Discrete Bayesian Sample Inference for Graph Generation
Authors: Ole Petersen
Marcel Kollovieh
Marten Lienen
Stephan Günnemann
Venue: ICLR 2026 Poster (2026)
Keywords: generative models
graph generation
diffusion models
bayesian flow networks
bayesian sample inference
molecule generation
Primary area: generative models
TL;DR: New discrete generative model based on Bayesian sample inference for graph generation.

Abstract: Generating graph-structured data is crucial in applications such as molecular generation, knowledge graphs, and network analysis. However, their discrete, unordered nature makes them difficult for traditional generative models, leading to the rise of discrete diffusion and flow matching models. In this work, we introduce GraphBSI, a novel one-shot graph generative model based on Bayesian Sample Inference (BSI). Instead of evolving samples directly, GraphBSI iteratively refines a belief over graphs in the continuous space of distribution parameters, naturally handling discrete structures. Further, we state BSI as a stochastic differential equation (SDE) and derive a noise-controlled family of SDEs that preserves the marginal distributions via an approximation of the score function. Our theoretical analysis further reveals the connection to Bayesian Flow Networks and Diffusion models. Finally, in our empirical evaluation, we demonstrate state-of-the-art performance on molecular and synthetic graph generation, outperforming existing one-shot graph generative models on the standard benchmarks Moses and GuacaMol.

Published as a conference paper at ICLR 2026
DISCRETE BAYESIAN SAMPLE GRAPH GENERATION
INFERENCE FOR
Ole Petersen" 1,4 Marcel Kollovieh" 1,2.3
Marten Lienen
Stephan Günneemann 122.3
School of Computation, Information and Technology. Technical University of Munich 2 Munich Center for Machine Learning
Munich Data Science Institute *Listen Labs
Correspondence to: m. w...llovieh@tum.cco
ABSTRACT
Generating graph-structured data is crucial in applications such as molecular gen- eration, knowledge graphs, and network analysis. However, their discrete, ordered nature makes them difficult for traditional generative models, leading to the rise of discrete diffusion and flow matching models. In this work, in- troduce GraphBSI, novel one-shot graph generative model based on Bayesian Sample Inference (BSI). Instead evolving samples directly, GraphBSI itera- tively refines belief over graphs in the continuous space of distribution param- eters, naturally handling discrete structures. Further, we state BSI stochastic differential equation (SDE) and derive noise-controlled family of SDEs that pre- serves the marginal distributions via an approximation the score function. Our theoretical analysis further reveals the connection to Bayesian Flow Networks and Diffusion models. Finally, in our empirical evaluation, we demonstrate state-of- the art performance molecular and synthetic graph generation, outperforming existing one-shot graph generative models on the standard benchmarks Moses and GuacaMol.
INTRODUCTION
Graph structures appear in various domains ranging from molecular chemistry to transportation and social networks. Generating realistic graphs enables simulation of real-world scenarios, augmenting incomplete datasets, and discovering new materials and drugs (Guo Zhao, 2022; Zhu et al., 2022). However, their unique and complex structure poses challenges to traditional generative models that are designed for continuous data such as images. This has resulted in a diverse landscape of graph generative models, featuring autoregressive models (You al., 2018) and one-shot models (Kipf & Welling, 2016), including a range of diffusion-basee models (Ho et al., 2020).
Recently, Bayesian Flow Networks (BFNs) (Graves et al., 2025) have emerged models that operate on the parameters a distribution over samples rather than on the samples them- selves. This approach is particularly appealing for discrete data, as the parameters of probability distribution evolve smoothly even when the underlying samples remain discrete. Graph generative models based BFNs have shown competitive performance in molecule generation (Song al., 2025). However, operating in parameter space and being motivated through information theory adds layer of complexity to the BFN framework that hinders its accessibility.
novel class of
Bayesian Sample Inference (BSI) (Lienen et al., 2025) offers a simplified interpretation and gener- alizes continuous BFNs by viewing generation as sequence Bayesian updates that iteratively refine a belief over the unknown sample. The model is trained by optimizing its corresponding ELBO.
