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

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

cs.LGOct 6, 2026

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

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

Organizations: Department of Neurophysics, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany · International Max Planck Research School on Cognitive NeuroImaging, Leipzig, Germany · Methods and Development Group Neural Data Science and Statistical Computing, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany · Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI), Dresden/Leipzig, Germany · Felix Bloch Institute for Solid State Physics, Faculty of Physics and Earth System Sciences, Leipzig University, Leipzig, Germany · Functional Imaging Laboratory, Department of Imaging Neuroscience, UCL Queen Square Institute of Neurology, University College London, London, UK

Abstract

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.

Figures & tables

Figure 1: Prior shift and test-time adaptation on the simple likelihood complex posterior (SLCP) task. (a) A concentrated target prior replaces the broad training prior. (b) The frozen model represents the base posterior under the training prior, while the target prior concentrates posterior mass on two of its four modes. (c) Both PriorGuide and Spectra recover the target modes. However, PriorGuide retains visible mass outside the target contours, whereas Spectra follows the target geometry more closely. The posterior panels show the

(\theta_{3},\theta_{4})

projection. Full results across tasks and observations are reported in Table 1 and Figure 3 .

		Single-factor, mild		Single-factor, strong		Mixture
Task	Method	C2ST	MMTV	C2ST	MMTV	C2ST	MMTV
Two Moons	Base	0.568 (0.036)	0.219 (0.087)	0.691 (0.095)	0.426 (0.192)	0.736 (0.074)	0.510 (0.147)
	SIR	0.525 (0.005)	0.075 (0.011)	0.544 (0.018)	0.070 (0.012)	0.543 (0.018)	0.062 (0.009)
	PriorGuide	0.528 (0.012)	0.123 (0.046)	0.556 (0.034)	0.165 (0.086)	0.561 (0.036)	0.176 (0.086)
	Spectra	0.521 (0.005)	0.093 (0.016)	0.532 (0.015)	0.102 (0.030)	0.521 (0.009)	0.073 (0.014)
SLCP	Base	0.765 (0.057)	0.255 (0.041)	0.917 (0.019)	0.466 (0.072)	0.946 (0.029)	0.498 (0.075)

Table 1: Posterior accuracy under single-factor and mixture prior shifts. Mean (standard deviation) across ten observations or 30 prior–observation pairs. Ideal values are

0.5

for C2ST and

0

for MMTV. Bold marks the mean closest to the ideal value within each task and setting.

Figure 2: Accuracy gap as the target prior narrows. Mean paired PriorGuide–Spectra difference with 95% paired

t

confidence intervals across ten observations. All experimental factors except target-prior width are held fixed. Positive gaps indicate better performance by Spectra.

Figure 3: Posterior geometry under strong single-factor prior shifts. Reference contours enclose 50% and 90% posterior mass. Markers show 1,000 samples from each method for one fixed observation, backbone, and sampling seed per task. Larger SIR markers indicate repeated resampling of the same Base sample. SLCP and GL-20D show two-dimensional projections. Table 1 reports full-dimensional accuracy.

Figure 4: Single-factor accuracy versus online sampling time under strong prior shifts. PriorGuide (VJP) uses a vector–Jacobian product for the same guidance update and preserves PriorGuide’s accuracy (Appendix M ). Timing covers the reverse sampling stage for 1,000 posterior samples after compilation. One-time setup costs are excluded.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$\pi_{\rm tr},\pi_{\rm new},r$	training prior, target prior, and their ratio
$p,q$	base and target posteriors
$q^{\phi}$	posterior induced by a single prior-ratio factor
$t$ , $\tau=\sigma(t)^{2}$	diffusion time and corresponding noise variance
$p_{\tau},s_{p},\widehat{s}_{p}$	Gaussian-smoothed base posterior, its exact score, and frozen score estimate
$\phi_{k},b_{k},q_{k}$	prior-ratio factor, its coefficient, and posterior component

Appendix

Table 2: Principal notation. Symbols specific to error analysis and sampling protocols are defined where they are used.

