Fixed-Diffusion Bayes Reverse

• Fixed-Diffusion Bayes Reverse is a framework that fixes the forward diffusion process while using Bayes’ rule to derive the reverse dynamics across classical stochastic and quantum systems.
• It underpins methods in generative modeling and Bayesian inversion by correcting the drift using density-dependent score terms and tailored Monte Carlo techniques.
• The approach unifies various applications—ranging from fixed diffusion priors to Petz maps and reverse-transition kernels—yielding practical insights for simulation and inversion problems.

Searching arXiv for the cited papers and closely related works to ground the article. Searching for the primary paper on the quantum/Petz interpretation. Searching for fixed forward diffusion and reverse-time SDE/ODE formulations in diffusion theory. Searching for fixed-diffusion Bayesian inverse problems with diffusion priors. Searching for reverse-diffusion Monte Carlo and RTK/SMC formulations. Fixed-diffusion Bayes reverse denotes a family of constructions in which a forward diffusion law is treated as fixed and a reverse-time dynamics is determined by Bayes’ rule, or by a closely related quantum, variational, or Monte Carlo analogue. In the classical continuous-time formulation, the forward diffusion tensor is kept unchanged while the reverse drift is corrected by a density-dependent term, yielding the reverse diffusion equation that underlies modern diffusion-based generative models (Fu et al., 21 May 2026). Recent work uses the same structural idea in several adjacent settings: semiclassical limits of Petz-reversed Lindblad dynamics (Nasu et al., 21 Oct 2025), Bayesian inversion with a fixed pretrained diffusion prior (Scimeca et al., 12 Mar 2025), fixed-bandwidth reverse-diffusion analogues in attention dynamics (Smart et al., 28 May 2026), and reverse-diffusion Monte Carlo samplers built around fixed Ornstein–Uhlenbeck references (Huang et al., 2024).

1. Classical fixed-forward diffusion and the Bayes reverse

The classical core is a forward Itô diffusion

$$dx_t^\mu = f^\mu(x_t,t)\,dt + \sum_{\nu=1}^n g^\mu_{\ \nu}(x_t,t)\,\eta^\nu(t)\,dt,$$

with diffusion matrix

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$

whose law $P(x,t)$ satisfies the forward Fokker–Planck equation

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$

Bayes’ rule links forward and reverse conditionals through

$$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$

and, together with the backward Fokker–Planck equation, yields a reverse-time diffusion in which the diffusion tensor is unchanged and only the drift is modified (Nasu et al., 21 Oct 2025).

For fixed forward dynamics, the reverse-time marginal $\bar P$ obeys

$$-\partial_t \bar P(x_t,t) = -\sum_\mu \partial_{x_t^\mu}\!\big(\bar f^\mu(x_t,t)\bar P(x_t,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x_t^\mu}\partial_{x_t^\nu}\!\big(G^{\mu\nu}(x_t,t)\bar P(x_t,t)\big),$$

with reverse drift

$$\bar f^\mu(x_t,t) = f^\mu(x_t,t) - \frac{1}{P(x_t,t)} \sum_\nu \partial_{x_t^\nu}\!\big(G^{\mu\nu}(x_t,t)P(x_t,t)\big).$$

This is the most literal meaning of fixed-diffusion Bayes reverse: the forward diffusion coefficients $f,G$ are fixed, and Bayes’ rule canonically determines the reverse diffusion in law (Nasu et al., 21 Oct 2025).

In the common constant-diffusion case $G^{\mu\nu}=\sigma^2\delta^{\mu\nu}$, the reverse drift becomes

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$0

so the Bayes correction reduces to a score term. This is the continuous-time backbone of score-based generative models and DDPMs (Nasu et al., 21 Oct 2025). A differential-equations treatment reaches the same conclusion from a fixed conditional Gaussian forward process, deriving the exact reverse SDE

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$1

and the reverse probability-flow ODE

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$2

both determined by the fixed forward coefficients $$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$3 and the marginal score $$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$4 (Fu et al., 21 May 2026).

2. Reverse diffusion in generative modeling

Within diffusion modeling, the fixed forward process is usually chosen first, and the reverse process is then interpreted as the Bayes-consistent inverse of that noising law. A standard construction begins from a conditional Gaussian forward kernel

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$5

or equivalently

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$6

which induces a forward SDE

$$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$7

The reverse-time SDE is again the forward drift plus a score correction, and the noise-prediction objective is equivalent to score matching up to an additive constant independent of the model parameters (Fu et al., 21 May 2026).

This viewpoint makes the phrase Bayes reverse highly literal. The forward SDE defines a generative law on paths, and the reverse SDE is the posterior law of paths conditioned on the endpoint at time $$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$8. In that sense, the score term $$G^{\mu\nu}(x,t)=\sum_\lambda g^\mu_{\ \lambda}(x,t)g^\nu_{\ \lambda}(x,t),$$9 is the Bayes update that converts the unconditional forward drift into a posterior-consistent reverse drift (Fu et al., 21 May 2026).