This work introduces GraphBSI, extending BSI to discrete graphs. Instead of operating on discrete states, GraphBSI evolves on the probability simplex of node and edge categories. We derive BSI for categorical data and show how to generate variably-sized graphs with it. Next, we formulate
Equull contribution

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Published as a conference paper at ICLR 2026
0.00
0.25
t 0.50
t= 0.75
1.00
p(x 2:) = softmax(z:). with dz: 3 ((()) l)dt + (t)ddW.
Figure 1: Ilusstattoon of GraphBSI's generative process. Nodes and edges are modeled as indepen- dent categorical variables. One edge-type used to represent the non-existence of an edge. The latent variable represents distribution over graphs rather than graph itself. The neural network fo smoothly steers this distribution from random initial distribution Zg distribution concen- trated on valid graphs 21, which is modeled as Stochastic Differential Equation (SDE).
categorical BSI as an SDE and, via the Pokker-Planck equation, derive noise-controlled family of SDEs that preserves marginals while interpolating between a deterministic probability ODE and highly stochastic sampler. Empirically, we demonstrate that GraphBSI achieves state-of-the- art results on the GuacaMol (Brown et al., 2019) and Moses (Polykovskiy et al., 2020) benchmarks for molecule generation. In extensive ablation studies, we show that noise control is crucial factor for optimizing performance. An overview of our method is shown Fig. 1.
Our main contributions be summarized follows:
We derive BSI for categorical data, enabling, among others, the generation of graphs and sequences. The result generalizes the Bayesian Flow Network (BFN) framework with a simplified interpretation while avoiding limit approximations in the Bayesian update.
We formulate categorical BSI SDE. Through the Tokker-Planck equation, we derive a generalized SDE with a noise- contrrconno parameter and identical marginals, allowing us to interpolate between a deterministic probability flow ODE and sampling scheme that overrides all previous predictions with the most recent one.
We demonstrate that GraphBSI achieves SOTA results across most metrics in the Moses and GuacaMol molccule generation benchmarks with as few as 50 function evaluations, and further gains substantial improvements with 500 function evaluations.
THE BAYESIAN SAMPLE INFERENCE FRAMEWORK FOR GRAPHS
Bayesian Sample Inference (BSI) (Lienen et al., 2025) is a novel generative modeling framework simplifying and generalizing Bayesian Flow Networks (BFNs) (Graves al., 2025). While BSI was originally presented for continuous data, we develop a theoretical framework extending BSI categorical data analogously. We start by introducing the required background knowledge. All proofs are shown in Sec. D.
Background. Bayesian Sample Inference (BSI) (Lienen et al., 2025) generates samples by iter- atively refining an initial belief p(x) about the sample x be generated through noisy measure- ments of x. The initial belief p(x 2n) follows broad isotropic Gaussian with parameters Zo = -(4g (H4,0o). op). The belief is then refined by a sequence of noisy measurements follow Gaussians centered around x. After receiving the measurement y. the information contained integrated into our next belief 2;+1 through a Bayesian update. Once the belief of x is suffi- ciently sharp, we return a sample from it. We train neural network fe predict the train sample x from the information collected about it the belief z, for each timestep i E O,... k 1. The samples during inference by creating the noisy
that
trained neural network allows us to generate measurements through an approximation &; = fo(2., i) of the sample x in each timestep i.
Extension to categorical data. Now. we will focus on the case that our data lies on the simplex, ie., we have categorical belief for over possible categories, i.e., A C [0, 1 If
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have access to noisy measurements y; Mx, Σ² 2 = II) of the sample x, can infer from the measurements using Bayes' theorem in a similar fashion to the continuous case. We start with an initial belief p(x Zo) Cat(softmax(zo)) where 2o Raxe are the logits of categorical distribution with n independent components. Then, we can update the belief parameters 2 after observing y: using Bayes' theorem.
Theorem 1. Given prior belief p(x z) Cat(x softmax(z)), after observing ~ (y μ x, Σ² = a II) at precision a, the posterior belief is p(x 2,y, = Cat((x softmax (ppst) with
post + oy.
