what is proved for the implemented algorithm and a mixing check on the ssMCMC sample at refresh times before accepting.","headline":"A genuinely new framework for surrogate-based Bayesian inference whose main gap is that the flagship theory is proved for an idealized version, not the exact implemented algorithm.","tokens_in":58765,"tokens_out":3514,"would_cite":true,"duration_ms":41529,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62L05","60G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"SALE makes the expected posterior—the average posterior over Gaussian-process sample paths—the single measure that chooses every likelihood evaluation, and proves explicit finite-budget bounds on the resulting posterior error.","keywords":["active learning","Bayesian inference","expensive likelihood","Gaussian process surrogate","expected posterior","total variation","Bayesian optimization","uncertainty reduction"],"falsifier":"Run SALE on one fixed problem while varying the inner sampler length L (e.g. 10, 50, 300) at fixed evaluation budget; if final TV error does not improve as L grows, or if L=10 samples give materially different query locations than near-exact EP draws, the implemented proxy is not operating in the regime the rate theorem assumes. A second check: compare the ideal UR rule (maximising Δ_t exactly) against the implemented proxy on a bimodal target where the surrogate starts poorly localised—the ablation predicts the proxy can beat the ideal rule at finite budget because the value-correction term h","tokens_in":57811,"feed_emoji":"🎯","tokens_out":7108,"duration_ms":74146,"temperature":0.7,"pith_summary":"When each likelihood evaluation requires a simulator, a Bellman solve, or a latent-model fit, the budget for Bayesian inference can be only a few hundred evaluations. Existing surrogate methods either optimize for the mode or reduce uncertainty over posterior-relevant regions, but with no common measure coordinating the two, they can waste evaluations and fail badly on hard geometries. SALE's proposal is to let one object—the expected posterior, the average of the normalized posterior laws induced by GP sample paths—drive every decision: it defines the search region, weights the uncertainty-reduction step, and supplies the candidate evaluations. The paper proves that this architecture carries budget-dependent guarantees: an annealed objective for optimization with a stability–bias trade-off quantified by a Bayesian regret bound, an explicit expected total-variation bound for the ideal EP-weighted uncertainty-reduction rule, and a rate for the implemented proxy with the same polynomial budget order. If correct, the practical payoff is a reliable posterior approximation from a small, fixed evaluation budget, with a finite-budget error certificate rather than only asymptotic guarantees.","feed_headline":"One posterior average steers every expensive likelihood query","feed_subtitle":"SALE balances locating and calibrating, cutting posterior error on every test problem it ran.","key_machinery":"The load-bearing object is the expected posterior π̄_t(dθ) = E_{f_t∼P_t}[π(dθ|f_t)], the average posterior induced by normalized GP sample paths; it minimizes conditional expected forward-KL loss and is a measure-valued martingale, so it drifts only when data arrive. Three mechanisms hang off it. (1) The variance identity E[V^{(θ)}_{t+1} | H_t] = V_t − Δ_t(θ): the one-step expected decrease of the π̄_t-weighted integrated variance equals an explicit GP-covariance functional, and TV error is bounded by (V_t/2)^{1/2}, so cutting V_t is literally cutting posterior error. (2) The annealed objective π_{τ,t}, whose query law is stable under W_{1,∞}-perturbations of the surrogate path law at positi","core_discovery":"The central claim is that posterior-guided sequential design need not choose between localizing and calibrating. SALE treats the expected posterior π̄_t—the pointwise average of normalized posterior densities induced by draws from the GP posterior over the log-posterior—as a common law that shapes every decision: samples from π̄_t carve out the search region for Bayesian optimization, and π̄_t weights the uncertainty-reduction criterion that refines the surrogate inside that region. Two theoretical results carry the argument: a Bayesian regret bound for the annealed objective (AO), which interpolates between π̄_t at τ=1 and Thompson sampling as τ→0 and is Lipschitz-stable in the surrogate pa","pith_inferences":["The architecture suggests a general principle: the object used to report uncertainty should be the same object used to design evaluations. If that principle is right, posterior-weighted acquisition criteria that use a different weighting measure (such as the plug-in mean) leave free information on the table.","The dimension-dependence of the AO stability mechanism—a narrow spurious spike must beat a broad mode by a height gap growing with the number of poorly resolved directions—predicts