A PDE formulation sharpens the same point. For the forward Ornstein–Uhlenbeck equation

$P(x,t)$0

the naive backward equation is ill-posed, but rewriting the anti-diffusion term via the score yields a stable reverse PDE

$P(x,t)$1

with corresponding reverse SDE

$P(x,t)$2

In this formulation, the reverse drift can be written through the conditional mean

$P(x,t)$3

so reverse diffusion becomes an explicit Bayesian inversion of the forward noise model (Cao et al., 28 Jan 2025).

The same framework also clarifies the relation between continuous and discrete samplers. DDPM training learns the same scaled score as the reverse-SDE formulation, DDPM sampling corresponds to discrete reverse-SDE sampling, and DDIM sampling corresponds to reverse-ODE sampling (Fu et al., 21 May 2026). A later numerical development adds an algebraically reversible solver for diffusion SDEs that can exactly invert real data samples into the prior distribution, giving a discrete pathwise realization of fixed-diffusion Bayes reversal (Blasingame et al., 12 Feb 2025).

3. Quantum generalization: Lindblad dynamics, Petz maps, and semiclassical reduction

A quantum version of fixed-diffusion Bayes reverse arises when the fixed forward dynamics is a Lindblad equation

$P(x,t)$4

and the reverse map is taken to be the Petz map relative to a reference trajectory $P(x,t)$5. For a channel $P(x,t)$6 and reference state $P(x,t)$7, the Petz map is

$P(x,t)$8

and it satisfies

$P(x,t)$9

For infinitesimal Lindblad evolution, this Petz construction yields a reversed Lindblad equation, so the Petz map plays the role of a canonical quantum Bayes reverse (Nasu et al., 21 Oct 2025).

The conceptual step in "Quantum Reversibility Meets Classical Reverse Diffusion" is to pass to phase space via the Wigner transform. Under a semiclassical approximation, the forward Lindblad equation yields a Wigner–Fokker–Planck equation

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$0

and the Petz-reversed Lindblad equation yields

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$1

with

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$2

The paper states that this equation takes the form of the reverse-time diffusion equation, establishing a direct correspondence between the Petz map and Bayes’ rule (Nasu et al., 21 Oct 2025).

In this semiclassical correspondence, the reference Wigner function $$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$3 plays the role of the forward density $$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$4. This unifies quantum reversibility and classical reverse diffusion at the level of generators: fixed Lindblad semigroup plus fixed reference trajectory on the quantum side, fixed forward diffusion plus forward marginal on the classical side (Nasu et al., 21 Oct 2025).

A more restrictive Gaussian analysis shows that this correspondence is not unconditional. For continuous-variable Gaussian Markov dynamics, complete positivity couples drift and diffusion at the generator level. For a one-mode quantum-limited attenuator with squeezed-thermal reference covariance $$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$5, the fixed-diffusion Wigner-score reverse is completely positive if and only if

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$6

and violates complete positivity iff

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$7

Any Gaussian completely positive repair must inject extra diffusion, implying

$$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$8

so score reversal is not free in Gaussian quantum dynamics (Fayad, 6 Mar 2026).

4. Fixed diffusion priors and Bayesian inversion

A second major usage of fixed-diffusion Bayes reverse appears in Bayesian inverse problems. Here the fixed object is not the forward noising law of a generative model, but a pretrained unconditional diffusion prior $$\partial_t P(x,t) = -\sum_\mu \partial_{x^\mu}\!\big(f^\mu(x,t)P(x,t)\big) +\frac12\sum_{\mu,\nu}\partial_{x^\mu}\partial_{x^\nu}\!\big(G^{\mu\nu}(x,t)P(x,t)\big).$$9, and the reverse is Bayes inversion with a known likelihood $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$0. The posterior is

$$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$1

and the task is to learn a posterior diffusion $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$2 such that

$$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$3

In this usage, fixed diffusion means that the pretrained prior diffusion is kept fixed; Bayes reverse means that this fixed generative model is combined with a likelihood to construct the posterior direction $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$4 (Scimeca et al., 12 Mar 2025).

"Solving Bayesian inverse problems with diffusion priors and off-policy RL" formulates this at trajectory level. If $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$5 is the fixed prior trajectory density and $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$6, Relative Trajectory Balance uses the objective

$$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$7

If this is driven to $$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$8, then the terminal marginal satisfies

$$P(x_t,t\mid x_s,s) = \frac{P(x_s,s\mid x_t,t)\,P(x_t,t)}{P(x_s,s)},$$9

so the learned reverse process is exactly the desired posterior sampler (Scimeca et al., 12 Mar 2025).

This formulation emphasizes that Bayes reversal can be imposed at the level of path probabilities, not only at the level of instantaneous drifts. It also makes the fixed-diffusion structure explicit: the prior trajectory law remains frozen, while a new posterior diffusion is trained relative to it. The same paper reports that existing training-free diffusion posterior methods struggle to perform effective posterior inference in latent space due to inherent biases, whereas RTB is asymptotically unbiased for the Bayesian posterior (Scimeca et al., 12 Mar 2025).

