Sample-efficient evidence estimation of score based priors for model selection

Frederic Wang; Katherine L. Bouman

cs.LGFeb 24, 2026

Sample-efficient evidence estimation of score based priors for model selection

Authors: Frederic Wang, Katherine L. Bouman

Organizations: Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA

Abstract

The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ that specify the prior and then selecting the one with the highest value. Diffusion models are the state-of-the-art approach to solving inverse problems with a data-driven prior; however, directly computing the model evidence with respect to a diffusion prior is intractable. Furthermore, most existing model evidence estimators require either many pointwise evaluations of the unnormalized prior density or an accurate clean prior score. We propose DiME, an estimator of the model evidence of a diffusion prior by integrating over the time-marginals of posterior sampling methods. Our method leverages the large amount of intermediate samples naturally obtained during the reverse diffusion sampling process to obtain an accurate estimation of the model evidence using only a handful of posterior samples (e.g., 20). We also demonstrate how to implement our estimator in tandem with recent diffusion posterior sampling methods. Empirically, our estimator matches the model evidence when it can be computed analytically, and it is able to both select the correct diffusion model prior and diagnose prior misfit under different highly ill-conditioned, non-linear inverse problems, including a real-world black hole imaging problem.

Figures & tables

Figure 1: Visualization of the unbiased estimators $\Theta_{high},\Theta_{low}$ of the likelihood score $\nabla_{{\bm{x}}_{t}}\log p({\bm{y}}\mid{\bm{x}}_{t})$ described in Lemma 2 for both in-distribution (left) and out-of-distribution (right) measurements ${\bm{y}}$ . At large diffusion time steps, the high noise estimator (top) uses the distance between $\tilde{{\bm{x}}}_{0}$ and $\mathbb{E}[{\bm{x}}_{0}\mid{\bm{x}}_{t}]$ . At small diffusion time steps, the low noise estimator (bottom) uses the likelihood score at $\tilde{{\bm{x}}}_{0}$ . The estimator is larger (longer arrows) when ${\bm{y}}$ is out-of-distribution, as there greater KL divergence from the posterior to the prior; this results in DiME computing a lower evidence. Note the scaling factors $\frac{a_{t}}{\sigma_{t}^{2}}$ and $\bm{\Sigma}_{{\bm{x}}_{0}\mid{\bm{x}}_{t}}$ of the estimator are not shown.

	In-distribution ${\bm{x}}^{*}$		Out-of-distribution ${\bm{x}}^{*}$		Saddle point ${\bm{x}}^{*}$
	Estimate	Rel. Error ↓	Estimate	Rel. Error ↓	Estimate	Rel. Error ↓
Ground truth $\log p({\bm{y}})$	$-128.7\pm 0.0$	$0.0$	$-1680.2\pm 0$	$0.0$	$-251.3\pm 0.0$	$0.0$
Naive MC (1000 samples)	$-3283\pm 130$	$2451\%$	$-41289\pm 577$	$2357\%$	$-6029\pm 170$	$2299\%$
Naive MC (10000 samples)	$-3041\pm 113$	$2263\%$	$-40123\pm 487$	$2288\%$	$-5666\pm 144$	$2155\%$
Original DAPS heuristic	$-316.2\pm 79.5$	$146\%$	$-1735.4\pm 4.7$	$3.3\%$	$-269.6\pm 3.3$	$7.3\%$
TI	$-132.8\pm 1.3$	$3.2\%$	$-1773.8\pm 11.8$	$5.6\%$	$-254.2\pm 3.2$	$1.2\%$

Table 1: Evidence estimates on Mixture of Gaussians study with analytic

\log p({\bm{y}})

and ground truth

{\bm{x}}^{*}

in-distribution (top), out-of-distribution (middle), and at a saddle point (bottom). Mean and standard deviation of 50 trials displayed each with 20 sample paths. When the analytic prior score is available, DiME performs comparably to the baselines for all cases, while DiME-PnPDM performs well when the ground truth is roughly in-distribution. *Important: as DiME-PnPDM outputs an unnormalized estimate, we normalize via the analytic constant factor for ease of comparison.

Figure 2: Model evidence confusion matrix for Gaussian phase retrieval ( left ) and Fourier phase retrieval ( right ) for each (ground truth measurement, model) pair of MNIST digits. Our method selects the correct model for all cases. Posterior samples shown for dotted matrix entries below. For Gaussian phase retrieval, DiME estimates higher model likelihood for visually similar digits, such as 4 and 9. For Fourier phase retrieval, both translations and reflections are invariances as seen in the posterior samples, so DiME estimates a high likelihood of model 9 given a measurement of a 6.

Figure 3: Model evidence estimates on real M87* observations across 5 different priors using exact DAPS ( left ) and Gaussian approximation DAPS ( right ). Our method concludes that, of these prior models, GRMHD is the most likely model. The violin plots show the distribution of evidences from 20 different posterior-sample paths from the DAPS sampling process, and the mean is the overall evidence estimate. Gaussian approximation DAPS gives highly accurate evidence estimates with 7x less compute than exact DAPS, but with slightly higher variance.

Figure 4: M87* reconstructions using the 5 priors using exact DAPS ( left ) and Gaussian approximation DAPS ( middle ). While all posterior samples have a data fit of around reduced $\chi^{2}\approx 1$ , they have varying path evidence estimates (shown in the top left of each image). Right: Example unconditional samples from each prior.

