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Spectra: Exact Component Transport for Test-Time Prior Adaptation in Simulation-Based Inference

Xin Zhao, Nico Scherf, Robert Trampel, Kerrin J. Pine, Nikolaus Weiskopf

arXiv:2610.08021v1cs.LGstat.ML

TL;DR

Amortized SBI reuses posterior models across observations, but the models remain tied to the training prior despite scientific needs to revise prior assumptions. Spectra adapts a frozen diffusion-SBI model by exact score transport for structured prior changes, without additional simulation or training. Across six benchmarks, it remains accurate under strong shifts while reducing online sampling time relative to strengthened PriorGuide.

  • Problem

    Amortized SBI reuses inference across observations, but scientific analyses may need revised or alternative priors beyond the one chosen for training.

    Source: §1 · p. 2

  • Method

    Spectra adapts a frozen diffusion-SBI model by transporting its score exactly for structured prior changes, including positive mixtures of transportable factors.

    Source: §1 · pp. 2–3

  • Results

    Across six SBI benchmarks, Spectra remains accurate under strong prior shifts and mixture priors, with 2.4–3.2× faster steady-state online sampling than strengthened PriorGuide.

    Source: §4 · p. 6, §4.4 · p. 9

  • Takeaways & Limitations

    A pretrained diffusion-SBI model can be reused for revised priors and prior-sensitivity analyses without repeating expensive simulation or training.

    Source: §5 · p. 9

    Exactness and low online cost depend on structure in the prior change; adaptation is limited to the training-prior support, and mixture representations add observation-specific weight-estimation setup cost.

    Source: §5 · p. 9

Abstract

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Simulation-based inference (SBI) has become a powerful approach to Bayesian inference in complex scientific models whose likelihoods are difficult or impossible to evaluate. Amortized SBI learns reusable inference models from simulated data, enabling rapid posterior inference for new observations, and modern generative models have made these models increasingly expressive. However, this reuse is limited to the prior distribution chosen during training, whereas scientific analyses often need revised priors as knowledge accumulates or alternative assumptions are tested. We introduce Spectra, a test-time adaptation method for diffusion-based SBI. Spectra uses an exact score-transport identity to obtain the adapted score from a frozen diffusion model in closed form for structured prior changes, without additional simulation or training. Across six SBI benchmarks, Spectra achieves accurate adaptation under strong prior shifts at low online sampling cost. This enables pretrained SBI models to incorporate updated prior information at test time.

1 INTRODUCTION

Amortized SBI reuses trained inference models across observations, but their posteriors remain tied to the training prior even when scientific analyses need to test revised assumptions. Spectra adapts frozen diffusion-SBI models through exact score transport for structured prior changes, without additional simulation or training.

1 INTRODUCTION

  • Scientific analyses may need multiple plausible priors as knowledge accumulates or researchers test alternative assumptions, but amortized SBI is trained under a chosen prior.Examples include population-informed priors for gravitational-wave events and prior-sensitivity analyses in cognitive science.Source: §1 · p. 2
  • Computing each revised-prior posterior separately or retraining forfeits amortization, while importance sampling can be inefficient when the target concentrates in poorly covered regions.Training across prior families or using in-context examples still constrains flexibility or requires suitable simulated context.Source: §1 · p. 2
  • Spectra obtains the revised-prior score from a transformed query to a frozen diffusion model; isotropic exponential–quadratic ratio factors admit exact transport, and positive mixtures extend it to richer changes.It leaves the simulator and trained model unchanged.Source: §1 · pp. 2–3
  • Across six SBI benchmarks, Spectra stays accurate under strong prior shifts and mixture priors, with its largest accuracy gains on nonlinear tasks and 2.4–3.2× faster online sampling than strengthened PriorGuide.Spectra remains comparable to PriorGuide on OUP and Gaussian Linear controls.Source: §1 · pp. 2–3

2 BACKGROUND

SBI learns posterior inference from simulated parameter–observation pairs when likelihood evaluation is impractical. Diffusion-based amortized SBI trains a conditional score model that can be reused for new observations, while retaining the training prior.

