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Calibrated Test-Time Guidance for Bayesian Inference
Daniel Geyfman, Felix Draxler, Jan Groeneveld, Hyunsoo Lee, Theofanis Karaletsos, Stephan Mandt
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 7/20/2026, 10:49:12 AM
Summary
The paper identifies that common test-time guidance methods for diffusion models produce miscalibrated Bayesian posteriors due to biased estimators of the diffused likelihood and incorrect handling of guidance scales. It proposes Calibrated Bayesian Guidance (CBG), a framework using consistent estimators (Gradient-Based and Gradient-Free) to enable accurate sampling from the true Bayesian posterior, achieving state-of-the-art results in black hole image reconstruction.
Entities (8)
Relation Signals (7)
Calibrated Bayesian Guidance → solves → miscalibrated inference
confidence 95% · We propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior.
Calibrated Bayesian Guidance → uses → Gradient-Based Calibrated Bayesian Guidance
confidence 95% · We call Equation 16 Gradient-Based Calibrated Bayesian Guidance.
Calibrated Bayesian Guidance → uses → Gradient-Free Calibrated Bayesian Guidance
confidence 95% · We call Equation 20 Gradient-Free Calibrated Bayesian Guidance.
Calibrated Bayesian Guidance → outperforms → Existing Methods
confidence 92% · We significantly outperform previous methods on a set of Bayesian inference tasks
Calibrated Bayesian Guidance → appliedto → Black Hole Image Reconstruction
confidence 90% · set a new state-of-the-art PSNR in black hole image reconstruction.
Test-Time Guidance → causes → miscalibrated inference
confidence 90% · Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference.
Posterior Mean Approximation → istypeof → biased estimator
confidence 88% · We find that the posterior mean approximation ... fails ... This means that Equation 7 is not exact.
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Abstract
Abstract:Test-time guidance is a widely used mechanism for steering pretrained diffusion models toward outcomes specified by a reward function. Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference. In this work, we show that common test-time guidance methods do not recover the correct posterior distribution and identify the structural approximations responsible for this failure. We then propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior. We significantly outperform previous methods on a set of Bayesian inference tasks, and set a new state-of-the-art PSNR in black hole image reconstruction.
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- Source: https://arxiv.org/abs/2602.22428v2
- Canonical: https://arxiv.org/abs/2602.22428v2
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Calibrated Test-Time Guidance for Bayesian Inference Daniel Geyfman Felix Draxler Jan Groeneveld Hyunsoo Lee Theofanis Karaletsos Stephan Mandt Abstract Test-time guidance is a widely used mechanism for steering pretrained diffusion models toward outcomes specified by a reward function. Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference. In this work, we show that common test-time guidance methods do not recover the correct posterior distribution and identify the structural approximations responsible for this failure. We then propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior. We significantly outperform previous methods on a set of Bayesian inference tasks, and set a new state-of-the-art PSNR in black hole image reconstruction. We publish our code at https://github.com/mandt-lab/Calibrated-Guidance. Machine Learning, ICML 1 Introduction Diffusion models (Sohl-Dickstein et al., 2015; Song and Ermon, 2019; Ho et al., 2020; Song et al., 2021) and their flow matching variants (Liu et al., 2023; Lipman et al., 2023; Albergo and Vanden-Eijnden, 2023) have achieved tremendous success in sampling high-quality yet diverse images (Dhariwal and Nichol, 2021; Rombach et al., 2022) and videos (Yang et al., 2023; Ho et al., 2022) as well as scientific applications such as molecule generation (Hoogeboom et al., 2022) and protein-ligand docking (Corso et al., 2023). A core benefit of diffusion models is their ability to sample from tilted distributions at test time: Take a pretrained model and guide the generation towards a desired outcome as specified by a potentially nonlinear reward function (Chung et al., 2023). This enables applying pretrained diffusion models to inverse problems such as super-resolution, deblurring, denoising, style guidance, image editing, and others, without any additional training. This approach is motivated by the goal of sampling from the Bayesian posterior, proportional to the diffusion model prior and a likelihood function that specifies the task (often called reward). Figure 1: We present a test-time guidance scheme to sample from calibrated Bayesian posteriors. (Left) Our framework accurately samples the correct posterior (blue). Posterior mean (Equation 7), posterior Gaussian (Equation 8), and optimal control approximations (Equation 9) to the diffused likelihood p(y∣xt)p(y x_t) yield uncalibrated samples (orange). (Right) Our framework can correctly sample from tempered posteriors p(x∣y,γ)∝p(x)p(y∣x)γp(x y,γ) p(x)p(y x)^γ (blue). Rescaling the noisy gradient by γ leads to biased samples, even if the diffused likelihood p(y∣xt)p(y x_t) is accurately estimated (orange). In this context, we contribute: • We find that existing test-time guidance methods do not sample from the true Bayesian posterior (Section 4), see Figure 1. We demonstrate that they use biased estimators for the diffused likelihood (Theorems 4.1, 4.1 and 4.3), suggesting why they usually do not converge to the true posterior even with more compute. • We propose Calibrated Bayesian Guidance (CBG), a new consistent yet tractable guidance framework for sampling the correct diffusion posterior in Section 5. Our framework supports non-differentiable objectives, and we propose practical approximations for fast sampling. • Experimentally, our estimators accurately sample the Bayesian posterior on Bayesian inference and set a new state-of-the-art PSNR on a black-hole imaging task. Together, we illustrate and address an important gap in the literature to accurately sample from Bayesian posteriors based on pretrained diffusion model priors. 2 Background 2.1 Diffusion Models Diffusion models represent a data distribution p(x)p(x) by iteratively transforming samples from a latent distribution p(x1)p(x_1) (usually a standard normal) through a stochastic or ordinary differential equation parameterized by a neural network sθ(xt,t)s_θ(x_t,t) towards data samples of pθ(x)p_θ(x). The idea is to learn a neural network to reverse the addition of noise to the training data x∼p(x)x p(x): p(xt∣x)=(xt;atx,bt2I).p(x_t x)=N(x_t;a_tx,b_t^2I). (1) We adopt the convention that t=0t=0 corresponds to the data distribution (so a0=1a_0=1 and b0=0b_0=0), and t=1t=1 is the noisy latent (a1=0a_1=0 and b1=1b_1=1). We write x=x0x=x_0 for noise-free data samples. The ordinary differential equation for sampling from a diffusion model is given by (Karras et al., 2022): dxdt=a˙tatx−bt2a˙tat∇xtlogp(xt)⏟learned sθ(xt,t). dxdt= a_ta_tx-b_t^2 a_ta_t _x_t p(x_t)_learned s_θ(x_t,t). (2) Diffusion models are learned by minimizing the following loss: t∼[0,1],x∼p(x),xt∼p(xt∣x)[‖atx−xtbt2−sθ(xt,t)‖2].E_t [0,1],x p(x),x_t p(x_t x)\! [ \| a_tx-x_tb_t^2-s_θ(x_t,t) \|^2 ]. (3) Other formulations of diffusion rewrite Equation 3 to instead predict the mean fθ(xt,t)≈[x∣xt]f_θ(x_t,t) [x x_t] or the instantaneous velocity vθ(xt,t)≈x|xt[x−ϵ∣xt]v_θ(x_t,t) _x|x_t[x-ε x_t], where xt=atx+btϵx_t=a_tx+b_tε. They also introduce weighting functions λ(t)λ(t) that rescale the loss for better convergence, or sample t∼p(t)t p(t). Similarly, different choices for the coefficients at,bta_t,b_t lead to different formulations of diffusion and flow matching. For the experiments in this paper, we choose the schedules ourselves, or leverage pretrained models, in which case we adopt their diffusion schedule. 