A related earlier line uses a fixed Ornstein–Uhlenbeck noising diffusion and its reverse-time law as a generic transport from a Gaussian reference to a target posterior or unnormalized density. Diffusion Schrödinger bridges then replace the long-time reverse diffusion by a finite-time entropic interpolation, while preserving the fixed-diffusion plus reverse-time viewpoint (Heng et al., 2023). This gives another concrete realization of fixed-diffusion Bayes reverse: start from a fixed forward diffusion, view the posterior as the initial law of that diffusion, and approximate the reverse-time transport back from Gaussian noise (Heng et al., 2023).

5. Reverse-transition, SMC, and fixed-bandwidth analogues

A third family of constructions treats fixed-diffusion Bayes reverse as a Monte Carlo design principle. In the Reverse Transition Kernel framework, reverse-time generation is decomposed into a small number of reverse transition kernel subproblems. For the Ornstein–Uhlenbeck forward SDE

$\bar P$0

the exact reverse kernel between two times is

$\bar P$1

so each reverse step is a Bayesian posterior with prior $\bar P$2 and Gaussian likelihood tied to the next state $\bar P$3 (Huang et al., 2024). The resulting RTK-MALA and RTK-ULD samplers replace many small Gaussian DDPM steps by a few larger, strongly log-concave Bayesian subproblems (Huang et al., 2024).

Reverse-diffusion Sequential Monte Carlo provides a complementary correction mechanism. There the reverse diffusion is used only as a proposal, while importance weights and intermediate targets correct for both score approximation and time-discretization error. The resulting sampler enables consistent sampling and unbiased estimation of the target’s normalization constant under mild conditions (Wu et al., 8 Aug 2025). A further exact Monte Carlo formulation embeds the target as the initial marginal of an OU diffusion over a finite horizon and derives a tractable Radon–Nikodym derivative for the reverse transition distribution with respect to that of an OU process; this yields samplers with neither time discretization error nor score function estimation, so that Monte Carlo variability is the only source of approximation (Batista et al., 3 Jun 2026).

A distinct but structurally related development appears in attention theory. "Attention as In-Context Empirical Bayes" studies a fixed Gaussian kernel bandwidth $\bar P$4 and finite depth horizon $\bar P$5, with mean-field dynamics

$\bar P$6

This is described as the fixed-bandwidth analogue of reverse diffusion: the score $\bar P$7 is replaced by the kernelized score $\bar P$8, and effective denoising does not require a time-varying noise schedule (Smart et al., 28 May 2026). Stage 1 refines a particle prior through interacting-particle dynamics; Stage 2 performs posterior averaging with the original noisy token as query. In that sense, a single attention step implements a kernel-weighted empirical Bayes posterior mean, while depth plays the role of reverse-diffusion time (Smart et al., 28 May 2026).

6. Conceptual status, limitations, and controversies

Current arXiv usage suggests that fixed-diffusion Bayes reverse is not a single formalism but a recurring structural pattern: hold the forward diffusion or diffusion prior fixed, and define the reverse through Bayes’ rule or a corresponding recovery principle. That structural unity coexists with several important limitations.

First, Bayes reversal need not be unique. "State retrieval beyond Bayes' retrodiction" argues that the Bayes-inspired reverse is just one case in a whole class of possible choices and can be optimized to retrieve the initial state more precisely than the Bayes rule (Surace et al., 2022). In the quantum setting, the same framework contains the Petz recovery map as a particular case, corroborating its interpretation as quantum analogue of the Bayes retrieval (Surace et al., 2022). This directly challenges the common intuition that the Bayes reverse is the only natural reverse map for a fixed irreversible dynamics.

Second, exact reverse diffusion does not by itself imply generalization. A PDE analysis of the Ornstein–Uhlenbeck forward/reverse pair proves that the reverse process’s distribution has its support contained within the original distribution. For a finite empirical distribution, the reverse dynamics are explicit and converge to the original samples; solving the minimization problem exactly is "too good for its own good" in the sense of yielding an overfitting regime (Cao et al., 28 Jan 2025). This suggests that practical generalization in diffusion models must arise from approximation, architecture, optimization, or discretization, rather than from the ideal reverse SDE alone.

Third, the fixed-diffusion principle may conflict with physicality in quantum Gaussian settings. As noted above, the Wigner-score reverse can fail complete positivity, and any Gaussian repair must add extra diffusion (Fayad, 6 Mar 2026). This separates the semiclassical Petz–Bayes correspondence from the stronger claim that a fixed-diffusion score correction is always physically realizable.

Finally, computational realizations inherit the usual trade-offs of reverse-time simulation. Reverse-diffusion samplers based on exact score fields can be expensive; posterior samplers based on fixed diffusion priors require nontrivial trajectory-level training (Scimeca et al., 12 Mar 2025); and path-space Monte Carlo methods replace discretization bias by estimator variance and acceptance-rate engineering (Wu et al., 8 Aug 2025, Batista et al., 3 Jun 2026). Even so, the fixed-diffusion Bayes reverse viewpoint remains a useful organizing idea because it isolates what is fixed, what is inferred, and where the reverse law obtains its canonical form.