2.1 SIMULATION-BASED INFERENCE

  • SBI learns a reusable posterior model from simulated parameter–observation pairs, avoiding direct likelihood evaluation when that likelihood is intractable or too costly.Amortized inference applies the trained model to new observations without repeating simulation or training.Source: §2.1 · p. 3

2.2 DIFFUSION MODELS

  • Diffusion models learn to reverse gradual Gaussian corruption, and their score—the gradient of the noisy log density—drives reverse-time denoising.The VE formulation uses increasing noise standard deviation σ(t), with σ(0) = 0.Source: §2.2 · pp. 3–4

2.3 DIFFUSION-BASED AMORTIZED SBI

  • In diffusion-based SBI, the parameter vector is the clean variable and the observation conditions the denoising model; its frozen learned score produces approximate base-posterior samples.The score still corresponds to the training prior πtr, although the model can be reused for new observations.Source: §2.3 · p. 4

3 SPECTRA: EXACT COMPONENT TRANSPORT

Spectra adapts a base posterior to a revised prior by transporting its noisy score through structured prior-ratio factors. Exact single-factor transport has a characterized scope, while positive mixtures enable richer changes with component weights estimated for each observation.

3.1 PRIOR ADAPTATION AS SCORE CORRECTION

  • The target noisy score equals the base score plus a correction involving the conditional expectation of the prior ratio given the noisy state, which the frozen model does not directly provide.PriorGuide approximates that conditional distribution as Gaussian; Spectra instead uses transformed queries for structured ratios.Source: §3.1 · p. 4

3.2 EXACT SINGLE-FACTOR TRANSPORT

  • For an isotropic exponential–quadratic prior-ratio factor, the target score is exact from one base-score query at a transformed state and noise level, under stated regularity and Dτ > 0.The base posterior may have arbitrary geometry; isotropy constrains the factor, not the posterior.Source: §3.2 · pp. 4–5

3.3 CHARACTERIZATION OF EXACT TRANSPORT

  • Within affine state transforms and scalar Gaussian variances, isotropic exponential–quadratic factors are the complete class of positive factors admitting exact kernel transport.The characterization also specifies the corresponding affine transform and variance, and conversely such factors transport when Dτ > 0.Source: §3.3 · p. 5

3.4 COMBINING POSTERIOR COMPONENTS

  • A positive weighted sum of transportable factors yields a mixture of posterior components, whose observation-dependent weights are determined by component normalizers.The coefficients of the prior-ratio representation are fixed independently of the observation.Source: §3.4 · p. 5

3.5 SAMPLING ONE COMPONENT PER TRAJECTORY

  • Spectra selects one component using its posterior mixture weight for each output trajectory, then uses that component’s transported score throughout reverse diffusion.This needs one base-score query per reverse step, independent of component count, instead of recombining component responsibilities at every step.Source: §3.5 · p. 6

3.6 ESTIMATING COMPONENT WEIGHTS

  • For multiple components, Spectra estimates observation-specific normalizers using transported component trajectories as importance-sampling proposals, since base-chain endpoints may have poor overlap.The finite-chain path likelihood ratio is tractable from known transition densities under common path support; estimates are an upfront calculation for each observation.Source: §3.6 · p. 6

4 EXPERIMENTS

Across six SBI benchmarks, Spectra adapts frozen diffusion models under single-factor and mixture prior shifts without retraining or additional simulator calls. It improves posterior agreement most clearly on nonlinear tasks and reduces steady-state sampling time, while results vary with posterior geometry and mixture complexity.

4.1 EXPERIMENTAL SETUP

  • The evaluation covers six tasks, mild and strong isotropic-Gaussian target-prior shifts, and two-component mixtures; baselines include PriorGuide, Base, and SIR.The study uses frozen Simformer models; C2ST measures joint agreement and MMTV measures average marginal discrepancy.Source: §4 · p. 6, §4.1 · pp. 6–7

4.3 MIXTURE PRIOR SHIFTS

  • Across 180 prior–observation pairs, Spectra improves reference-posterior agreement over PriorGuide on all six mixture-shift tasks, with clearest gains on Two Moons and SLCP.For 162 pairs one component accounts for more than 95% of posterior mass; the advantage persists in the 18 pairs where both components retain at least 5%.Source: Table 1 · p. 7, §4.3 · p. 8