2.2 Bayesian Inference We are interested in sampling from the Bayesian posterior: p(x∣y)∝p(x)p(y∣x).p(x y) p(x)p(y x). (4) Here, the prior p(x)p(x) is usually given by training a diffusion model on data; in our experiments in Section 6.1, we find that it is even useful for analytic priors with tractable noising convolutions. The “observation” y is an abstract placeholder for specifying the task: it could be a noisy measurement, but also covers arbitrary reward functions r(x;y)r(x;y), potentially without an external signal r(x;y)≡r(x)r(x;y)≡ r(x), so that p(y∣x)=exp(r(x;y))p(y x)= \! (r(x;y) ). Sampling from Equation 4 is a hard task in general. Diffusion models can be used here by learning them as a prior distribution p(x)p(x). The core insight is that given a pretrained diffusion model, we can modify its prediction to correct from the diffusion prior to the Bayesian posterior via Bayes’ rule: ∇xtlogp(xt∣y)=∇xtlogp(xt)⏟pretrained model+∇xtlogp(y∣xt)⏟diffused likelihood. _x_t p(x_t y)= _x_t p(x_t)_pretrained model+ _x_t p(y x_t)_diffused likelihood. (5) While this modification is formally correct, the general form of the likelihood p(y∣x)p(y x) in Equation 4 makes it impossible to analytically compute the diffused likelihood p(y∣xt)p(y x_t) on noisy data as required for Equation 5, even if we have a closed-form expression for p(y∣x)p(y x). Instead, we need to approximate this integral, which evaluates to: p(y∣xt)=∫p(x∣xt)p(y∣x)x,p(y x_t)= p(x x_t)p(y x)dx, (6) where p(x∣xt)p(x x_t) samples from the diffusion posterior. The central goal of this paper is to identify valid approximations of the integral in Equation 6 that lead to sampling from the true Bayesian posterior p(x∣y)p(x y) as defined in Equation 4. 2.3 Diffused Likelihood Approximation Figure 2: Approximations of diffused likelihoods. (Left) The posterior mean approximation in Equation 7 looks up the likelihood value at the mean of the diffusion posterior. (Center) Gaussian approximations to the posterior lead to inconsistent estimates that cannot be corrected by sampling more points. (Right) Our method relies on the true diffusion posterior p(x∣xt)p(x x_t), yielding arbitrary precision to determine the diffused likelihood p(y∣xt)p(y x_t) and its gradients. Equation 6 is usually approximated to save costs. In this section, we show how these approximations prevent sampling from the correct posterior. The common approximations in the literature are: The Posterior Mean Approximation computes the cost of the mean, instead of the mean of the cost (Chung et al., 2023): p(y∣xt)≈p(y∣x=[x∣xt]).p(y x_t)≈ p(y x=E[x x_t]). (7) Intuitively, this is justified late in sampling when the diffusion posterior has small support relative to the curvature of the reward. Evaluate Equation 6 with a Posterior Gaussian Approximation to the denoising distribution p(x∣xt)p(x x_t), as proposed as Π (Song et al., 2023a): p(y∣xt)≈∫p(y∣x)(x;[x∣xt],σt2I)x.p(y x_t)≈ p(y x)N(x;E[x x_t], _t^2I)dx. (8) The standard deviation is typically chosen as σt2=bt2/(at2+bt2) _t^2=b_t^2/(a_t^2+b_t^2), the correct value if the prior p(x)p(x) is a standard normal distribution. Optimal control formulations such as nonlinear diffusion trajectory matching (NDTM) (Pandey et al., 2025b) steer the trajectory through a local optimization problem: x~t=xt+ut,where ut=argminutℒ(ut,xt) x_t=x_t+u_t,where u_t=argmin_u_tL(u_t,x_t) (9) The loss function ℒ(ut,xt)L(u_t,x_t) usually regularizes finite control utu_t plus maximizes the posterior mean approximation in Equation 7. The optimization over utu_t broadens the design space of the guidance framework. While this usually leads to higher rewards, it does not aim for calibrated posterior sampling. We analyze the shortcomings of these estimators in Section 4 and propose a well-calibrated alternative in Section 5. 3 Related Work Our training-free guidance framework is based on diffusion models. Diffusion models learn to generate data by iteratively denoising samples from a simple noise distribution, with foundational formulations ranging from discrete-time diffusion processes (Sohl-Dickstein et al., 2015; Ho et al., 2020) to continuous-time and score-based (Song and Ermon, 2019; Song et al., 2021) as well as velocity-based perspectives (Liu et al., 2023; Lipman et al., 2023; Albergo and Vanden-Eijnden, 2023). These methods have demonstrated strong empirical performance across modalities. Regarding the calibration of these models, existing work analyzes epistemic uncertainty (Jazbec et al., 2025), and improves tail behavior (Pandey et al., 2025a). The test-time guidance framework we develop in this paper leverages pretrained diffusion models and adapts them to novel tasks. The first approaches to solving inverse problems using diffusion models consider linear tasks such as inpainting, super-resolution, and colorization (Kadkhodaie and Simoncelli, 2021; Song et al., 2021; Chung et al., 2022b; Kawar et al., 2022; Chung et al., 2022a). The methods strictly adhere to the task specification, not aiming to recover the Bayesian posterior. Diffusion Posterior Sampling (DPS) (Chung et al., 2023) generalized this to solve nonlinear inverse problems using the posterior mean approximation. Several follow-up works extend DPS to more tasks, such as Universal Guidance for Diffusion Models (Bansal et al., 2024) and FreeDoM (Yu et al., 2023a). We find that the posterior mean approximation can yield high adherence to the reward, but the sampling does not follow the true Bayesian posterior diffusion. To combat the inaccuracies in the posterior mean approximation, Pseudoinverse-Guided Diffusion Models (Π ) (Song et al., 2023a) and C-Π (Pandey et al., 2024) approximate the diffusion posterior p(x∣xt)p(x x_t) by a normal distribution. This approximation is also used in Loss-Guided Diffusion (LGD) (Song et al., 2023b) as well as Diffusion Policy Gradient (DPG) (Tang et al., 2023). Again, we find this can be biased in general. MPGD (He et al., 2024) projects the posterior mean to the data manifold and evaluate the likelihood there. This alleviates the problem that the reward function might be ill-defined off the data manifold, a common scenario for posterior approximations. However, this method is still limited by a single point estimate for the diffusion posterior. Unified Training-Free Guidance (TFG) (Ye et al., 2024) unifies several design decisions underlying previous methods. Similarly, Nonlinear Diffusion Trajectory Matching (NDTM) (Pandey et al., 2025b) extends the design space, but ultimately both rely on the posterior mean approximation. Divide-and-Conquer Posterior Sampling (DCPS) (Janati et al., 2024) performs local Monte Carlo updates at every diffusion step. In contrast to all these works, we formulate a consistent diffusion posterior sampling, where increased computational budget leads to calibrated samples. In concurrent work, Potaptchik et al. (2026) train few-step models to evaluate the same estimators as in Equations 20 and 16. While the focus of that work is on image synthesis, we focus on Bayesian inference. Also, by using the renoise trick, we can leverage pretrained one-step or few-step models. A complementary body of work learns the diffused likelihood directly, which requires retraining for new tasks. This is achieved in classifier-free guidance (Ho and Salimans, 2021), which requires the likelihood to be representable as an input to the model. Other methods leverage reinforcement learning techniques to learn the diffused likelihood or finetune the teacher to a given reward (Li et al., 2024; Fan et al., 2023; Black et al., 2024; Uehara et al., 2024; Pandey et al., 2026). 4 Shortcomings of Existing Estimators We now derive that the approximations of the diffused likelihoods in Section 2.3 yield uncalibrated Bayesian posteriors. In particular, we show that the approximations are always biased, apart from trivial cases. 