Task	Reference	Validation
GL-10D/20D	closed-form posterior	analytic
Two Moons	exact-likelihood quadrature on a $4096\times 4096$ grid	grid refinement changes the log normalizer by at most $2\times 10^{-4}$ nats
OUP	exact-likelihood quadrature on a $4096\times 4096$ grid	grid refinement changes the log normalizer by less than $10^{-12}$ nats; target mass on the grid boundary at most $4\times 10^{-10}$ (both shifts)
SLCP	tempered SMC, 40,000 particles, four independent runs	cross-run C2ST at most $0.509$ (strong) and $0.521$ (mild)
BCI	tempered SMC, 8,000 particles, four independent runs; Gauss–Hermite likelihood of order 301	cross-run C2ST at most $0.509$ (both shifts and mixtures); quadrature-order check

Appendix

Table 3: Posterior references and validation. Reported validation statistics are worst-case values over the evaluated single-factor observations; BCI also includes the mixture pairs. Cross-run C2ST compares independent reference runs of the same posterior (ideal value

0.5

		Mild			Strong
Task	Method	C2ST	MMTV	RMSE	C2ST	MMTV	RMSE
Two Moons	Base	0.568 (0.036)	0.219 (0.087)	0.397 (0.197)	0.691 (0.095)	0.426 (0.192)	0.174 (0.106)
	SIR	0.525 (0.005)	0.075 (0.011)	0.340 (0.147)	0.544 (0.018)	0.070 (0.012)	0.100 (0.061)
	PriorGuide (exact ratio)	0.522 (0.010)	0.105 (0.040)	0.371 (0.173)	0.554 (0.032)	0.167 (0.085)	0.136 (0.063)
	PG-FullCov	0.528 (0.015)	0.123 (0.057)	0.379 (0.180)	0.552 (0.038)	0.159 (0.093)	0.142 (0.070)
	Spectra	0.512 (0.006)	0.072 (0.007)	0.346 (0.151)	0.519 (0.009)	0.068 (0.012)	0.102 (0.059)

Appendix

Table 4: Single-factor comparison at

(N,L)=(100,0)

. Mean ( standard deviation ) across ten observations, after averaging sampling seeds within each backbone and then averaging the three backbones within each observation. C2ST is ideal at

0.5

and MMTV at zero. Bold indicates the displayed mean closest to the ideal value within each task and shift, including rounded ties. RMSE to the generating parameter is a supplementary diagnostic and is not bolded.

Shift	Task	PriorGuide $-$ Spectra	PG-FullCov $-$ Spectra
Mild	Two Moons	$+0.0097\ [+0.0018,+0.0177]$	$+0.0151\ [+0.0039,+0.0264]$
	SLCP	$+0.0019\ [-0.0070,+0.0108]$	$+0.0031\ [-0.0026,+0.0087]$
	OUP	$+0.0014\ [-0.0025,+0.0052]$	$+0.0011\ [-0.0011,+0.0034]$
	BCI	$+0.0009\ [-0.0022,+0.0040]$	$-0.0026\ [-0.0047,-0.0004]$
	GL-10D	$+0.0113\ [+0.0044,+0.0182]$	$-0.0049\ [-0.0074,-0.0024]$
	GL-20D	$+0.0309\ [+0.0130,+0.0488]$	$-0.0073\ [-0.0109,-0.0038]$

Appendix

Table 5: Paired C2ST differences at

(N,L)=(100,0)

. Entries are baseline minus Spectra, with 95% paired Student-

t

confidence intervals across ten observations. Positive values correspond to lower C2ST for Spectra.

	Two Moons		SLCP		GL-20D
Method	C2ST	MMTV	C2ST	MMTV	C2ST	MMTV
PriorGuide (exact ratio)	0.560 (0.039)	0.174 (0.099)	0.617 (0.029)	0.115 (0.030)	0.531 (0.025)	0.035 (0.004)
PG-FullCov	0.567 (0.047)	0.192 (0.118)	0.620 (0.030)	0.117 (0.032)	0.530 (0.023)	0.035 (0.004)
Spectra	0.534 (0.017)	0.103 (0.033)	0.586 (0.038)	0.100 (0.041)	0.531 (0.024)	0.035 (0.004)

Appendix

Table 6: Held-out single-factor comparison with Langevin corrections. Strong shifts at

(N,L)=(25,8)

. Mean ( standard deviation ) across seven held-out observations after averaging three sampling seeds within each of three backbones. C2ST is ideal at

0.5

and MMTV at zero. Bold indicates the displayed mean closest to the ideal value within each task, including rounded ties.