4.2 SINGLE-FACTOR PRIOR ADAPTATION

  • Under strong single-factor shifts, Spectra agrees better with reference posteriors than PriorGuide on Two Moons, SLCP, and BCI, while results are similar on OUP and Gaussian Linear controls.The advantage grows as the target prior narrows on Two Moons but stays essentially flat on OUP when other factors are fixed.Source: Table 1 · p. 7, §4.2 · p. 7
  • Representative posterior plots show Spectra following target geometry closely while PriorGuide has off-target mass on nonlinear examples; SIR becomes inefficient when Base overlap deteriorates.The plots use 1,000 samples per method for one fixed observation, backbone, and seed per task; SLCP and GL-20D are shown as two-dimensional projections.Source: Figure 3 · p. 8, §4.2 · p. 8

4.4 ACCURACY AND COMPUTATIONAL COST

  • Spectra reduces steady-state online sampling time by 2.4–3.2× across five timing tasks, even against PriorGuide implemented with a vector–Jacobian product.Timing covers reverse sampling for 1,000 posterior samples after compilation and excludes one-time setup costs.Source: Figure 4 · p. 9, §4.4 · p. 9
  • With mixtures, sampling time remains unchanged from K = 1 to K = 64 on Two Moons, while observation-specific path-space weight estimation grows linearly with K.Weight estimation can dominate initial cost for a new observation, but additional samples for that observation remain inexpensive.Source: §4.4 · p. 9

5 DISCUSSION

Spectra reuses a pretrained diffusion-SBI model when prior assumptions change, provided the prior update has suitable structure. Its exactness concerns score transport, not the complete numerical sampler, and richer changes require structured representations and may add setup costs.

5 DISCUSSION

  • Structured prior changes let Spectra propagate revised assumptions through a frozen model, avoiding repeated simulation or training while supporting prior-sensitivity analyses.Composition extends this approach to richer prior changes while retaining one frozen-score evaluation per step.Source: §5 · pp. 9–10
  • Exact transport is restricted by prior-change structure and training-prior support; anisotropic or correlated Gaussian changes require structured representations such as positive mixtures.Multi-component representations add observation-specific weight estimation and setup cost.Source: §5 · p. 9
  • The transported score is exact under the theory, but implemented sampling still inherits learned-score, terminal-initialization, and finite-integration errors.The conclusion is therefore scoped to suitable prior updates rather than unrestricted prior flexibility.Source: §5 · pp. 9–10

AI USE DISCLOSURE

The included material records the authors’ AI-use practices, outlines the appendix, reviews related work, and states notation and regularity assumptions. The assumptions include target-prior support within the training-prior support and distinguish exact score identities from approximate implemented sampling.

AI USE DISCLOSURE

  • AI tools assisted with brainstorming, derivations and proofs, implementation, experimental analysis, and manuscript preparation under author direction.The authors state they determined the scientific and methodological decisions, validated results, and take responsibility for the work.Source: AI USE DISCLOSURE · p. 10

APPENDIX

  • The appendix collects related work, notation, derivations, proofs, error bounds, normalizer estimation, correlated-prior approximations, and experimental protocols and results.Appendices J–N cover protocols and full results.Source: APPENDIX · p. 15

A EXTENDED RELATED WORK

  • Related methods adapt frozen models using prior ratios, train target-specific corrections, train across prior families, or impose new priors through likelihood-based SBI; Spectra instead keeps a single-prior posterior model frozen.The related-work discussion also distinguishes proposal changes during sequential simulation and robustness objectives on unlabeled observations.Source: A EXTENDED RELATED WORK · p. 15

B NOTATION AND ASSUMPTIONS

  • The target prior is assumed absolutely continuous with respect to the training prior, and exact score transport does not by itself guarantee exact posterior sampling.The implementation uses a learned score, finite solver, and approximate terminal distribution; regularity assumptions include finite nonzero normalizers and differentiable Gaussian convolutions.Source: B NOTATION AND ASSUMPTIONS · pp. 15–16

C DERIVATION OF THE NOISY PRIOR-REWEIGHTING IDENTITY

The derivation relates the score under a reweighted posterior to the base-posterior score plus a correction. It expresses that correction using an expectation over the clean parameter conditional on the noisy state and observation.