4.1 Inconsistency of Diffused Likelihood Estimators The estimators introduced in Equations 7, 8 and 9 in Section 2.3 have one common pitfall: They are inconsistent estimators, meaning that even increasing the computational budget will not approximate the true diffused likelihood in Equation 6. Instead, they converge to a biased estimate. First, we find that the posterior mean approximation in Equation 7 fails once the likelihood has a nontrivial maximizer on the support of the prior, which is common for example in the context of inverse problems: Theorem 4.1 (Informal). Fix 0<t<10<t<1. If the likelihood p(y∣x)p(y x) has a nontrivial maximizer, then, unless the likelihood is constant in x, there exists some xtx_t at which the posterior mean approximation in Equation 7 fails. This means that Equation 7 is not exact. The detailed statement is given in Section A.1, together with a mild regularity assumption. Intuitively, the proof shows that the maximum of the diffused likelihood is not the same as the maximum of the original likelihood: maxxtp(y∣xt)≠maxxp(y∣x) _x_tp(y x_t)≠ _xp(y x), contradicting Equation 7. Similarly, the Gaussian posterior approximation can be incorrect, and we give a concrete example below: Example 4.2. Let p(x)=12((x;μ,1)+(x;−μ,1))p(x)= 12(N(x;μ,1)+N(x;-μ,1)) and p(y∣x)=(x;0,ρ2)p(y x)=N(x;0,ρ^2) for μ,ρ2>0μ,ρ^2>0. Then, for every t, there exists xt∈ℝx_t such that Equation 8 does not hold. Please see Section A.2 for the detailed derivation. Figure 1 (left) shows how this leads to methods relying on these estimates to sample from incorrect posteriors: Every step in the guided diffusion process in Equation 5 follows an approximation of the guidance term that ultimately leads to bad calibration. Figure 2 visualizes how the different approaches approximate the diffused likelihoods. In Section 5, we will derive novel estimators for the diffused likelihoods that are consistent: Increasing the computational budget for the diffused likelihood makes the estimator converge to the true value. 4.2 Bias of Guidance Scales However, even if we have access to a correct p(y∣xt)p(y x_t), there is another common bias introduced when guiding diffusion models: It occurs when one tries to vary the importance of the prior p(x)p(x) relative to the likelihood p(y∣x)p(y x) using a parameter γ: p(x∣y,γ)∝defp(x)p(y∣x)γ.p(x y,γ) def p(x)p(y x)^γ. (10) This is referred to as a tempered likelihood posterior in Bayesian inference (Wuttke et al., 2014; Mandt et al., 2016; Friel and Pettitt, 2008; Pawlowski et al., 2017). In the context of diffusion models, γ is referred to as the classifier or the guidance scale (Ho and Salimans, 2021; Dhariwal and Nichol, 2021). It is usually implemented by rescaling the likelihood term in Equation 5: ∇xtlogp(xt)+γ∇xtlogp(y∣xt). _x_t p(x_t)+γ _x_t p(y x_t). (11) Assume now that we have access to the ground truth p(y∣xt)p(y x_t) for γ=1γ=1. Equation 11 would imply that the tempered diffused likelihood reads: p(y∣xt,γ)=?p(y∣xt)γ.p(y x_t,γ) ?=p(y x_t)^γ. (12) However, this is not correct. According to the definition in Equation 6, the guidance scale needs to be applied within the integral: p(y∣xt,γ)∝∫p(x∣xt)p(y∣x)γx.p(y x_t,γ) p(x x_t)p(y x)^γdx. (13) In fact, we can show that Equation 12 is only correct in the trivial case where the likelihood is constant, in which case guidance has no effect: Theorem 4.3 (Informal). Fix 0<t<10<t<1 and γ∈ℝ+∖0,1γ _+ \0,1\. Unless p(y∣x)p(y x) is constant, there exists some xtx_t at which the naive tempered diffused likelihood Equation 12 fails. The detailed statement and proof are given in Section A.3. A similar result by Sun et al. (2024) previously showed that the diffusion trajectory under Equation 12 is inconsistent with the true diffusion forward process. So whenever the likelihood is nontrivial, picking Equation 12 instead of Equation 13 results in an error. Figure 1 (right) shows how the incorrect Equation 11 as opposed to the correct Equation 13 leads to sampling from an incorrect tempered posterior. Notably, this restriction not only applies to existing test-time guidance methods, but it also applies to classifier-free guidance (Ho and Salimans, 2021), for which we can only extract the untempered diffused likelihood gradient: ∇xtlogp(y∣xt)≈∇xtlogpθ(xt∣y)−∇xtlogpθ(xt). _x_t p(y x_t)≈ _x_t p_θ(x_t y)- _x_t p_θ(x_t). (14) Only a tempered conditional model pθ(xt∣y,γ)p_θ(x_t y,γ) can extract the correct tempered diffused likelihood gradient. This means rescaling the diffused likelihood gradient is not enough for calibrated tempered inference. Together, these results imply that no amount of additional computation can correct the bias in existing methods: they converge to the wrong distribution. We now derive a consistent test-time guidance framework that is immune to the shortcomings in Sections 4.1 and 4.2. 5 Calibrated Bayesian Guidance (CBG) In the light of the shortcomings of existing estimators in Section 4, we now present a novel guidance framework that enables consistent sampling from the true Bayesian posterior, both with and without tempering the likelihood. The central idea is to directly approximate the integral in Equation 6. 5.1 Differentiable Rewards Assuming p(y∣x)p(y x) is differentiable everywhere, we can compute the gradient of the diffused likelihood using the reparameterization trick: ∇xtlogp(y∣xt) _x_t p(y x_t) =1p(y∣xt)∇xt∫p(x∣xt)p(y∣x)x = 1p(y x_t) _x_t p(x x_t)p(y x)dx =1p(y∣xt)∫∇xtp(y∣x=gt(xt;ϵ))p(ϵ)ϵ. = 1p(y x_t) _x_tp(y x=g_t(x_t;ε))p(ε)dε. (15) Here, gt(xt;ϵ)g_t(x_t;ε) samples from p(x∣xt)p(x x_t) such that it is differentiable with respect to xtx_t, and ϵε is the randomness that is used to sample from it. This makes the gradient of the diffused likelihood straightforward to estimate: ∇xtlogp(y∣xt) _x_t p(y x_t) ≈1∑ip(y|x(i))∑i=1K∇xtp(y∣x=gt(xt;ϵ(i))). ≈ 1 _ip(y|x^(i)) _i=1^K _x_tp(y x=g_t(x_t;ε^(i))). (16) Here, ϵ(i)∼p(ϵ)ε^(i) p(ε) samples the randomness in the sampler g for samples i=1,…,Ki=1,…,K. While naive diffusion sampling such as DDPM (Ho et al., 2020) can be used to sample from x∼p(x∣xt)x p(x x_t), we use few- or one-step models as detailed in Section 5.3. We call Equation 16 Gradient-Based Calibrated Bayesian Guidance. In practice, we work with log-likelihoods logp(y∣x) p(y x) and average using softmax(wi)softmax(w_i) for numeric stability. Equation 16 yields a consistent sampling procedure: as K→∞K→∞, any bias vanishes. This stands in contrast to the estimators considered in Section 2.3, which remain biased even asymptotically. While consistency requires averaging over multiple samples to obtain a reliable guidance signal, the additional computation can be viewed as the necessary cost of eliminating systematic bias rather than a fundamental limitation of the estimator itself. 5.2 Non-Differentiable Rewards The reparameterization estimator Equation 16 has the major drawback that it is not compatible when computing gradients through the sampling process p(x∣xt)p(x x_t) and/or the likelihood p(y∣x)p(y x) is computationally expensive or intractable. To this end, we propose a REINFORCE estimator (Williams, 1992) to compute the gradient of the diffused likelihood in Equation 5: ∇xtlogp(y∣xt) _x_t p(y x_t) =1p(y∣xt)∇xt∫p(x∣xt)p(y∣x)x = 1p(y x_t) _x_t p(x x_t)p(y x)dx (17) =1p(y∣xt)∫p(x∣xt)p(y∣x)∇xtlogp(x∣xt)x. = 1p(y x_t) p(x x_t)p(y x) _x_t p(x x_t)dx. Inserting into Equation 5, we absorb ∇xtlogp(xt) _x_t p(x_t) into the integral and find for the diffused posterior score: ∇xtlogp(xt∣y) _x_t p(x_t y) (18) =1p(y∣xt)∫p(x∣xt)p(y∣x)∇xtlogp(xt∣x)x. = 1p(y x_t) p(x x_t)p(y x) _x_t p(x_t x)dx. By Equation 1, we can replace ∇xtlogp(xt∣x)=atx−xtbt2. _x_t p(x_t x)= a_tx-x_tb_t^2. (19) Evaluated using a finite set of samples x(i)∼p(x∣xt)x^(i) p(x x_t) for i=1,…,Ki=1,…,K, we find our estimator: ∇xtlogp(xt∣y)≈1∑iwi∑i=1Kwiatx(i)−xtbt2. _x_t p(x_t y)≈ 1 _iw_i _i=1^Kw_i a_tx^(i)-x_tb_t^2. (20) The likelihood enters as weights wi=p(y∣x(i))w_i=p(y x^(i)). We call Equation 20 Gradient-Free Calibrated Bayesian Guidance. Equation 20 is again a consistent estimator, meaning that it reduces bias with more samples. Again, the estimator relies on samples x(i)∼p(x∣xt)x^(i) p(x x_t). In the next section, we discuss how to obtain these samples efficiently. 