that Thompson-sampling surrogates will degrade relative to AO precisely on high-dimensional, scale-separated problems; the Rosenbrock and Funnel results already point that way.","The TV bound's dependence on the realized number of UR evaluations suggests a practical diagnostic: track the empirical UR fraction and warn when it falls below a threshold, since the theory shows the error bound degrades when P(N^UR_T < qT) is large. The paper does not itself propose this as a monitoring rule."],"forward_implications":["A practitioner can run a single sequential design under a fixed evaluation budget and receive a final surrogate posterior with a stated expected-TV error bound, rather than only an asymptotic guarantee.","Because UR provably receives a positive fraction of evaluations asymptotically, the method keeps calibrating after localization instead of degenerating into pure optimization.","The EP's forward-KL optimality means the final object is, among all H_T-measurable laws, the best conditional estimator of the posterior in expected forward-KL; the TV bounds attach a budget-dependent certificate to that object.","The same EP-driven design transfers across problem classes—analytic targets, simulator likelihoods, Bellman-solver likelihoods, and conditional-INLA likelihoods—since the GP surrogate, refresh schedule, and stopping rule are not problem-specific."],"supporting_citations":[{"why":"Supplies the expected posterior as an uncertainty-propagated inferential object and its forward-KL optimality, which Proposition 3 applies conditionally on H_t.","marker":"Roberts et al. (2026a)"},{"why":"Introduces the EP as an uncertainty-propagated posterior estimator whose properties SALE reuses as a design measure.","marker":"Reiser et al. (2025)"},{"why":"Provides the regularised maximal information gain γ_T(υ) that enters the annealed-objective Bayesian regret bound in Theorem 10.","marker":"Whitehouse et al. (2023)"},{"why":"Supplies the information-gain orders for SE and Matérn kernels that convert γ_T into the explicit rates in Theorems 10 and 17.","marker":"Iwazaki (2025)"},{"why":"Provides the sequential determinant identity used to bound the cumulative posterior-variance sum along the adaptive query sequence (Lemma 28).","marker":"Srinivas et al. (2012)"},{"why":"The exchange algorithm used by ESA to cancel the intractable normalising constant in the annealed-objective target.","marker":"Murray et al. (2006)"},{"why":"Matheron's representation, used with random Fourier features to draw the GP posterior sample paths that define the EP and feed ESA.","marker":"Wilson et al. (2021)"},{"why":"The excursion probability p*_t whose cheap local-spacing proxy drives SALE's BO–UR allocation rule.","marker":"Wilson (2024)"}],"fun_headline_variants":["One posterior mean directs all query choices","Expected posterior steers both search and calibration","A single posterior average guides every expensive likelihood query","Posterior mean powers adaptive active learning for Bayesian inference","SALE: one expected posterior for all sequential decisions"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The theory for the implemented uncertainty-reduction proxy assumes that the surrogate-posterior samples used at each refresh are exact draws from the expected posterior; the implementation instead draws them from a short stochastic surrogate Markov chain with only ten inner Metropolis updates per path, and the paper's own Online Appendix remarks that the resulting errors are not controlled by the convergence theorem.","fun_headline_variants_meta":{"raw":{"variants":["One posterior mean directs all query choices","Expected posterior steers both search and calibration","A single posterior average guides every expensive likelihood query","Posterior mean powers adaptive active learning for Bayesian inference","SALE: one expected posterior for all sequential decisions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000129,"raw_usage":{"total_tokens":953,"prompt_tokens":734,"completion_tokens":219,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":149}},"tokens_in":478,"tokens_out":219,"duration_ms":3449,"temperature":1.0,"reasoning_tokens":149,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T00:11:21.753920+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run SALE on one fixed problem while varying the inner sampler length L (e.g. 10, 50, 300) at fixed evaluation budget; if final TV error does not improve as L grows, or if L=10 samples give materially different query locations than near-exact EP draws, the implemented proxy is not operating in the regime the rate theorem assumes. A second check: compare the ideal UR rule (maximising Δ_t exactly) against the implemented proxy on a bimodal target where the surrogate starts poorly localised—the ablation predicts the proxy can beat the ideal rule at finite budget because the value-correction term h","supporting_citations":[],"review_version":1}