C DERIVATION OF THE NOISY PRIOR-REWEIGHTING IDENTITY

  • The derivation begins with the noisy posterior under the training prior and applies the same forward diffusion to the new posterior.The supplied passages state these steps but do not provide a complete displayed expression for either distribution.Source: C DERIVATION OF THE NOISY… · p. 16
  • Bayes’ rule gives the reverse conditional under the base posterior, which is substituted into the reweighted expression.The derivation then uses that the normalizer does not depend on the noisy state.Source: C DERIVATION OF THE NOISY… · pp. 16–17
  • The adapted noisy score equals the base score plus the gradient of the log expected prior ratio under the clean-parameter conditional.The expectation is conditioned on the noisy state and observation.Source: C DERIVATION OF THE NOISY… · p. 17

D.1 PROOF OF PROPOSITION 1

The appendix establishes exact transport for isotropic exponential–quadratic prior-ratio factors and characterizes when an affine Gaussian-kernel mapping is possible. It extends transport to positive mixtures, with static component sampling preserving the target mixture.

D.1 PROOF OF PROPOSITION 1

  • The transported score is exact without a Gaussian assumption on the base posterior or a derivative of its score.The proof obtains the score by transforming the query state and noise level.Source: §D.1 · p. 18, §E.1 · p. 19

D.2 LEARNED-SCORE ERROR AND ADMISSIBLE NOISE LEVELS

  • For κ < 0, every sampler noise variance must satisfy τ < −1/κ, and transformed noise levels must remain within the learned model’s supported range.A multiplier below one does not guarantee smaller learned-score error because the transformed query may have different base-score error.Source: §D.2 · p. 18

D.3 SPECIAL CASES AND INTERPRETATION

  • A linear exponential tilt shifts the score query by τa and adds correction a; isotropic Gaussian localization moves the query toward its center and lowers its noise variance.The linear tilt leaves the noise level unchanged, whereas the localization variance is τλ/(τ + λ).Source: §D.3 · p. 18

E FORMAL CHARACTERIZATION OF EXACT TRANSPORT

  • An affine state map with scalar isotropic variance gives an exact kernel identity only for positive isotropic exponential–quadratic factors.The characterization holds at a single fixed noise level; across a schedule, the factor coefficients remain fixed while the kernel transform changes.Source: §E.1 · p. 19, §E.3 · p. 20

F.1 NORMALIZING THE POSITIVE SUM

  • For a positive sum of factors, the target posterior is a mixture whose component weights depend on the observation through the factor normalizers.Those static weights persist under Gaussian smoothing, while noisy-state responsibilities vary with the noisy state.Source: §F.1 · p. 20, §F.2 · p. 21

F.3 CORRECTNESS OF SAMPLING THE COMPONENT ONCE

  • Drawing one component label and keeping it fixed through reverse diffusion recovers the target mixture, using one base-score query per reverse step.Each component’s transported score comes from a transformed base-score query.Source: §F.3 · p. 21

F.4 DERIVING THE MARGINAL MIXTURE SCORE

  • The full-mixture score requires noisy-state responsibilities and normalized base-density values unavailable from local score queries, whereas static component sampling avoids that recombination.The responsibilities generally require density values at different transformed states and noise levels.Source: §F.4 · p. 22

G.1 MIXTURE ERROR BOUND

These appendices distinguish prior-ratio, weight-estimation, and component-sampling errors and describe normalizer estimation and correlated-Gaussian approximations. Their diagnostics have specific overlap scopes and do not by themselves establish posterior accuracy.

G.2 BINARY COMPONENT-WEIGHT SENSITIVITY

  • A given absolute component-weight error matters most when component distributions are well separated and has little effect when they overlap strongly.For small log-odds error, the weight itself is most sensitive near α = 1/2.Source: §G.2 · p. 25

G.3 APPROXIMATE-RATIO POSTERIOR TV

  • TV(qr,⋆, q˜r,⋆) ≤ ε/Zr, where ε is the expected absolute prior-ratio approximation error under the observation-specific base posterior.An error measured only under the training prior does not by itself control every observation-specific target posterior.Source: §G.3 · p. 25