5.3 Efficient Approximate Diffusion Posterior p(x∣xt)p(x x_t) Both estimators in Equations 16 and 20 rely on samples from the diffusion posterior x∼p(x∣xt)x p(x x_t). The straightforward way to obtain them is stochastic diffusion sampling such as DDPM (Ho et al., 2020). However, this scales poorly: drawing K samples from xtx_t down to x with an N-step sampler, across an outer denoising loop of N noise levels, requires KN(N−1)/2∈(KN2)KN(N-1)/2 (KN^2) model evaluations. This can quickly run into computational bottlenecks, especially for large K and large models. One mitigation is to limit the number of DDPM iterations for the per-step sample generation to a maximum of M. This reduces the quadratic complexity to a linear dependence O(MNK)O(MNK) on the total number of diffusion steps. Empirically, we found that this yields better quality for the same compute than reducing K instead. To reduce compute further, we propose to either use an analytic expression for p(x∣xt)p(x x_t) if available (such as in the Bayesian inference tasks in Section 6.1), or to use a pretrained one-step or few-step model (Song et al., 2023c; Draxler et al., 2024; Geng et al., 2025; Frans et al., 2025) to sample from it. These approaches reduce the cost to (KN)O(KN). In detail, we propose to use deterministic one-step models x=ft(xt)x=f_t(x_t) to sample from the stochastic posterior p(x∣xt)p(x x_t) via renoising: x(i)=ft(atft(xt)+btϵ(i)),ϵ(i)∼(0,I).x^(i)=f_t\! (a_t\,f_t(x_t)+b_t\,ε^(i) ), ε^(i) (0,I). (21) Here, at,bta_t,b_t are the schedule coefficients from Equation 1, and ϵ(i)ε^(i) is independent noise. Equation 21 applies ftf_t to obtain a deterministic prediction x^=ft(xt) x=f_t(x_t), perturbs it back to noise level t with independent noise ϵ(i)ε^(i), and denoises once more. The renoising step itself has been used previously in other contexts (Wang et al., 2023; Yu et al., 2023b; Lugmayr et al., 2022). Even though this construction is heuristic, we empirically observe identical performance to using inner diffusion rollouts at orders of magnitude faster inference (e.g. Table 2). Concurrent to this work, Potaptchik et al. (2026) propose to learn a stochastic one-step sampler x=ft(xt,ϵ)x=f_t(x_t,ε) for p(x∣xt)p(x x_t) instead. However, at the time of writing, these models are not readily available for many modalities, while deterministic one-step models often are. 5.4 Comparing Gradient-Free and Gradient-Based Methods One might think that being based on REINFORCE, the gradient-free estimator in Equation 20 has larger variance than the gradient-based estimator in Equation 16, as discussed in (Paisley et al., 2012; Ruiz et al., 2016; Miller et al., 2017). Figure 3: Empirical performance on Bayesian Inference tasks. Performance is measured in C2ST (lower is better, ↓ ) (Friedman, 2004), comparing the distribution of guided samples to those of ground truth samples. Our methods improve performance with more compute, while other test-time adaptation methods are limited due to their approximations to diffused likelihood gradients. Table 1: Best method performance on Bayesian Inference C2ST (↓ ), standard deviations in parentheses. Algorithm Task 1 Task 2 Task 3 Task 4 Task 5 Average Diffusion Methods CBG (gradient-free, ours) 0.505 (0.004) 0.513 (0.009) 0.584 (0.063) 0.507 (0.006) 0.525 (0.028) 0.527 CBG (gradient-based, ours) 0.507 (0.005) 0.589 (0.048) 0.789 (0.089) 0.667 (0.019) 0.852 (0.038) 0.681 DPS (Chung et al., 2023) 0.633 (0.024) 0.641 (0.032) 0.998 (0.002) 0.812 (0.023) 0.929 (0.045) 0.803 LGD (Song et al., 2023b) 0.621 (0.049) 0.558 (0.013) 0.894 (0.042) 0.820 (0.028) 0.854 (0.117) 0.750 DPG (Tang et al., 2023) 0.621 (0.078) 0.758 (0.027) 0.968 (0.015) 0.920 (0.023) 1.000 (0.000) 0.872 SCG (Huang et al., 2024) 0.603 (0.037) 0.664 (0.028) 0.832 (0.027) 0.860 (0.017) 0.714 (0.131) 0.735 Langevin+Noisy Classifier (Song and Ermon, 2019) 0.506 (0.002) 0.520 (0.005) 0.633 (0.017) 0.642 (0.083) 0.538 (0.030) 0.568 Langevin+REINFORCE (Song and Ermon, 2019) 0.502 (0.003) 0.514 (0.004) 0.604 (0.009) 0.705 (0.015) 0.540 (0.013) 0.573 MCMC No-U-Turn Sampler (Hoffman and Gelman, 2014) 0.514 (0.005) 0.511 (0.004) 0.627 (0.005) 0.512 (0.007) 0.603 (0.046) 0.553 Likelihood-free Methods NLE (Papamakarios et al., 2019) 0.515 (0.009) 0.506 (0.004) 0.699 (0.069) 0.731 (0.023) 0.668 (0.094) 0.624 NPE (Papamakarios and Murray, 2016) 0.506 (0.004) 0.509 (0.005) 0.831 (0.052) 0.555 (0.015) 0.542 (0.024) 0.589 NRE (Hermans et al., 2020) 0.536 (0.031) 0.631 (0.034) 0.919 (0.039) 0.734 (0.019) 0.629 (0.056) 0.690 REJ-ABC (Pritchard et al., 1999) 0.802 (0.023) 0.909 (0.028) 0.961 (0.020) 0.772 (0.034) 0.664 (0.040) 0.822 SMC-ABC (Beaumont et al., 2009) 0.726 (0.032) 0.794 (0.041) 0.963 (0.018) 0.664 (0.011) 0.663 (0.048) 0.762 SNLE (Papamakarios et al., 2019) 0.519 (0.008) 0.509 (0.005) 0.578 (0.029) 0.624 (0.089) 0.571 (0.050) 0.560 SNPE (Greenberg et al., 2019) 0.507 (0.004) 0.507 (0.005) 0.666 (0.061) 0.533 (0.014) 0.530 (0.026) 0.545 SNRE (Hermans et al., 2020) 0.515 (0.004) 0.536 (0.007) 0.721 (0.058) 0.542 (0.016) 0.563 (0.032) 0.575 Interestingly, this general rule does not apply to our estimators, since they involve a self-normalization term of dividing by the sum of the weights. Empirically, we find that the variance of the gradient-free estimator in Equation 20 is in general lower than that of the gradient-based estimator in Equation 16. See Section C for a concrete example. The reason is that the two estimators behave quite differently in the presence of sharp likelihoods: The gradient-free estimator reduces to a weighted sum, as can be seen by extracting Tweedie’s estimate from Equation 20: [x∣xt,y]≈∑iwi∑jwjx(i).E[x x_t,y]≈ _i w_i _jw_jx^(i). (22) As the weights wiw_i are typically of different orders of magnitude in practice, it often considers only the subset of samples with relatively high likelihood. The diffusion process will then move towards these samples in a region of high likelihood, discarding the others. For the gradient-based estimator in Equation 16, the estimate is dominated by gradients associated with the highest-likelihood samples. These gradients can still be large, since small gradients occur only in a narrow neighborhood around the likelihood maximizer. Moreover, they are effectively amplified because the normalization term is determined by weights that are smaller than the (unobserved) maximum likelihood value: ∇xtlogp(y∣xt)≈∑i∇xtwi∑jwj _x_t p(y x_t)≈ _i _x_tw_i _jw_j (23) As a consequence, a large gradient can induce a disproportionately large update. Reducing this effect requires a larger number of samples, so that local gradients average out into a stable global direction and the maximum observed weight becomes a less noisy estimate of the true tail behavior. From a practical perspective, the reparameterization estimator also requires computing gradients through the diffusion sampling, which increases computational cost and increases memory consumption. We therefore do not consider it for high-dimensional experiments. 6 Experiments We now confirm the quality of our framework for practical Bayesian inference tasks. We find that CBG outperforms other test-time guidance methods on tasks where the prior and the likelihood are given as closed-form expressions, as well as on a scientific inverse problem where the prior is given by a pretrained diffusion model of black hole images. 6.1 Bayesian Inference Benchmark In this experiment, we evaluate a diverse collection of diffusion guidance methods on a benchmark of Bayesian inverse problems proposed by Lueckmann et al. (2021). This benchmark is designed to test methods on their fit to the true Bayesian posterior p(x∣y)p(x y) at varying computational budgets. To this end, the benchmark provides reference samples from the true posterior. For the diffusion posterior p(x∣xt)p(x x_t) in Equations 16 and 20 as well as for the posterior mean [x∣xt]E[x x_t] in Equations 7 and 8, we leverage closed-form expressions from the true prior p(x)p(x) of these tasks (see Section F). This is possible because the prior distributions are analytic expressions such as normal and uniform distributions. This makes the experiments blazingly fast: A single inference takes on the order of milliseconds. Figure 3 shows how both our gradient-free and gradient-based estimators improve their C2ST towards the optimal value of 0.50.5 as the compute budget is increased. Other test-time guidance methods generally do not improve performance with more compute. Additionally, Table 1 shows the best classifier score for each method over all hyperparameters. The gradient-free method achieves the best distributional fit for every task by far compared to all other test-time guidance methods. We also evaluate against a collection of likelihood-free methods that rely only on samples y(i)∣xy^(i) x and a target y, instead of an exact likelihood p(y∣x)p(y x). Our CBG method also outperforms these likelihood-free methods on average. Together, we see that our estimators perform well on Bayesian inverse tasks, and outperform a significant collection of likelihood-based and likelihood-free methods. 