(1)
Now, we can iterate over multiple noisy measurements and update our belief until p(x .1.....3. y.) identifies with high probability. Through Theorem 1, encode the information contained in all these measurements in our updated belief parameters 2₁ as p(x yl,...,ys) = p(x Zx) ~
(st(sottmax((z)) with n Ze E2 a.y... a.y..
We process each observation y, sequentially, inducing a notion of time. We measure at time At E [0, 1] with At 1/(k + 1)), and the subsequent Bayesian update takes us f,+1- To control the total amount of information added to the belief p(x 2,) up to time t, we define a monotonically increasing precision schedule 3: [0. - R. The measurement y, contains the information added in the time interval [t,, f++ and therefore we choose = B(t++1) ) (t+))
R
Note that the update the logits in Theorem BSI. Here, the belief components accumulate in each update, whereas in the continuous case, the update is interpolated with its previous state.
fundamentally different than that of continuous
Generative model construction. We build a generative model for categorical data given the above procedure, similarly as done for BSI with continuous data (Lienen et al., 2025). We begin with a logit zu defining the initial belief of the sample that will generate in the end, with 2g N(po 8o) As x is unknown a priori, we cannot measure it, so instead we estimate it from the information
sampled from a simple prior distribution.
have gathered so far encoded in latest belief. Let fo : Rxxx x [0, 1] + Ag_ be a neural network with parameters 0 estimating the unknown sample & fe(z,, 4), followed by noisy measurement y; N(; Σ a, centered around X; with precision a;- Then, we update our belief with y; via Theorem 1. Now, we repeatedly predict &, measure y,, and update the belief parameters Z <- z, + o;y; until our belief is sufficiently sharp at = Finally, we return a sample from Cat(x softmax(z;)) See Alg. for a formal description.
behind our observations given our current belief z, and time We estimate
Evidence Lower Bound. To train our neural network, we interpret CatBSI a hierarchical la- tent variable model to derive an evidence lower bound (ELBO) of the sample likelihood (Kingma & Welling, 2013), providing a natural training target. As latent variables, we choose the beliefs Zo. Their distribution in Alg. factorizes, allowing us to write
p(x)
E
lp(x 2x)]
p(2.) IT. (2..12).1.8)
As encoding distribution qZz,, ....... 2i x), we choose the distribution induced under Alg. with
fixed reconstruction fz(2. 1) x.
Thanks the simple form of Theorem 1, it is straightforward
to compute the marginal g(Z: x):
2: = Zo +
a,y;~ ~ + (())) E = -00+00 B(4))
(3)
1=0
Equipped with this, we can derive the following ELBO:
Theorem 2. For categorical BSI, the log-likelihood of x under Alg. is lower bounded by
k
logp(x)
E 21 2x/0(2/xx11)
Dogp(x zx)l
E [(3(1++1) -8(4)))l.fo(z, x113],
(4)
2
1-110,1-1
2,
a.>g(2/x..t.
where q(2) x,() -V2 140 3(1xx Bo + B(1)T) and p(x (R) Cat(x softmax(zz))
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Algorithm 1 Sampling with Categorical BSI
Algorithm 2 Training Categorical BSI
Require: reconstructor Se. discretization k,
while not converged do
precision schedule 3: (0, 1| + R
x ~ p(x)
Zo ~ [(uo (b1)
20 N [No: BoT)
for i = 0, O....,k k do
t ~ u(0, 1)
& - fo(z,, t2)
a 3(t) 3(0)
0; + B(1++1) ((4))
y ~ Ni = x, x, =1/a-11
yr ~ Ni = K,, E² = ai .I.
Z 2o + Fa.y
Z,+1 2, + a,yi
X fa(z.t)
end for
(1)1 || x11
Cat (al(soflmax(zz)
0 = 0 VoL
return x:
end while
The first term does not depend on 0 and therefore cannot be optimized; only need to minimize
the second term. For k + oo, we have that ((((t++1) - 3(t,)) + 3 (ttt) since At = t,+1 t = 1/(k + 1) = 1/k, and t 4400 1). Maximizing the ELBO for k oo over the dataset above is therefore equivalent to minimizing
C
E x>p(x)
[B((t/2 ((f))2--lfo(z,tt x
(5)
t~(1011)
2=x(2
The loss above immediately yields the training procedure Alg. 2. This matches the continuous-ting time categorical BFN loss up a constant when 30 + i.e., the prior is Dirae delta at / O.