H NORMALIZER-ESTIMATION DETAILS

  • The path-space normalizer estimator uses change of measure on reverse trajectories, with a separate transition count from the output sampler.Its proposal path law may differ from the path law used for final posterior sampling.Source: H NORMALIZER-ESTIMATION DETAILS · p. 25, §H.1 · p. 26

H.1 FINITE-CHAIN PATH-SPACE IDENTITY

  • Path-space ESS diagnoses overlap between the finite-chain proposal and endpoint-tilted path law, not discrepancy from the exact Bayesian posterior.The proposal path law for normalizer estimation may also differ from the final sampler’s path law.Source: §H.1 · p. 27

H.2 DIRECT ESTIMATION AND OVERLAP

  • Direct estimation reuses a bank of base samples across factors, but its ESS cannot detect important regions the entire bank failed to sample.A low ESS indicates that only a small part of the bank dominates the estimate.Source: §H.2 · p. 27

H.3 COMPLETE SPECTRA INFERENCE PROCEDURE

  • The complete sampler estimates component weights, draws one label per output, then runs reverse sampling with that component’s transported score.The procedure checks factor form and admissible transformed noise levels before sampling.Source: §H.3 · p. 27

I.1 CONTINUOUS MIXTURE AND POSITIVE QUADRATURE

I.2 FROM PRIOR QUADRATURE TO PRIOR-RATIO COEFFICIENTS

  • The quadrature component count grows exponentially with dimension, so this construction is used only in low-dimensional demonstrations.An arbitrary training prior need not preserve the isotropic factor family after division by that prior.Source: §I.1 · p. 28, §I.2 · p. 29

J.1 TASKS AND PRIOR SHIFTS

The appendix specifies the six-task single-factor and two-component mixture benchmarks, evaluation procedures, sampling settings, and reference construction. Controlled comparisons find Spectra's strongest gains on nonlinear tasks under strong shifts, while overlap diagnostics show severe SIR degeneration on Gaussian Linear tasks.

J.1 TASKS AND PRIOR SHIFTS

  • The six tasks use Gaussian or uniform training priors, with mild and strong target widths set to 0.5 and 0.2 times the training-prior standard deviation.The ten-observation single-factor settings use separately drawn centers and observations at each shift; the mixture benchmark uses two strong-shift Gaussian components and ten mixture priors per task.Source: §J.1 · p. 29
  • Each task uses three trained backbones, and the reported hierarchy averages sampling seeds, then backbones within observations, then across observations.Single-factor results use ten observations; mixture results use 30 prior–observation pairs, with three observations per mixture prior.Source: §J.1 · p. 29, §J.2 · p. 30

J.2 METRICS AND STATISTICAL AGGREGATION

  • C2ST measures joint agreement with ideal value 0.5, while MMTV averages marginal total variation distances with ideal value zero.RMSE to the generating parameter is supplementary rather than a distributional accuracy metric because multimodal posteriors need not concentrate around that parameter.Source: §J.2 · p. 30

J.4 POSTERIOR REFERENCES AND SUPPORT CHECKS

  • Posterior references are analytic for Gaussian Linear tasks and numerically constructed for the others; on uniform-support tasks, support-restricted references preserve all method orderings in the tested comparisons.Across the support-mismatch check, all 12 C2ST/MMTV method orderings remain unchanged and the largest aggregate-metric change is 0.0013.Source: §J.4 · pp. 31–32

J.5 PRIORGUIDE PROTOCOLS

  • PriorGuide receives the exact analytic prior ratio in controlled comparisons, and its VJP implementation changes only the guidance-update implementation.At (25, 8), the VJP and benchmarked implementations differ by at most 5×10−4 in absolute mean C2ST and below 10−4 in MMTV on the paired validation tasks.Source: §J.5 · p. 32

J.6 PRIORGUIDE-FULLCOV

  • PG-FullCov replaces PriorGuide's scalar covariance with a moment-consistent full covariance, but retains a Gaussian approximation to the reverse conditional.It uses a Tweedie moment-projection approach; unlike the scalar-covariance comparison, this baseline requires the full denoiser Jacobian.Source: §J.6 · pp. 32–33