6.2 Black Hole Imaging Figure 4: Uncurated comparison of CBG+DDPM with other test-time guidance methods on the black-hole imaging task proposed by Lueckmann et al. (2021). Despite computing no likelihood gradient, CBG is able to reconstruct the ground truth samples well. In this experiment, we apply our method to scientific inference, where having a calibrated posterior is important. In particular, we choose the black hole imaging task proposed by Zheng et al. (2025) with a diffusion prior trained on images by Mizuno (2022). The task provides 100 problem instances, each consisting of a reference solution x∗x^* as well as a corresponding radio-telescope measurement y. The likelihood function p(y∣x)p(y x) measures the compatibility of an image x with a given observation y. The task is to reconstruct the image from that measurement. We use the gradient-free estimator in Equation 20, as we find the gradient-based estimator to require many samples. To evaluate the estimator, we compare sampling x∼p(x∣xt)x p(x x_t) using the pretrained diffusion model with a fixed number of inner steps M and using the renoise trick (compare Section 5.3). For the latter, we train a mean-flow model (Lu et al., 2026) using samples from the original diffusion model. See Appendix H for details. Table 2 shows that our method matches the state of the art at similar wall-clock time, and significantly outperforms them as the number of per-step samples K is increased. Figure 4 visualizes uncurated qualitative reconstruction results. While some of the baselines produce results that are either noticeably blurred or fail to faithfully follow the ground-truth, our method yields results that are visually consistent with the ground truth. This confirms that our method scales well to high-dimensional scientific inverse tasks. Table 2: Peak signal-to-noise ratio (PSNR, higher is better) on 100 black hole imaging tasks. Standard deviations over tasks in parentheses. Per-sample inference time measured on a single NVIDIA TITAN RTX (24 GB). SMILI and EHT-Imaging are CPU-based methods whose runtime can’t be reliably compared. For our method, DDPM denotes a DDPM inner loop, while renoise variants use the one-step sampler from Section 5.3. Methods PSNR (↑ ) Time Traditional Methods SMILI (Chandra et al., 2018) 22.67 (3.13) – EHT-Imaging (Chael et al., 2018) 24.28 (3.63) – Test-time Guidance Methods DPS (Chung et al., 2023) 25.86 (3.90) 24 s LGD (Song et al., 2023b) 21.22 (3.64) 36 s RED-diff (Mardani et al., 2024) 23.77 (4.13) 16 s PnP-DM (Wu et al., 2024) 26.07 (3.70) 45 s DAPS (Zhang et al., 2024) 25.60 (3.64) 27 s DiffPIR (Zhu et al., 2023) 25.01 (4.64) 21 s CBG (gradient-free, ours) + DDPM (K=512K=512) 26.10 (3.83) ∼ 23 h + renoise (K=128K=128) 25.36 (3.99) 48 s + renoise (K=512K=512) 26.10 (3.84) 2.9 min + renoise (K=4096K=4096) 26.39 (3.81) 22.9 min + renoise (K=65536K=65536) 27.08 (3.74) ∼ 6 h Figure 5: Increasing the number of candidate x samples K per denoising step in the gradient-free CBG estimator consistently increases PSNR (higher is better) on black hole imaging. Error bars are 95% confidence intervals over 100 tasks. 6.3 Image Inverse Problems Figure 6: We show that CBG is also applicable in the image domain. (a) For image super-resolution, the proposed method effectively reconstructs high-resolution images (256×256256× 256) from the low-resolution inputs (64×6464× 64). (b) For prompt alignment, we consider text-to-image generation with exact object-count constraints. Our method satisfies the prompt while preserving image quality, demonstrating the effectiveness of calibrated guidance. In contrast, DPS (Chung et al., 2023) produces incorrect counts or distorted images. In the domain of natural images, the generation of visually pleasing results that score high in likelihood is often the primary goal, while distributional aspects like compliance with the true posterior are secondary. Still, as a proof of concept, we conduct qualitative experiments demonstrating that the proposed method also applies to the image domain. Specifically, we consider two tasks: (a) super-resolution, and (b) prompt alignment. For super-resolution, we consider a 4×4× upscaling setting, reconstructing 256×256256× 256 images from 64×6464× 64 low-resolution inputs on natural images from the ImageNet test set (Deng et al., 2009). We use a pixel mean flow model (Lu et al., 2026) trained on ImageNet, and compute the likelihood by measuring the difference between the low-resolution input and the downsampled version. For prompt alignment, we consider a scenario in which a pretrained SANA model (Xie et al., 2025) is prompted to generate images with precise object counts specified in the text prompt. To compute the likelihood, we use a pretrained vision-language reward model (Wang et al., 2025), following the setup of FMTT (Sabour et al., 2025). Detailed experimental settings are described in Appendix I. As shown in Figure 6, in the super-resolution task, CBG successfully reconstructs the high-resolution ground truth images. In the prompt alignment task, our method consistently generates images with the correct object count, whereas the DPS baseline (Chung et al., 2023) results in suboptimal outcomes. These highlight the practical benefit of CBG in image-domain inverse problems. 7 Conclusion In this work, we address an important gap in the literature of test-time guidance for diffusion models: The common approximations for the diffused likelihoods as well as gradient rescaling can be biased, preventing sampling from the true Bayesian posterior. This may not pose problems for applications such as natural image inverse synthesis, where maximal adherence to the likelihood is preferred over calibrated inference in a Bayesian sense. For scientific scenarios however, proper uncertainty calibration is essential. We propose two novel estimators that allow arbitrary reduction of this bias by leveraging the strong signal from evaluating the likelihood function on actual diffusion samples. By increasing computational resources, our method converges to the true Bayesian posterior. The gradient-free estimator is particularly easy to adapt to new scenarios as it does not require computing likelihood and diffusion model gradients. Limitations. Our method requires p(y∣x)p(y x) to overlap with the prior in measure; if it concentrates on a measure-zero manifold, samples from p(x∣xt)p(x x_t) will almost surely miss it. In such cases, methods that over-optimize the likelihood can outperform our calibrated sampler on task metrics. Projecting x∼p(x∣xt)x p(x x_t) onto the manifold of high likelihood can be a potential solution (Sorrenson et al., 2024), but is out of scope for this work. Acknowledgements Stephan Mandt acknowledges funding from the National Science Foundation (NSF) through an NSF CAREER Award IIS-2047418, IIS-2007719, the NSF LEAP Center, and the Hasso Plattner Research Center at UCI, and the Chan Zuckerberg Initiative. 