Figure 5: Left: GRMHD model validation results on M87* observations by comparing to evidence of in-distribution measurements ${\bm{y}}$ . Our method shows that the evidence of M87* observations have a $z$ -score of about -0.81 compared to the evidence distribution of GRMHD measurements, indicating that M87* is statistically in-distribution of GRMHD. The evidence of simulated measurements of out-of-distribution images are also shown, demonstrating that measurements from out-of-distribution (OOD) images have OOD evidence. Right: Mean reconstruction and posterior samples of the OOD images with the highest and lowest distance to the GRMHD prior. The blurred ground truth image, or the maximum resolution that can be obtained from measurements, is also displayed.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Task	lr	Max steps
Mixture of Gaussians (in-distribution)	$5\times 10^{-4}$	$2{,}000$
Mixture of Gaussians (out-of-distribution)	$2\times 10^{-5}$	$10{,}000$
Mixture of Gaussians (saddle point)	$2\times 10^{-5}$	$10{,}000$
Fourier phase retrieval	$1\times 10^{-3}$	$1{,}000$
Gaussian phase retrieval	$5\times 10^{-4}$	$1{,}000$
M87* black hole imaging	$1\times 10^{-5}$	$1{,}000$

Appendix

Table 7

Figure 6: Model evidence confusion matrix using sequential Monte Carlo (SMC) for Gaussian phase retrieval ( left ) and Fourier phase retrieval ( right ) for each (ground truth measurement, model) pair of MNIST digits. When SMC uses the learned diffusion score $\nabla_{{\bm{x}}}\log p({\bm{x}}_{t})$ at $t=0.1$ as the prior score, it often fails to select the correct model.

	$\sigma_{\min}=0.1$			$\sigma_{\min}=0.05$
SMC schedule	GT 8 Model 6	GT 8 Model 7	GT 8 Model 8	GT 8 Model 6	GT 8 Model 7	GT 8 Model 8
Linear	$-182\pm 9$	$-212\pm 9$	$-210\pm 11$	$-279\pm 26$	$-352\pm 3$	$-426\pm 40$
Exponential	$-183\pm 7$	$-224\pm 5$	$-239\pm 6$	$-292\pm 20$	$-383\pm 9$	$-468\pm 12$
Sigmoidal	$-175\pm 5$	$-219\pm 13$	$-191\pm 2$	$-273\pm 15$	$-374\pm 9$	$-480\pm 17$
Adaptive 0.5	$-195\pm 31$	$-202\pm 23$	$-261\pm 1$	$-296\pm 52$	$-346\pm 16$	$-489\pm 33$
Adaptive 0.6	$-184\pm 6$	$-189\pm 7$	$-259\pm 4$	$-279\pm 45$	$-373\pm 17$	$-485\pm 26$

Appendix

Table 2: SMC model evidence estimates on Gaussian phase retrieval.

	$\sigma_{\min}=0.1$			$\sigma_{\min}=0.05$
SMC schedule	GT 8 Model 6	GT 8 Model 7	GT 8 Model 8	GT 8 Model 6	GT 8 Model 7	GT 8 Model 8
Linear	$-416\pm 11$	$-431\pm 4$	$-409\pm 1$	$-485\pm 25$	$-577\pm 2$	$-563\pm 2$
Exponential	$-422\pm 20$	$-427\pm 2$	$-467\pm 11$	$-527\pm 10$	$-582\pm 12$	$-585\pm 9$
Sigmoidal	$-415\pm 9$	$-451\pm 3$	$-411\pm 7$	$-520\pm 13$	$-580\pm 7$	$-575\pm 5$
Adaptive 0.5	$-409\pm 4$	$-409\pm 6$	$-460\pm 8$	$-524\pm 33$	$-563\pm 18$	$-567\pm 10$
Adaptive 0.6	$-431\pm 20$	$-437\pm 11$	$-458\pm 8$	$-510\pm 9$	$-566\pm 14$	$-562\pm 13$

Appendix

Table 3: SMC model evidence estimates on Fourier phase retrieval.

Bayesian Inference

Bayesian inference updates probability distributions over unknowns with evidence to quantify uncertainty and guide prediction and decisions.

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Optimizing Diffusion Priors in Image Reconstruction from a Single Observation

Frederic Wang, Katherine L. Bouman

Caltech, Pasadena CA 91125, USA

Bayesian Inverse Problems Image Reconstruction Diffusion Model Fine-Tuning Diffusion-Based Inverse Problems Diffusion Posterior Sampling

Open paper AskPDF arXiv

While diffusion priors generate high-quality posterior samples across many inverse problems, they are often trained on limited training sets or purely simulated data, thus inheriting the errors and biases of these underlying sources. Current approaches to finetuning diffusion models rely on a large number of observations with varying forward operators, which can be difficult to collect for many applications, and thus lead to overfitting when the measurement set is small. We propose a method for tuning a prior from only a single observation by combining existing diffusion priors into a single product-of-experts prior and identifying the exponents that maximize the Bayesian evidence. We validate our method on real-world inverse problems, including black hole imaging, where the true prior is unknown a priori, and image deblurring with text-conditioned priors. We find that the evidence is often maximized by priors that extend beyond those trained on a single dataset. By generalizing the prior through exponent weighting, our approach enables posterior sampling from both tempered and combined diffusion models, yielding more flexible priors that improve the trustworthiness of the resulting posterior image distribution.

Abstract

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