Adaptation for graphs. We represent graphs with nodes as tuples (X, A), where E AN C [0, 11]xxxx are the one-hot encoded categories of each node and A E ANN C 1j the one-hot encoded categories of each edge, with the first category denoting the absence of edge. We treat each node and edge belief, allowing us to apply the categorical BSI framework to graphs. Note that dependence between edges is introduced via our network f- We choose a permutation invariant reconstruction network fe. resulting in a permutation invariant generative model when the noise is isotropic.
CA-1
independent component of the categorical
To enable a varying number of nodes in the graph, we first sample a number of nodes N from the marginal node count distribution, and subsequently generate the node and edge values. In practice, this is achieved by masking out inactive nodes and edges for train graphs with fewer nodes.
Adaptation for sequences a general discrete generative model, Categorical BSI is applicable for sequence generation, too. Here, sequence S of length / with vocabulary size e is represented in the one-hot-encoded format S E A, C 10 We include an exemplary implementation trained on DNA sequences in Sec. B.
CATEGORICAL BSI AS A STOCHASTIC DIFFERENTIAL EQUATION
In this section, we analyze the update equation in Theorem and take the infinite-step limit, ob- taining SDE. We then introduce parameter that controls the stochasticity and yields a family of SDES with identical marginals.
SDE Dynamics. First, we notice that as the number of steps k increases, i.c., At = 1/(k+1) + 0, the updates in Theorem converge the following SDE. Theorem 3. As At the update equation Theorem converges to the following SDE:
dz, ft( t)dt + (Br())W.
(6)
where dW, is Wiener process and 2n N No T).
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0.00
0.33
t= 0.67
[ == 1.00
p(x zi) Y = 0.0 (ODE)
y = 0.1
7 = 1.0 (Vanilla)
2.0
Figure 2: Trajectories of the SDE Theorem for different values of with three classes and fixed reconstruction fo(2,, t) = é2. At 7 O, the sampler resembles a probability flow ODE as in flow matching. Increasing leads to noisier trajectories. At 1, the original SDE in Theorem is recovered, and increasing the noise further makes the trajectories even more volatile. The density function of the marginal distribution p(x zz) (shown in the background) is identical for all 7
Note that while the distribution of Zg is not required to be normal for Theorem 3 itself, it is necessary for the following steps. Phrasing the evolution of the latent variable 2, as an SDE enables the use of more advanced sampling schemes and allows us to derive generalized SDE family. The original discrete update in Theorem recovered by applying an Fuler-Maruyama discretization of Eq. (6).
Generalized SDE. We now generalize Eq. (6) to family that preserves the marginal probability paths Pr(z.) while controlling stochasticity via the parameter ", similar to Karras et al. (2022):
Theorem The SDE in Theorem is generalized by the following family of SDEs with equal marginal densities (z:):
dz. = B t)dt 7 7-1 Vl) (z)d (iBi
2
where dW, is a Wiener process and Zo ~ NN 90-0,,0-1) 1).
Setting 0 yields a deterministic probability flow ODE, equivalent to Xue et al. (2024). Unlike BFNs, however, CatBSI samples the prior belief p(z t= 0) rather than choosing a fixed prior, naturally avoiding the discontinuity around 0. Further, choosing y 1 recovers the original SDE in Theorem 3, and larger produces more stochastic trajectories. We visualize in Figs. 2 and how varying 7 affects the dynamics for toy examples. Although the marginal distributions for all in theory, the empirical performance varies as Vz. log Pr (z,) is not available in closed form. Higher stochasticity allows the model correct errors made in previous sampling steps but requires finer discretization (see Sec. 4.3). In the limit y + oo, the sampler effectively overwrites the current state completely in every step (see Sec. C.3). To turn Eq. (7) into a practical sampling algorithm, we approximate the score function Vz, log (z,), as described in the following.