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(24) Assume that S∩μt(ℝd)≠∅S∩ _t(R^d)≠ and ℙx∼p(x)[x∈S]<1P_x p(x)[x∈ S]<1. For example, if supppsuppp is compact, x↦p(y∣x)x p(y x) is continuous, and the likelihood attains its maximum at an interior point that lies in μt(ℝd) _t(R^d), then this assumption is fulfilled. Detailed restatement. Under Section A.1, for every 0<t<10<t<1, unless the likelihood p(y∣x)p(y x) is constant in x, there exists some xt∈ℝdx_t ^d at which the posterior mean approximation in Equation 7 fails. Proof. If the likelihood p(y∣x)p(y x) is constant in x, then p(y∣xt)=∫p(x∣xt)p(y∣x)x=∫p(x∣xt)p(y∣x=0)x=p(y∣x=0)∫p(x∣xt)x=p(y∣x=0).p(y x_t)= p(x x_t)p(y x)dx= p(x x_t)p(y x=0)dx=p(y x=0) p(x x_t)dx=p(y x=0). (25) We now prove the theorem by contradiction under Section A.1, assuming that equality holds in Equation 7 for all xt∈ℝdx_t ^d. First, the diffusion posterior p(xt∣x)p(x_t x) forms an exponential family: p(x∣xt)∝p(x)exp(−12bt2‖xt−atx‖2).p(x x_t) p(x) \! (- 12b_t^2\|x_t-a_tx\|^2 ). (26) With the natural parameter η=atbt2xtη= a_tb_t^2x_t and h(x)=p(x)exp(−at2/(2bt2)‖x‖2)h(x)=p(x) (-a_t^2/(2b_t^2)\|x\|^2), we get the exponential family: fX(x∣θ)=h(x)exp(η⋅x−A(η)).f_X(x θ)=h(x) (η· x-A(η)). (27) Since the Gaussian kernel is strictly positive, p(x∣xt)p(x x_t) and p(x)p(x) have the same support for all xt∈ℝdx_t ^d. The natural parameter η is a bijection of xtx_t (on 0<t<10<t<1), and so the exponential family is minimal. Therefore, the function μt(xt)=x∼p(x∣xt)[x]=∇ηA(η) _t(x_t)=E_x p(x x_t)[x]= _ηA(η) is a bijection onto its image (Wainwright and Jordan, 2008, Proposition 3.2). Let x⋆∈S∩μt(ℝd)x ∈ S∩ _t(R^d). By bijectivity onto the image, there exists xt⋆∈ℝdx_t ^d with μt(xt⋆)=x⋆ _t(x_t )=x . Since x⋆∈Sx ∈ S, it holds that p(y∣x⋆)=Mp(y x )=M. Under our contradiction assumption, Equation 7 holds at xt⋆x_t , and so: p(y∣xt⋆)=p(y∣x=μt(xt⋆))=p(y∣x⋆)=M.p(y x_t )=p(y x= _t(x_t ))=p(y x )=M. (28) On the other hand, p(y∣xt⋆)=∫p(x∣xt⋆)p(y∣x)x.p(y x_t )= p(x x_t )p(y x)dx. (29) Since ℙx∼p(x)[x∈S]<1P_x p(x)[x∈ S]<1, the set A=suppp∖S=x∈suppp:p(y∣x)<MA=suppp S=\x :p(y x)<M\ (30) has positive prior probability. Since p(x∣xt⋆)∝p(x)p(xt⋆∣x)p(x x_t ) p(x)p(x_t x) and the Gaussian factor p(xt⋆∣x)p(x_t x) is strictly positive for all x, the set A also has positive posterior probability under p(x∣xt⋆)p(x x_t ). Therefore, p(y∣xt⋆)=∫p(x∣xt⋆)p(y∣x)x<∫p(x∣xt⋆)Mx=M,p(y x_t )= p(x x_t )p(y x)dx< p(x x_t )Mdx=M, (31) contradicting the previous equality. Hence Equation 7 must fail at xt⋆x_t . ∎ A.2 Derivation of Section 4.1 Consider the one-dimensional symmetric two-component Gaussian mixture prior p(x)=12(x;−μ,1)+12(x;μ,1),μ>0,p(x)= 12N(x;-μ,1)+ 12N(x;μ,1), μ>0, (32) and the Gaussian likelihood slice centered at the symmetry point p(y=0∣x)=(0;x,ρ2),ρ2>0.p(y=0 x)=N(0;x,ρ^2), ρ^2>0. (33) Fix t∈(0,1)t∈(0,1), let s=1−ts=1-t, and set σt2=t2s2+t2,d=t2s2+t2μ>0. _t^2= t^2s^2+t^2, d= t^2s^2+t^2μ>0. (34) We claim that the Gaussian proxy identity fails already at xt=0x_t=0. At xt=0x_t=0, symmetry gives posterior weights 12,12 12, 12, and the posterior of each component is Gaussian with variance σt2 _t^2 and mean ±d± d. Hence p(x∣xt=0)=12(x;−d,σt2)+12(x;d,σt2).p(x x_t=0)= 12N(x;-d, _t^2)+ 12N(x;d, _t^2). (35) Its posterior mean is therefore [X∣Xt=0]=0.E[X X_t=0]=0. (36) So the Gaussian proxy is (x;0,σt2).N(x;0, _t^2). (37) Now use the Gaussian convolution identity ∫(0;x,ρ2)(x;ν,σt2)x=(0;ν,ρ2+σt2). (0;x,ρ^2)N(x;ν, _t^2)dx=N(0;ν,ρ^2+ _t^2). (38) Therefore the exact posterior expectation equals 12(0;−d,ρ2+σt2)+12(0;d,ρ2+σt2)=(0;d,ρ2+σt2), 12N(0;-d,ρ^2+ _t^2)+ 12N(0;d,ρ^2+ _t^2)=N(0;d,ρ^2+ _t^2), (39) whereas the proxy expectation equals (0;0,ρ2+σt2).N(0;0,ρ^2+ _t^2). (40) Since d>0d>0, the Gaussian density (0;ν,ρ2+σt2)N(0;ν,ρ^2+ _t^2) is strictly maximized at ν=0ν=0, so (0;d,ρ2+σt2)<(0;0,ρ2+σt2).N(0;d,ρ^2+ _t^2)<N(0;0,ρ^2+ _t^2). (41) Hence ∫p(y=0∣x)p(x∣xt=0)x≠∫p(y=0∣x)(x;[X∣Xt=0],σt2)x. p(y=0 x)p(x x_t=0)dx≠ p(y=0 x)N (x;E[X X_t=0], _t^2 )dx. (42) Thus the Gaussian posterior proxy fails for this example. A.3 Proof for Theorem 4.3 Detailed restatement. Fix 0<t<10<t<1 and γ∈ℝ+∖0,1γ _+ \0,1\. If there is no constant c∈ℝ+c _+ such that p(y∣x)=cp(y x)=c for p(x)p(x)-almost every x, then there exists some xt∈ℝdx_t ^d at which the naive tempered diffused likelihood Equation 12 fails. Proof. Let ϕ(u)=uγφ(u)=u^γ for u≥0u≥ 0. For γ∈(0,1)γ∈(0,1), ϕφ is strictly concave; for γ>1γ>1, ϕφ is strictly convex. For fixed xtx_t, define x∼p(x∣xt)x p(x x_t) and z=p(y∣x)∈[0,∞)z=p(y x)∈[0,∞). Then ∫p(x∣xt)p(y∣x)γx=(∫p(x∣xt)p(y∣x)x)γ p(x x_t)p(y x)^γdx= ( p(x x_t)p(y x)dx )^γ (43) is exactly [ϕ(z)]=ϕ([z])E[φ(z)]=φ(E[z]). By Section A.3, this equality holds if and only if z is almost surely constant under p(x∣xt)p(x x_t). Assume now that the naive identity Equation 12 holds for all xtx_t. Fix any xt⋆∈ℝdx_t ^d. Then there exists c∈ℝ+c _+ such that p(y∣x)=cp(y x)=c for p(x∣xt⋆)p(x x_t )-almost every x. Since p(x∣xt⋆)∝p(x)exp(−‖xt⋆−atx‖22bt2),p(x x_t ) p(x) \! (- \|x_t -a_tx\|^22b_t^2 ), (44) and the Gaussian factor is strictly positive, p(x∣xt⋆)p(x x_t ) and p(x)p(x) have the same null sets. Hence p(y∣x)=cp(x)-almost surely.p(y x)=c p(x)-almost surely. (45) Therefore, if no such constant c exists p(x)p(x)-almost surely, the naive identity cannot hold for all xtx_t. Hence there must exist some xtx_t at which Equation 12 fails. ∎ Lemma A.2. Let ϕ:[0,∞)→ℝφ:[0,∞) be strictly convex or strictly concave, and let Z be an integrable random variable with values in [0,∞)[0,∞) such that ϕ(Z)φ(Z) is integrable. Then [ϕ(Z)]=ϕ([Z])⟺Z is almost surely constant.E[φ(Z)]=φ(E[Z]) Z is almost surely constant. (46) Proof. If Z is almost surely constant, the identity is immediate. We now prove the converse. First assume that ϕφ is strictly convex and that Z is not almost surely constant. Let m=[Z]m=E[Z] and define the events A=Z<m,C=Z≥m.A=\Z<m\, C=\Z≥ m\. (47) Since Z is not almost surely constant and m is its mean, both ℙ(A)P(A) and ℙ(Z>m)P(Z>m) are positive. In particular, with λ=ℙ(A)∈(0,1)λ=P(A)∈(0,1), m−=[Z∣A],m+=[Z∣C]m_-=E[Z A], m_+=E[Z C] (48) satisfy m−<m<m+m_-<m<m_+ and m=λm−+(1−λ)m+.m=λ m_-+(1-λ)m_+. (49) Applying Jensen’s inequality conditionally on A and C, we obtain [ϕ(Z)] [φ(Z)] =λ[ϕ(Z)∣A]+(1−λ)[ϕ(Z)∣C] =λ\,E[φ(Z) A]+(1-λ)\,E[φ(Z) C] (50) ≥λϕ(m−)+(1−λ)ϕ(m+) ≥λ\,φ(m_-)+(1-λ)\,φ(m_+) (51) >ϕ(λm−+(1−λ)m+) >φ(λ m_-+(1-λ)m_+) (52) =ϕ(m). =φ(m). (53) The strict inequality uses strict convexity and m−≠m+m_-≠ m_+. This contradicts [ϕ(Z)]=ϕ([Z])E[φ(Z)]=φ(E[Z]). Therefore, if ϕφ is strictly convex, equality can only hold when Z is almost surely constant. If ϕφ is strictly concave, then −ϕ-φ is strictly convex, and applying the previous argument to −ϕ-φ gives the same conclusion. ∎ Appendix B Algorithm Algorithm 1 Calibrated Bayesian Guidance Input: Initial noise xT∼(0,1)x_T (0,1) for t=Tt=T to 11 do Draw samples from p(x∣xt)p(x x_t) Compute ∇xtlogp(xt∣y) _x_t p(x_t y) with Equations 5 and 16 or Equation 20 Compute xt−1x_t-1 with an Euler step along Equation 2 end for Appendix C Gradient-Free and Gradient-Based Methods We consider the case where we have a simple prior p(x)=(x;2,1)p(x)=N(x;2,1) and a simple likelihood p(y|x)=(y;0,0.42)p(y|x)=N(y;0,0.4^2). Denote x^(xt) x(x_t) as the exact analytic estimate for [x|xt,y]E[x|x_t,y], and x^grad-free(xt),x^grad-based(xt) x_grad-free(x_t), x_grad-based(x_t) our estimates from the CBG gradient-free and CBG gradient-based algorithms using Tweedie’s formula. We plot the variances [(x^grad-based(xt)−x^(xt))2]E [ ( x_grad-based(x_t)- x(x_t) )^2 ] and [(x^grad-free(xt)−x^(xt))2]E [ ( x_grad-free(x_t)- x(x_t) )^2 ] as we vary xtx_t and t. Both estimators use K=1000K=1000 samples, with the variance expectation estimated over 100100 algorithm calls. Looking at Figure 7, there is a clear region to the left of t≈0.3t≈ 0.3 where the gradient-based method performs better, but noticeably not by much. Figure 7: Variance comparison for gradient-free and gradient-based methods. In some regions, the gradient-based method has lower variance. Appendix D Bias and Variance of Diffusion Methods We consider Task 4 of the Bayesian Inference Benchmark, specifically estimating [x∣xt,y]E[x x_t,y]. We initialize an xtx_t from p(xt|y,t=0.5)p(x_t|y,t=0.5) and run one step of each diffusion sampling algorithm to estimate [x∣xt,y]E[x x_t,y], with varying values for K. We repeat this process 100100 times to estimate the bias and variance of each method compared to an analytic solution. The results are shown in Figure 8. We see that all methods converge to 0 variance as K increases, but only our CBG methods (both gradient-free and gradient-based) converge to the correct analytic solution. Notably, the gradient-free version of CBG converges much faster than the gradient-based version, which is consistent with the variance comparison in Figure 7. Figure 8: Bias and variance of [x∣xt,y]E[x x_t,y] as a function of K for Bayesian Inference Task 4. While the variance of all methods decreases to 0 with more compute, only our CBG methods decrease to 0 bias. Appendix E Bayesian Inference Benchmark Guidance Methods E.1 Diffusion Process For all guidance methods, the flow matching forward process was used: p(xt|x)=(xt;(1−t)x,t2I)p(x_t|x)=N(x_t;(1-t)x,t^2I) For generation, t is scheduled to linearly decrease from 11 to 0. E.2 Diffusion Posterior Sampling Diffusion Posterior Sampling (DPS) (Chung et al., 2023) approximates the score ∇xtlogp(y|xt) _x_t p(y|x_t) as ∇xtlogp(y|xt)≈∇xtlogp(y|[x|xt]) _x_t p(y|x_t)≈ _x_t p (y|E [x|x_t ] ) E.3 Loss Guided Diffusion Loss Guided Diffusion (LGD) (Song et al., 2023b) approximates the score ∇xtlogp(y|xt) _x_t p(y|x_t) as ∇xtlogp(y|xt)≈∇xtlog∫p(y|x)q(x|xt)x _x_t p(y|x_t)≈ _x_t p(y|x)q(x|x_t)dx where q(x|xt)=([x|xt],t2(1−t)2+t2I)q(x|x_t)=N (E [x|x_t ], t^2(1-t)^2+t^2I ) approximates the true diffusion posterior distribution. Evaluated using a finite set of samples x(i)∼q(x|xt)x^(i) q(x|x_t) for i=1,⋯,Ki=1,·s,K, we have: ∇xtlogp(y|xt)≈∇xtlog[1K∑ip(y|x(i))] _x_t p(y|x_t)≈ _x_t [ 1K _ip (y|x^(i) ) ] E.4 Diffusion Policy Gradient Diffusion Policy Gradient (DPG) (Tang et al., 2023) approximates the score ∇xtlogp(y|xt) _x_t p(y|x_t) as ∇xtlogp(y|xt)≈Cs(xt)‖s(xt)‖22 _x_t p(y|x_t)≈ C s(x_t)\|s(x_t)\|_2^2 Thus: s(xt)=x∼q(x|xt)[(p(y|x)−b)∇xt(−∥x−[x|xt]∥22)]s(x_t)=E_x q(x|x_t) [ (p(y|x)-b ) _x_t (-\|x-E [x|x_t ]\|^2_2 ) ] where q(x|xt)=([x|xt],t2(1−t)2+t2I)q(x|x_t)=N (E [x|x_t ], t^2(1-t)^2+t^2I ) approximates the true diffusion posterior distribution. Evaluated using a finite set of samples x(i)∼q(x|xt)x^(i) q(x|x_t) for i=1,⋯,Ki=1,·s,K, we approximate as: s(xt)≈1K∑i(p(y|x(i))−b(i))∇xt(−∥x(i)−[x|xt]∥22)s(x_t)≈ 1K _i (p(y|x^(i))-b^(i) ) _x_t (-\|x^(i)-E [x|x_t ]\|^2_2 ) where b(i)b^(i) is computed as b(i)=1K−1∑j=1,j≠iKp(y|x(j))b^(i)= 1K-1 _j=1,j≠ i^Kp (y|x^(j) ) E.5 Stochastic Control Guidance Stochastic Control Guidance (SCG) (Huang et al., 2024) chooses an x^t′ x_t to transition to, given the current state xtx_t, as follows: x^t′=argmaxxt′p(y|[x|xt′]) x_t = *arg\,max_x_t p (y|E[x|x_t ] ) Approximating p(xt′|xt)p(x_t |x_t) as q(xt′|xt)=(t′txt+(1−t′t)[x|xt],σ2I),σ2=(t−t′)2(1−t)2+t2q(x_t |x_t)=N ( t tx_t+ (1- t t )E[x|x_t],σ^2I ),σ^2= (t-t )^2(1-t)^2+t^2 we can evaluate this using a finite set of samples xt′(i)∼q(x|xt)x_t ^(i) q(x|x_t) for i=1,⋯,Ki=1,·s,K as: x^t′≈argmaxip(y|[x|xt′(i)]) x_t ≈ *arg\,max_ip (y|E [x|x_t ^(i) ] ) E.6 Langevin + Noisy Classifier Langevin + Noisy Classifier performs annealed Langevin Sampling (Song and Ermon, 2019) on p(xt)p(y|x=xt)p(x_t)p(y|x=x_t), where a noisy image xtx_t is passed to the classifier p(y|x)p(y|x). E.7 Langevin + REINFORCE Langevin + REINFORCE performs annealed Langevin Sampling (Song and Ermon, 2019) using an estimate of ∇xtlogp(xt∣y) _x_t p(x_t y) generated by the REINFORCE estimator in our CBG gradient-free algorithm (20). E.8 MCMC No-U-Turn Sampler MCMC No-U-Turn Sampler uses the No-U-Turn Sampler sampler from (Hoffman and Gelman, 2014) to sample directly on the analytic unnormalized distribution p(x|y)∝p(x)p(y|x)p(x|y) p(x)p(y|x). Appendix F Bayesian Inference Benchmark Guidance Tasks F.1 Task 1 Inference on the mean of a 10-dimensional Gaussian, with a Gaussian prior. Prior: p(x)=(x;0,0.1I)p(x)=N (x;0,0.1I ) Diffusion Posterior: p(x∣xt)=(x;1−t10t2+(1−t)2xt,t210t2+(1−t)2I).p(x x_t)=N\! (x;\ 1-t10t^2+(1-t)^2\,x_t,\ t^210t^2+(1-t)^2\,I ). Likelihood: p(y|x)=(x;y,0.1I)p(y|x)=N (x;y,0.1I ) Dimensionality: x∈ℝ10x ^10 F.2 Task 2 Inference on the mean of a 10-dimensional Gaussian, with a uniform prior. Prior: p(x)=(−1,1)p(x)=U(-1,1) Diffusion Posterior: p(x∣xt)∝x∈[−1,1]10(x;xt1−t,t2(1−t)2I).p(x x_t) 1\x∈[-1,1]^10\\,N\! (x;\ x_t1-t,\ t^2(1-t)^2\,I ). Likelihood: p(y|x)=(y;x,0.1I)p(y|x)=N (y;x,0.1I ) Dimensionality: x∈ℝ10,y∈ℝ10x ^10,y ^10 F.3 Task 3 Inference on the parameters of a 10-dimensional Gaussian with nonlinear mean and variance functions, with a uniform prior. Prior: p(x)=(−3,3)p(x)=U(-3,3) Diffusion Posterior: p(x∣xt)∝x∈[−3,3]5(x;xt1−t,t2(1−t)2I).p(x x_t) 1\x∈[-3,3]^5\\,N\! (x;\ x_t1-t,\ t^2(1-t)^2\,I ). Likelihood: p(y|x)=∏i=14(y(i);μ(x),Σ(x))p(y|x)= _i=1^4N (y^(i);μ(x), (x) ) where μ(x)=[x1x2],Σ(x)=[s12ρs1s2ρs1s2s22],s1=x32,s2=x42,ρ=tanhx5μ(x)= bmatrixx_1\\ x_2 bmatrix, (x)= bmatrixs_1^2&ρ s_1s_2\\ ρ s_1s_2&s_2^2 bmatrix,s_1=x_3^2,s_2=x_4^2,ρ= x_5 Dimensionality: x∈ℝ5,y∈ℝ4×2x ^5,y ^4× 2 F.4 Task 4 Inference on the shared mean of a mixture of two 2-dimensional Gaussians, one much broader than the other, with a uniform prior. Prior: p(x)=(−10,10)p(x)=U(-10,10) Diffusion Posterior: p(x∣xt)∝x∈[−10,10]2(x;xt1−t,t2(1−t)2I).p(x x_t) 1\x∈[-10,10]^2\\,N\! (x;\ x_t1-t,\ t^2(1-t)^2\,I ). Likelihood: p(y|x)=0.5(y;x,I)+0.5(y;x,0.01I)p(y|x)=0.5N (y;x,I )+0.5N (y;x,0.01I ) Dimensionality: x∈ℝ2,y∈ℝ2x ^2,y ^2 F.5 Task 5 Inference on a two-moons distribution with both bimodality and local structure, with a uniform prior. Prior: p(x)=(−1,1)p(x)=U(-1,1) Diffusion Posterior: p(x∣xt)∝x∈[−1,1]2(x;xt1−t,t2(1−t)2I).p(x x_t) 1\x∈[-1,1]^2\\,N\! (x;\ x_t1-t,\ t^2(1-t)^2\,I ). Likelihood: Let s(x)=[−|x1+x2|2−x1+x22]s(x)= bmatrix- |x_1+x_2| 2\\ -x_1+x_2 2 bmatrix, z=y−s(x)z=y-s(x), u=z1−0.25u=z_1-0.25, v=z2v=z_2, ρ=u2+v2ρ= u^2+v^2, ϕ=atan2(v,u)φ=atan2(v,u). Then, for ρ>0ρ>0, p(y∣x)=ϕ∈(−π2,π2)⋅1πρ⋅(ρ;0.1,0.012)Φ(10).p(y x)=1\! \φ∈ (- π2, π2 ) \· 1π\,ρ· N(ρ;0.1,0.01^2) (10). Appendix G Experimental Details for Bayesian Inference Benchmark Table 3 contains the set of hyperparameters tried for all methods, and table 4 contains the chosen hyperparameters to maximize distributional fit. All combinations of hyperparameters were tried, restricting to combinations satisfying N⋅K≤105N· K≤ 10^5, which restricts the total number of likelihood evaluations. CBG does not tune a guidance scale since we aim to sample from the true distribution p(x|y)p(x|y) at γ=1γ=1. Additionally, some other diffusion models do not use the maximum number of likelihood evaluations since they converge to the wrong distribution, and happen to get better results with less evaluations. 10410^4 reference samples are given, and the methods generate 10410^4 samples as well. For experimental details on likelihood-free methods, see Lueckmann et al. (2021). Table 3: Bayesian inference benchmark hyperparameter sweep. Methods N K γ C CBG (gradient-free) [101,105][10^1,10^5] [101,105][10^1,10^5] N/A N/A CBG (gradient-based) [101,105][10^1,10^5] [101,105][10^1,10^5] N/A N/A DPS [101,105][10^1,10^5] N/A [10−3,103][10^-3,10^3] N/A LGD [101,105][10^1,10^5] [101,105][10^1,10^5] [10−3,103][10^-3,10^3] N/A DPG [101,105][10^1,10^5] [101,105][10^1,10^5] [10−3,103][10^-3,10^3] [10−1,101][10^-1,10^1] SCG [101,105][10^1,10^5] [101,105][10^1,10^5] [10−3,103][10^-3,10^3] N/A Table 4: Bayesian inference benchmark chosen hyperparameters. Task 1 Task 2 Task 3 Task 4 Task 5 Methods N K γ C N K γ C N K γ C N K γ C N K γ C CBG (grad-free) 100 1000 – – 100 1000 – – 100 1000 – – 100 1000 – – 100 1000 – – CBG (grad-based) 100 1000 – – 100 1000 – – 1000 100 – – 1000 100 – – 100 1000 – – DPS 100 – 0.022 – 100 – 0.022 – 100 – 0.005 – 100 – 0.005 – 10000 – 0.001 – LGD 100 1000 2.154 – 10 100 10.000 – 10 1000 0.464 – 10 10 0.100 – 10 10000 0.464 – DPG 10 1000 10.000 10.000 10 1000 2.154 10.000 100 100 0.001 0.100 100 100 0.464 1.000 100 10 215.443 10.000 SCG 10 10 – – 100 10 – – 10 100 – – 10 100 – – 10 100 – – Appendix H Experimental Details for Black Hole Imaging We evaluate two configurations of CBG on the black-hole benchmark of InverseBench (Zheng et al., 2025). In both cases, we use the gradient-free estimator from Equation 20. We tune hyperparameters on the validation split and report all final numbers on 100 test samples. For baselines, we adopt the hyperparameter settings reported in Table 12 of the original InverseBench paper to ensure fair comparison. For the full setting in Table 2, we use the pretrained diffusion model released with InverseBench to sample from p(x∣xt)p(x x_t) with a DDPM inner loop. The outer denoising process uses N=1000N=1000 steps. We select the inner-loop length as M=50M=50 and the number of per-step samples based on validation performance. We use guidance scale γ=0.003γ=0.003 for the reported full-model result. Table 2 reports the TITAN RTX runtime used for the main comparison. As Table 2 shows, the compute time with the pretrained DDPM model is still too long to be practical. We therefore additionally evaluate the renoising strategy from Sections 5.3 and 21 using a one-step mean-flow model. Because the training dataset for the black-hole images is not publicly released, we use the pretrained diffusion model above to generate ∼140 000 140\,000 synthetic training samples, and train a mean-flow model (Lu et al., 2026) on this dataset. It has the same architecture as the DDPM model. In this renoise setting, proposal generation requires one model evaluation per sample instead