Task	Width	Difference [95% CI]	Favor	Difference [95% CI]	Favor
		C2ST		MMTV
Two Moons	0.50	$+$ 0.0075 [ $+$ 0.0006, $+$ 0.0144]	10/10	$+$ 0.0297 [ $+$ 0.0062, $+$ 0.0532]	10/10
	0.40	$+$ 0.0191 [ $+$ 0.0032, $+$ 0.0350]	9/10	$+$ 0.0596 [ $+$ 0.0203, $+$ 0.0990]	10/10
	0.30	$+$ 0.0434 [ $+$ 0.0179, $+$ 0.0688]	10/10	$+$ 0.1140 [ $+$ 0.0532, $+$ 0.1748]	10/10
	0.25	$+$ 0.0571 [ $+$ 0.0280, $+$ 0.0861]	9/10	$+$ 0.1486 [ $+$ 0.0788, $+$ 0.2185]	10/10
	0.20	$+$ 0.0727 [ $+$ 0.0414, $+$ 0.1039]	10/10	$+$ 0.1828 [ $+$ 0.1065, $+$ 0.2590]	10/10

Appendix

Table 7: Fixed-center shift-severity sweep. Entries are mean paired differences (PriorGuide minus Spectra) across ten observations, with 95% paired Student-

t

confidence intervals after averaging over three sampling seeds and three backbones within each observation. Width is the target-prior marginal standard deviation relative to the training prior. Positive differences favor Spectra. Favor gives the number of observations with a positive paired difference.

C2ST
					Spectra
Task	Base	SIR	PriorGuide	PG-FullCov	Direct	PS	Ref
Two Moons	0.736 (0.074)	0.543 (0.018)	0.552 (0.035)	0.538 (0.021)	0.514 (0.007)	0.515 (0.007)	0.515 (0.008)
SLCP	0.946 (0.029)	0.796 (0.086)	0.611 (0.043)	0.587 (0.042)	0.554 (0.040)	0.555 (0.040)	0.554 (0.040)
OUP	0.703 (0.063)	0.529 (0.011)	0.517 (0.009)	0.507 (0.008)	0.506 (0.006)	0.507 (0.006)	0.506 (0.006)
GL-10D	0.996 (0.003)	0.999 (0.001)	0.598 (0.035)	0.539 (0.025)	0.570 (0.094)	0.536 (0.023)	0.531 (0.025)

Appendix

Table 8: Mixture posterior inference across 180 prior–observation pairs. Mean ( standard deviation ) across 30 pairs per task, after averaging sampling seeds within each backbone and then the three backbones within each pair. All output samplers use

(N,L)=(100,0)

. PriorGuide receives the exact prior ratio. Direct, PS, and Ref are the Spectra variants of Appendix J.4 . Ref uses reference component weights and is included only as a diagnostic. C2ST is ideal at

0.5

and MMTV at zero. Bold indicates the displayed mean closest to the ideal value among methods that do not use reference weights, including rounded ties.

	$(100,0)$		$(25,8)$
Method	C2ST	MMTV	C2ST	MMTV
PriorGuide	0.568 (0.043)	0.112 (0.038)	0.549 (0.038)	0.100 (0.044)
Spectra-Direct	0.591 (0.123)	0.158 (0.171)	0.595 (0.122)	0.161 (0.170)
Spectra-PS	0.529 (0.022)	0.072 (0.023)	0.534 (0.026)	0.074 (0.024)
Spectra-Ref	0.519 (0.019)	0.049 (0.012)	–	–

Appendix

Table 9: Mixtures with substantial mass in both posterior components. Mean ( standard deviation ) across the 18 pairs for which each component carries at least 5% reference posterior mass. Weight estimates are fixed across output configurations. Spectra-Ref uses reference component weights and is shown only as a diagnostic at

(N,L)=(100,0)

	C2ST			MMTV
$K$	Spectra-PS	Spectra-Ref	PriorGuide	Spectra-PS	Spectra-Ref	PriorGuide
2	0.517 (0.008)	0.518 (0.009)	0.573 (0.062)	0.064 (0.016)	0.070 (0.011)	0.211 (0.152)
4	0.521 (0.013)	0.517 (0.008)	0.586 (0.049)	0.074 (0.024)	0.075 (0.011)	0.260 (0.134)
8	0.520 (0.010)	0.519 (0.009)	0.555 (0.049)	0.062 (0.004)	0.069 (0.004)	0.178 (0.141)
16	0.507 (0.001)	0.509 (0.003)	0.545 (0.038)	0.063 (0.011)	0.070 (0.002)	0.170 (0.115)

Appendix

Table 10: Accuracy as the number of prior components increases on Two Moons. Mean ( standard deviation ) across three observations at

(N,L)=(25,8)

after averaging three sampling seeds within each backbone. Spectra-PS uses one backbone, while Spectra-Ref and PriorGuide use three backbones. All methods are evaluated against the support-matched reference.