equal
Theorem 5. The BSI loss Eq. (5) also is a score matching loss with the score model sq sg(z, t) param- eterized as
se (0(2)) 140 f(,,))f(z,t))2 2 V pz(z)
3(1) + 30
Discretization and integration. As the SDE is not solvable in closed form, we resort to numerical sampling. While a simple Murer-Maruyama (EM) approach performs well on sufficiently fine time grids, we find that integrating a locally linearized SDE within each step can improve sample quality for low numbers of neural function evaluations (see Sec. 4.3). More specifically, we freeze the reconstructor & = fo (Z,, !) over the time interval [t, L AtJ. representing an Ornstein-Ullenbeck process. This allows us to solve the SDE analytically within this interval.
Theorem 6. Fixing the prediction x folzz, and the values B(t+ At/2), B (t++ in Eq. (7) in a time interval [t. t + At] yields an Ommstein-Ullenber Uhlenbeck (OU) process with the exact marginal
7/3' (1 c 2xdr)) -N(0,
Zt4 At ~ m + (z. mme- +
2x
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Algorithm 3 Euler-Maruyand Sampling
Algorithm 4 Ornstein-Uhlenben Sampling
Require: reconstructor fo. discretization At,
Require: reconstructor fu. discretization Al,
precision schedule 3: [0, +
precision schedule 3 [0, 10 RR .7 1
(uo, Bb1)
z ~ (no, Bo1)
for t = 0, At,
At do
for = At/2, At + At/2, 1 do
& fo(2,t)
X fo(z, t)
(1-1))0(1)
8g Bit)+Po
K - 2[c++((e]1
μ 5 (t)
8-1 50)
Mu (3(t) (tt)/K)&
2
- (t)
f- 23
2x (1 - -2x2t)
z + HÂt + LV I)
(z m)e xat +V22.N
end for return (uaatize([o(zz 1])
end for
return Quantize(So((2
17-1)8 where x 2[ m = + ( /'K)
Note that the OU discretization converges towards the EM scheme for At - 0 (see Sec. C.1).
is sufficiently sharp, the final sampling step in Alg. 1 is de facto deterministic. However, this presents an opportunity to improve sampling efficiency:
Quantizing instead of sampling. If the belief precision
the last few steps, simply sampling from the belief would yield too noisy samples, but the belief contains enough information so that the reconstructor can make perfect reconstruction of it (see Fig. 5). Therefore, we can instead stop at lower final precision and return reconstruction projected on the sample space through quantization. Employing the discretization schemes yields Algs. 3 and
We also allow nonuniform time grid. Following Karras al. (2022), we introduce parameter that controls the distribution of function evaluations over the time grid:
t =
i= 0, k.
(10)
Here, o 1 recovers a uniform grid; larger p concentrates steps near the beginning (t 0), whereas smaller concentrates them near the end (t x 1).
4 EXPERIMENTS
In this section, we present our empirical results. We benchmark our model against state-of-the-arr baselines from the diffusion and flow-matching literature on unconditional molecular and synthetic graph generation. The GuacaMol and Moses benchmarks for molecular generation (Brown et al., 2019; Polykovskiy et al., 2020) serve as our primary evaluation datasets. Additionally, we conduct ablation studies to analyze the impact of various components and hyperparameters on the model's performance. Further, we report results on the synthetic planar, tree, and stochastic block model graph generation tasks (Bergmeister et al., 2024; Martinkus et al., 2022).
4.1 EXPERIMENTAL SETUP
Datasets. To test performance on real-world graphs, we train GraphBSI on the Moses (Polykovskiy et al., 2020) and GuacaMol (Brown et al., 2019) datasets for molecular generation. We extract graphs out of the dataset smiles with RDKit RDKit (2025) and construct the node fea- tures X and adjacency matrix in the format described in Sec. 2, where atom- and bond types correspond to node- and edge categories, respectively. Further, we include results for the planar, tree, and stochastic block model (Martinkus et al., 2022; Bergmeister et al., 2024) synthetic graph generation datasets. Find summary in Tab. 8.