K.1 CONTROLLED COMPARISON WITHOUT CORRECTOR UPDATES

  • Under strong shifts, Spectra improves on PG-FullCov on Two Moons and SLCP, is similar on OUP and BCI, while PG-FullCov has a small C2ST advantage on the Gaussian Linear controls.Under mild shifts, Spectra is better on Two Moons, while PG-FullCov's C2ST advantage on BCI and both Gaussian Linear tasks is at most 0.008.Source: §K.1 · p. 33

K.2 PAIRED COMPARISONS AND OVERLAP DIAGNOSTICS

  • Under strong shifts, median SIR ESS fractions are approximately 45%, 6.0%, and 1.8% on Two Moons, SLCP, and BCI, versus 8 × 10−5 and 5 × 10−5 on GL-10D and GL-20D.The Gaussian Linear fractions correspond to roughly two effective samples or fewer from the full bank; the overlap collapse is consistent with severe SIR degeneration.Source: §K.2 · p. 33

K.3 PRACTICAL SAMPLING CONFIGURATIONS

The practical sampler configuration yields similar OUP accuracy for PriorGuide and Spectra, while held-out results favor Spectra on Two Moons and SLCP. A fixed-center sweep finds that the gap grows with narrower priors on Two Moons but remains small and trendless on OUP.

K.3 PRACTICAL SAMPLING CONFIGURATIONS

  • Under the practical (25, 8) configuration, PriorGuide and Spectra have similar OUP accuracy, with strong-shift C2ST at 0.505 for both.PriorGuide's C2ST changes from 0.521 at (100, 0) to 0.505 at (25, 8), while Spectra changes from 0.506 to 0.505.Source: §K.3 · p. 35
  • On the seven held-out observations, Spectra has lower C2ST and MMTV than both PriorGuide and PG-FullCov on Two Moons and SLCP, with 95% paired intervals excluding zero.The held-out subset belongs to the ten-observation practical benchmark, so this is a configuration-selection check rather than an independent replication.Source: §K.3 · p. 35, Table 6 · p. 35

K.4 FIXED-CENTER SHIFT-SEVERITY SWEEP

  • On Two Moons, the gap grows systematically as the target prior narrows, with positive paired differences for at least nine of ten observations at every tested width.On OUP, all paired differences remain below 0.002 in magnitude and show no systematic trend; the sweep fixes the prior center, observations, backbones, and sampler.Source: §K.4 · p. 35

L.1 SYSTEMATIC MIXTURE BENCHMARK

Across the two-component benchmark, Spectra-Direct and Spectra-PS are nearly identical on four tasks, while Spectra-PS is better on Gaussian Linear tasks. Diagnostics show mixture accuracy depends on component-weight estimation, and both variants remain accurate as mixtures grow in a small composition study.

L.1 SYSTEMATIC MIXTURE BENCHMARK

  • At the practical settings, Spectra-Direct and Spectra-PS give nearly identical results on Two Moons, SLCP, OUP, and BCI, while Spectra-PS has lower mean C2ST and MMTV on the Gaussian Linear tasks.The larger Gaussian Linear difference is on GL-10D; the controlled comparison at (100, 0) is reported separately.Source: §L.1 · p. 35, Table 8 · p. 36

L.2 COMPONENT ALLOCATION AND WEIGHT ESTIMATION

  • On the 18 pairs with at least 5% posterior mass in each component, Spectra-PS has lower mean C2ST and MMTV than Spectra-Direct but remains short of Spectra-Ref.Spectra-PS mean C2ST is 0.529 versus 0.519 for Spectra-Ref, and its MMTV is higher on all 18 pairs, averaging 0.072 versus 0.049.Source: §L.2 · pp. 36–37, Table 9 · p. 37
  • For the 30 remaining BCI pairs, none has at least 5% posterior mass in both components; Spectra-Direct and Spectra-PS select the reference-dominant component in all 90 pair–backbone combinations.The largest minority weight is 7 × 10−4, so this regime says little about estimating very small minority weights.Source: §L.2 · p. 37

L.3 COMPOSITION BEYOND TWO COMPONENTS

  • Both Spectra variants remain accurate as the number of mixture components increases in the Two Moons composition study.At K = 16, the effective number of components ranges from 2.5 to 4.5, five components exceed 5% posterior mass for every observation, and path-space weights differ from reference weights by at most 1.6 × 10−2.Source: §L.3 · p. 37, Table 11 · p. 38