of a 5050-step DDPM inner loop, plus one model call to denoise x^=ft(xt) x=f_t(x_t). We also reduce the outer denoising process to N=100N=100, which we found sufficient in practice, and sweep different values of K. The rows labeled renoise in Table 2 report this K-sweep, and the additional PSNR and χ2χ^2 ablations in Figures 5 and 9 use the same setup. (a) Closure-phase χ2χ^2 (lower is better). (b) Closure-amplitude χ2χ^2 (lower is better). Figure 9: χ2χ^2 data-consistency metrics for the K-ablation in Figure 5. Both scale log-log in K and improve monotonically past K=512K=512. Markers show per-K means with 95% confidence intervals over 100 tasks; red lines are log-log OLS fits to the means. Appendix I Experimental Details for Image Inverse Problems Super-resolution. We target 4× super-resolution on ImageNet (Deng et al., 2009) as a stress test of the Calibrated Bayesian Guidance (CBG) estimator on a high-dimensional inverse problem. For each ground-truth 256×256256×256 image we form a 64×6464×64 observation by bicubic downsampling; the likelihood is the Gaussian y∼(bicubic4×(x),σy2I)y \! (bicubic_4×(x),\, _y^2I ) with σy=0.01 _y=0.01, matching the SR3 benchmark used by Pandey et al. (2025b) and the baselines below. Super-resolution then reduces to drawing from the Bayesian posterior with this likelihood and an ImageNet prior. While guidance methods for ImageNet inverse problems typically use the classifier-guided diffusion prior of Dhariwal and Nichol (2021), the number of inner diffusion steps required for each candidate makes the overall budget prohibitive at the candidate counts CBG requires. We instead take the pixel mean flow (pMF) model of Lu et al. (2026) as the prior; specifically, the released class-conditional pMF-B/16 checkpoint at 256×256256×256. The model is one-step, and we can generate candidates with the fast renoising procedure of Section 5.3. Because pMF-B/16 is class-conditional but super-resolution is not, we feed the bicubic LR through the noise-aware 64×6464×64 ImageNet classifier of Dhariwal and Nichol (2021) (evaluated at t=0t=0) and condition pMF on the classifier’s top-1 prediction. Classifier-free guidance is used with ω=3.0ω=3.0. While CBG can be applied directly to super-resolution with the per-image MSE log-likelihood as the reward, that reward signal is very sparse: a single scalar per candidate must summarise the agreement between a full 256×256256×256 image and the LR observation. The gradient-free estimator then struggles to reconstruct fine detail unless compute is increased dramatically, because most candidates contribute essentially the same global reward and the per-pixel direction is averaged out. Intuitively, think of an image such as the White Stork in Figure 10 with the stork in the center, grass behind, and mud below. It is unlikely that a single sampled candidate will sample all three regions in a single composition; far more often, different candidates recover different parts. The job of the estimator is to recombine these partially correct candidates into one globally correct direction. Fortunately, the SR MSE loss decomposes as a sum of local pixel losses, so we can compute and weight the estimator on a per-tile basis. For 4×4× SR from 64×6464×64 to 256×256256×256 we use one tile per LR pixel, so each tile covers a 4×4=164×4=16-pixel patch of the HR image, giving 64×64=409664×64=4096 tiles in total. For each tile we evaluate the per-tile log-likelihood of every candidate and compute a tile-local softmax-weighted target; the global drift direction is then the soft stitch of these per-tile targets. This decoupling lets entirely different weightings be used for different regions of the image and is what makes the estimator usable on SR at a competitive compute budget. Table 5: 4× super-resolution on ImageNet (256×256256×256), evaluated on the 1000-image SR3 test list of Pandey et al. (2025b) (likelihood noise σy=0.01 _y=0.01). Paper rows are reproduced from Pandey et al. (2025b) (Tables 2 and 12); FID/KID are computed with clean-fid in legacy_pytorch mode, matching the paper’s evaluation code. Best per column in bold. Method PSNR↑ SSIM↑ LPIPS↓ FID↓ KID↓ Ours (pMF tiled gradient-free CBG with renoise) 24.41 0.668 0.298 40.74 0.0033 DPS 23.61 0.676 0.195 30.67 0.0021 DDRM 24.15 0.701 0.325 52.76 0.0151 RED-diff 24.06 0.685 0.354 51.83 0.0084 C-Π 23.20 0.631 0.270 39.96 0.0024 MPGD 21.83 0.587 0.215 37.39 0.0017 RB-Modulation 23.41 0.674 0.211 35.26 0.0032 NDTM 23.12 0.674 0.158 28.75 0.0011 Table 5 reports our run on the 1000-image SR3 list of Pandey et al. (2025b). CBG was run with K=2048K=2048 candidates per step, 8 outer denoising steps, and 64×6464×64 tiles, for a total of 16 38416\,384 pMF evaluations per image. Super-resolving a single image took on average 74s74\,s on 44 workstation-class GPUs (NVIDIA Titan RTX). Our run leads in PSNR and is competitive on SSIM, but trails the strongest perceptual baselines (NDTM, DPS, RB-Modulation) on LPIPS and FID. The fact that the prior model needs to be different between our method and the baselines, and that the number of diffusion steps is very different, remains as a limitation to compare the numbers. Figure 10: Super-resolution of an image of a White Stork. The ground truth image is 256×256256× 256 and gets downscaled to 64×6464× 64. For each of the eight denoising steps, four out of the 2048 candidates are shown. Out of these candidates, the tiled gradient-free estimator assembles a “direction”, and the noise is updated towards it. It is visible that the tiled gradient-free estimator can assemble an image early that follows the down-scaled version. The fact that the direction provides a strong signal allows for only 8 denoising steps. The rightmost column visualizes the per-pixel standard deviation of the proposals, where a brighter color channel signals higher diversity. Consequently, the area will be blue if the blue color channel disagrees, white if all disagree, and black when the candidates agree on that pixel. As the denoising process progresses, the samples are highly diverse, but converge towards the final image. Prompt alignment. We use the pretrained SANA-1600M model (Xie et al., 2025) with the classifier-free guidance scale of 4.5 and generate images with resolution of 512×512512× 512. We use a VLM-based reward function computed with the pretrained Skywork-VL-Reward-7B model (Wang et al., 2025). For each text prompt, we construct a corresponding binary visual question that checks whether the generated image satisfies the prompt-specific constraint. For instance, for the prompt “a photo of six dogs,” we ask: “Hint: Please answer the question with exactly one word, Yes or No. Question: Does this image depict exactly six dogs? Criteria: - Count all visible dogs in the image - The total number must be exactly six (no more, no less) Choices: (A) no (B) yes”. Following FMTT (Sabour et al., 2025), we define the reward function value as σ(logits[Yes]−logits[No])σ(logits[Yes]-logits[No]), where σ(⋅)σ(·) denotes the sigmoid function. We set N=20N=20, M=20M=20, K=256K=256, and select γ in the range [100,500][100,500]. Appendix J Additional Analysis for Black Hole Imaging Figure 9 shows the additional analysis of the relationship between K and performance, in terms of χ2χ^2-distance metric. We also visualize several samples for two random black hole imaging tasks in Figures 11 and 12. Figure 13 shows five samples that were generated by the mean-flow variant with renoise. Figure 11: Qualitative analysis of the generative results of the different methods with a random index, five samples were taken per method. The numbers are the PSNR to the ground truth for each task. CBG is the implementation based on the pretrained diffusion model (1000 steps outer loop, 50 steps inner loop, 512 particles) Figure 12: Qualitative analysis of the generative results of the different methods with a different random index, five samples were taken per method. The numbers are the PSNR to the ground truth for each task. CBG is the implementation based on the pretrained diffusion model (1000 steps outer loop, 50 steps inner loop, 512 particles) Figure 13: Blackhole reconstruction with the gradient-free pixel mean flow model compared to the baselines. Reconstructions were done with K=65 536K=65\,536