Evaluation metrics. We follow the standard evaluation practices as established by Polykovskiy al. (2020); Brown al. (2019); Preuer al. (2018) for molecule generation and Martinkus et al.

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Published as a conference paper at ICLR 2026
(2022); Bergmeister et al. (2024) for synthetic graph generation. Find detailed description in Tabs. 9 and 10.
Practical considerations. The reconstruction network fa parameterized using the same graph transformer architecture as Qin et al. (2025); Vignac et al. (2023), with the node- and edge logits and class probabilities, entropy, random walk features, and sinusoidal embeddings (Vaswani al., 2017) of the timestep t with frequencies proposed by Lienen et al. (2024) as features. Empirically, we find that an exponential precision schedule with final precision that allows for a near-perfect reconstruction maximizes performance (see Tab. 7 and Figs. 5 and 7). For both latent node- and edge classes, we choose normal prior with the marginal distribution over the dataset and a small variance of 1.0. Finally, apply a preconditioning scheme where the neural network predicts the difference between the belief and the true sample, setting fo(2, t) softmax(2 + fo(2,t)..
Evaluation After training to convergence, we evaluate the benchmark metrics for both discretiza- report results with
tion schemes Algs. 3 and 4. For both molecule generation benchmarks, compute budget of 50 and 500 discretization steps. In each of the four configurations (2 discretiza- tion schemes, 2 numbers of steps), we optimize the noise level y and report the best result. Find the final configurations in Tab. 7. For the synthetic graph generation benchmarks, we report results with the best-performing noise level and the Ornstein- -hheebee discretization with 1000 function evaluations.
4.2 RESULTS
Molecule Generation. As illustrated in Tab. 1, GraphBSI is competitive with 50 steps with both discretization schemes for both molecule benchmarks, achieving tfat--of-the-ar results on the ma- jority of the metrics. Notably, GraphBSI outperforms DeFoG with both discretization schemes on all metrics except novelty on Moses. On most metrics, the OU discretization performs better than the EM scheme. At the full 500 steps, GraphBSI with the OU discretization outperforms all existing models on all metrics on GuacaMol, saturating validity and consistently exceeding the state-of-the- art. The EM scheme performs slightly worse than OU most metrics, but remarkably surpasses the state-of-the-an on the FCD metric, reducing it from 1.07 to 0.72 on Moses. Find an extended comparison in Tab. 5.
Table Results on the GuacaMol and Moses benchmarks for molecular generation with 50 and 500 (OU) discretization.
sampling steps and the Malle-Maruyama (EM) and
GuacaMol
Moses
Model
Steps
Val.
V.U. 1
V.U.N.
KL 1
FCD
Val.
Uniq.
Nov. 1
Filters
FCD
SNN
Scaf
Train Set
100.0
100.0
0.0
99.9
92.8
100.0
100.0
0.0
100.0
0.01
0.64
99.1
DeFoGi
50
91.7
91.7
91.2
92.3
57.9
83.9
99.9
96.9
96.5
1.87
0.50
23.5
GraphBSI (EM)
50
27.5
97.5
27.2
90.7
65.6
99.3
100.0
26.5
96.9
1.06
0.50
15.2
GraphBSI (OU)
50
99.2
99.2
98.7
93.7
71.3
99.7
100.0
94.6
98.2
1.19
0.52
15.1
DiGress
500
85.2
85.2
85,1
92.9
68.0
85.7
100.0
95.0
97.1
1.19
0.52
14.8
DisCo
500
86.6
86.6
86.5
92.6
59.7
88.3
100.0
97.7
95.6
1.44
0.50
15.1
Cometh
500
98.9
98.9
97.6
96.7
72.7
90.5
99.9
92.6
99.1
1.27
0.54
16.0
DeFoG
500
99.0
99.0
97.7
73.8
92.8
92.1
98.9
1.95
0.55
14.4
GraphBFN
98.5
98.3
1.07
0,59
10.0
GraphBSI (EM)
94.6
99.8
100.0
99.1
14.3
GraphRSI (OU)
99.6
80.3
99.9
100.0
12.7
Synthetic Benchmarks. As shown Tab. 2, GraphBSI achieves competitive results on the syn- thetic graph generation benchmarks. Our model saturates validity on the planar- and tree graph generation tasks, and achieves adequate validity on the stochastic block model graphs. The mean ratio measure of distribution similarity is competitive on all three datasets, even though the metric should be taken with grain of salt due to the small dataset size of only 128 graphs, resulting in high uncertainty in the evaluation.