$K$	Effective components	Components with at least 5%	Effective components	Largest weight error
	Reference weights		Path-space estimate
2	1.00 / 1.02 / 1.00	1 / 1 / 1	1.00 / 1.02 / 1.00	$3.4\times 10^{-4}$
4	1.19 / 1.36 / 1.12	2 / 2 / 2	1.19 / 1.35 / 1.12	$2.2\times 10^{-3}$
8	1.21 / 3.09 / 2.18	2 / 3 / 2	1.21 / 3.07 / 2.18	$1.6\times 10^{-2}$
16	2.50 / 4.54 / 4.10	5 / 5 / 5	2.50 / 4.52 / 4.15	$8.4\times 10^{-3}$

Appendix

Table 11: Component allocation and path-space weight accuracy. For each of the three observations, the table reports the effective number of components

1/\sum_{k}\alpha_{k}^{2}

and the number carrying at least 5% of the posterior mass. The final column gives the largest absolute difference between path-space and reference component weights across all components and observations.

Figure 5: Online sampling time across the five timed tasks. Each task uses the sampler configuration of the main benchmark and the timing protocol above. Labels show the runtime ratio of PriorGuide (VJP) to Spectra.

Figure 6: Single-factor MMTV versus online sampling time. The figure adds PG-FullCov to the methods shown in Figure 4 . Filled circles mark

(25,8)

, squares mark

(100,0)

, and hollow circles mark the remaining sampler configurations. Error bars are observation-level standard errors. Both axes are logarithmic.

Figure 7: Cost of the first 1,000 mixture posterior samples. Stacks separate precomputation, component-weight estimation, and online sampling for PriorGuide, PG-FullCov, Spectra-Direct, and Spectra-PS. Labels give the total displayed cost. Panels use different linear time scales.

$K$	1	2	4	8	16	32	64
Spectra	0.1166	0.1165	0.1165	0.1165	0.1165	0.1165	0.1164
PriorGuide	0.4464	0.4440	0.4442	0.4446	0.4456	0.4466	0.4482

Appendix

Table 12: Online sampling time as the number of prior components increases on Two Moons. Median seconds for 1,000 posterior samples at

(N,L)=(25,8)

under the timing protocol of Appendix M .

C2ST
$\gamma$	Orient.	$K$	SIR	PriorGuide	Spectra-Ref	Spectra-PS
1	iso	1	0.613 (0.069)	0.621 (0.055)	0.548 (0.016)	0.548 (0.016)
2	$+$	52	0.589 (0.090)	0.621 (0.092)	0.528 (0.013)	0.570 (0.143) $\dagger$
2	$-$	52	0.578 (0.044)	0.604 (0.063)	0.528 (0.012)	0.535 (0.027)
4	$+$	104	0.599 (0.070)	0.609 (0.084)	0.531 (0.023)	0.571 (0.149) $\dagger$
4	$-$	104	0.590 (0.073)	0.600 (0.084)	0.527 (0.018)	0.530 (0.021)

Appendix

Table 13: Anisotropy sweep on Two Moons. Mean (standard deviation) over ten observations, scored against support-matched references for the exact target. Each observation’s value averages three backbones, and for PriorGuide and Spectra three sampling seeds per backbone. Sampling uses

(N,L)=(25,8)

, except

(100,0)

for SIR. At

\gamma=1

K=1

and Spectra-Ref and Spectra-PS coincide.

Observation	Approx. TV	PriorGuide	PG-FullCov	Spectra-Ref	Spectra-PS
1	0.00306	$0.535\pm 0.003$	$0.527\pm 0.007$	$0.507\pm 0.006$	$0.513\pm 0.006$
2	0.00456	$0.519\pm 0.011$	$0.519\pm 0.013$	$0.518\pm 0.008$	$0.518\pm 0.006$
3	0.00285	$0.549\pm 0.008$	$0.526\pm 0.009$	$0.530\pm 0.009$	$0.529\pm 0.010$

Appendix

Table 14: Correlated-prior approximation on Two Moons. Approx. TV is the posterior total variation distance induced by the

K=64

finite quadrature, evaluated on a

4096\times 4096

grid. C2ST is mean

\pm

standard deviation over three sampling seeds. Approx. TV quantifies posterior error from the finite prior approximation, while C2ST evaluates the sampled posterior against the exact target. PriorGuide and PG-FullCov use the exact ratio; Spectra-Ref and Spectra-PS use the quadrature approximation with reference and path-space component weights, respectively.

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