M COMPUTATIONAL COST

  • Across 22 timed configurations, PriorGuide (VJP) is 2.2–3.2× slower than Spectra, compared with 3.5–19.7× for the benchmarked implementation.The timings are specific to the implementations and hardware used in the study.Source: M COMPUTATIONAL COST · p. 38

M.1 SINGLE-FACTOR ACCURACY AND ONLINE COST

  • The single-factor timing comparison covers five tasks at the main-benchmark sampler settings, with both benchmarked and VJP PriorGuide implementations shown.Figure 6 adds PG-FullCov to the MMTV–time trade-off and uses development observations for configuration selection.Source: §M.1 · p. 38, Figure 5 · p. 38, Figure 6 · p. 39

M.2 MULTI-COMPONENT FIRST-USE COST

Spectra's component sampler is inexpensive, but mixture-weight estimation can dominate first-use cost, and path-space weight estimation is the most expensive first-use method in the reported tasks. Online Spectra sampling changes little as component count grows in the Two Moons scaling study.

M.2 MULTI-COMPONENT FIRST-USE COST

  • On SLCP, Spectra-PS takes 0.11 seconds to draw 1,000 samples once weights are available, but 18.09 seconds including weight estimation, versus 3.28 seconds for PriorGuide including precomputation.Spectra-PS remains the most expensive first-use method on all five tasks, even when PriorGuide's online stage uses VJP.Source: §M.2 · p. 39, Figure 7 · p. 39
  • Estimated normalizers can be reused for the same observation and component set when only the prior mixture coefficients change.The reported first-use measurements cover a single adaptation and do not quantify amortized cost over a sequence of prior changes.Source: §M.2 · p. 39

M.3 SCALING WITH THE NUMBER OF COMPONENTS

  • Spectra's online sampling time varies by less than 0.2% from K = 1 to K = 64 because each trajectory follows only the transported score of its selected component.PriorGuide combines all K components at each guidance step, but its time changes by less than 1% on this two-dimensional task; the time ratio stays between 3.81 and 3.85.Source: §M.3 · p. 40, Table 12 · p. 40

N ADAPTATION BEYOND THE EXACT SINGLE-FACTOR CLASS

  • The out-of-class study applies positive-mixture approximations to anisotropic and rotated correlated Gaussian targets on Two Moons.These target priors are outside the exactly transportable single-factor class.Source: N ADAPTATION BEYOND THE EXACT… · p. 40

N.1 CONTROLLED ANISOTROPY BOUNDARY

The anisotropy sweep compares Spectra with alternative methods while tracking approximation and component-weight errors; a separate correlated Two Moons example tests quadrature with 64 factors but remains low-dimensional.

N.1 CONTROLLED ANISOTROPY BOUNDARY

  • The sweep fixes the target center and determinant while increasing anisotropy; the construction uses 52 factors at γ=2 and 104 at γ=4.Source: §N.1 · p. 40
  • Approximation error grows with anisotropy: prior TV is 0.00149 at γ=2 and 0.03312 at γ=4, while mean posterior TV at γ=4 is 0.02360 and 0.02889 across orientations.Source: §N.1 · p. 40
  • On one observation in the + orientation, Spectra-PS worsens C2ST versus Spectra-Ref by 0.417 at γ=2 and 0.405 at γ=4; the other nine differ by at most 0.003.Source: §N.1 · pp. 40–41
  • Across all four anisotropic settings, PriorGuide is worse than Spectra-Ref in both metrics on every observation and worse than Spectra-PS on at least nine of ten.Source: §N.1 · p. 40
  • SIR is worse than Spectra-Ref in C2ST on every observation, but has lower MMTV on five to eight of ten observations.Source: §N.1 · pp. 40–41

N.2 ROTATED CORRELATED GAUSSIAN EXAMPLE

  • Across three observations and three backbones, path-space weights are within 0.026–0.092 L1 of reference weights; replacing them changes C2ST by +0.0013 ± 0.0020 and MMTV by +0.0035 ± 0.0018.Source: §N.2 · p. 41
  • The K=64 correlated-prior result is a low-dimensional demonstration, not evidence of reliable path-space weight estimation at large K.Source: §N.2 · p. 41

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