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Table Results the synthetic graph generation benchmarks. Like DeFoG, we generate 40 graphs five times and report the mean and standard deviation over the runs.
Planar
Tree
SBM
Model
Steps
V.U. 1
Ratio
V.U.
Ratio
V.U.
Ratio
Train Set
100.0
1,0
100.0
1.0
85,9
1.0
DeFoG
50
95.0 + 3.2
3.2 1,1
73.5 a: 9.0
2.5 1.0
86.5 #.3
2.2 # 0.3
GraphBSI (EM)
50
7.5 1.0
47.3 4.3
7.0
2.1 0.8
4.2 1.4
GraphBSI (OU)
50
18.5 8.6
3.2
96.0 2: 1,2
2.5 0.9
53.0 a: 7.5
51.4 4.0
HSpectre
256
95.0
2.1
100.0
4.0
75.0
10.5
DiGress
1000
71.5
5.1
90.0
1.6
60.0
1.7
DeFoG
1000
99.5 # 1.0
1,6 1.0
96.5 2.6
1.6 + 0.4
4,9
GraphBPN
1000
96.7
87.5
GraphBSI (EM) 1000
100.0
3.8 1.0
2
1.3 0.4
50.5
11.3 1.4
GraphBSI (OU) 1000
100.0
3.2
100.0 0.0
1.8 0.5
77.5
4.6
FCD (4)
SNN Score (t)
100% Filters (L)
2
Normalized Score
2
20
100
250
1000
20
100
250
1000
20
100
250
1000
Noise Level (n)
Noise Level (7)
Noise Lovel (?)
Figure 3: Normalized metries (zero mean, unit variance) noise level for different numbers of function evaluations (FE) and discretization schemes. Our custom
discretiza- tion scheme denoted OU, while the standard Euler-Maruyama scheme written Euler. Some values for the Euler scheme are missing since the sampler becomes unstable if ? At becomes too large (see Sec. C.2).
4.3 ABLATION STUDIES
Noise level. To test the effect of the compute budget, noise level, and discretization scheme on performance, we conduct a grid search over the number of function evaluations (NFEs) in (25,50, 50, 100,200 500). noise levels in {0.0, 5.0, 20.0, 100.0, 250.0, 1000.0), and both discretiza- tion schemes on the Moses dataset. As shown in Fig. 3, performance in both discretization schemes expected since both discretize the same SDE.
closely related low noise levels, which Higher compute budgets lead to better performance. However, the Fuler-Maruyama scheme be- comes unstable at higher noise levels, leading to significant drop in performance (see Sec. C.2). In contrast, the Ornstein- Uhlennb--Uleeeekeeeee scheme remains stable, and both the SNN score and Filters met- ric benefit from higher noise levels. The FCD metric optimal 20 and 100. With a few exceptions. the PrnsteinUUhlebbec. scheme matches outperforms the Euler-Maruyama scheme at all compute budgets and noise levels. Novelty suffers from increased noise levels and compute budgets, which is consistent with the model generating samples closer to the training data distribution. Notably, all metrics perform poorly at noise level of 0.0, which corresponds the probability flow ODE (equivalent to Xue et al. (2024)). Fig. 8 illustrates that optimizing the noise level is a key driver in the performance gains of our model.
medium noise level between
Non-uniform timesteps. To test whether a fine discretization is more important at some timesteps compared to others, we analyze the effect of non-uniform timestepping, putting a finer discretization at either earlier or later timesteps. As shown in Fig. 4, SNN and Moses Filters remain mostly unaffected by the choice of p; only the FCD displays clear trend. A finer discretization later timesteps (p < 1) improves the FCD at 50 function evaluations in both discretization schemes and 500 evaluations in the Omsteenn Uhlenbeck scheme.
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50 Steps
500 Steps
15
Change p Lin
5
FCD
Filt f
-5-
OU
-15
0.5
1.0
1.5
2.0
Performance change for changes in the non-uniform timestepping parameter / in 4, (i/k) for 0, o......, &o compared to the uniform case P 1. results in a finer discretization at later timesteps, while > corresponds to finer discretization at earlier steps.
Figure 4:
Precision schedule. We find that while an exponential precision schedule yields the best results, the difference compared to a simple linear schedule is negligible (see Tab. 5). One parameter that significantly affects performance is the final precision [(t = 1). As illustrated in Fig. 7, an exces- sively large final precision wastes sampler iterations in the final steps, and a too small final precision results in noisy samples. Ideally, the reconstructor is just able to predict the train samples flaw- lessly at (t = 1). Finally, we isolate the effect of sampling the belief at = O instead of taking fixed value, as with BFNs, by training a new model with a smaller initial variance of compared to the standard 3p = 1.0. Tab. 5 shows that for both values of 3a, the OU sampler outper- forms the Flowback (Song et al., 2025) sampler on most metrics. Surprisingly, the performance of the Flowback sampler drops significantly when Sp is increased, while a higher value of 3o improves performance for the OU sampler.
0.05,
We conclude that two key factors are crucial for the performance gains of GraphBSI: First, the noise control, and second, a final precision that is just high enough for perfect reconstruction. The exact precision schedule and non-uniform time-stepping show only marginal contribution.
RELATED WORK
Graph generation presents three main challenges compared to image and text generation: (1) graphs discrete structures, unlike images, which are continuous; (2) graphs have a variable shape, with both the number and arrangement of nodes and edges changing across samples, unlike the fixed dimensions of images; and (3) nodes in graphs lack a natural order, in contrast to text, where tokens follow a well-defined sequence. Various approaches have been proposed to tackle these challenges.
Autoregressive models have proven successful in text generation by sequentially predicting the next token based on previous ones (Brown et al., 2020). Applied to graphs, these models generate nodes and edges by one, maintaining the graph structure they proceed. This approach has been used for tasks such as molecule and social-network generation (You et al., 2018; Liao et al., 2020). However, autoregressive models violate permutation invariance by relying on a specific node ordering.
One-shot models address the ordering challenge by generating the entire graph in single step, without relying on specific node ordering. Examples include Variational Autoencoders (Kingma & Welling, 2013), GANs (Cao Kipf, 2022), normalizing flows (Liu et al., 2019), and discrete flow matching (Gat et al., 2024; Qin et al., 2025).
Diffusion models have emerged as a powerful class of one-shot generative models for continuous data such as images (Sohl-Dickstein et al., 2015; Ho et al., 2020). Their core idea is to learn a gen- crative process that gradually transforms noise into clean data by reversing a diffusion process with neural network. Noise is typically applied independently to each pixel in images to each node in graphs, naturally resulting in a permutation-ing invaaiiaann model when combined with a Graph Neural Network (GNN) (Niu et al.. 2020). A variable number of nodes diffusion process on the node count, e.g., by first sampling a node mask and then applying diffusion to the masked graph (Niu et al., 2020; Qin al., 2025). To improve scalability, hybrid methods that
be handled by conditioning the

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reverse coarsening process and generate local structures with diffusion model have also been proposed (Bergmeister et al., 2024).
Discrete diffusion addresses the discreteness of graphs. The most straightforward approach relaxes discrete data to a continuous space, applies diffusion, and quantizes the generated outputs back to the discrete space in final step (Niu al., 2020; Jo et al., 2022; 2024). Alternatively, use discrete diffusion which the state is perturbed via a Markovian transition matrix in discrete time steps (often including an absorbing state) (Austin al., 2023); this has been applied graphs (Vignac et al., 2023; Haefeli et al., 2023). A related recent approach uses a continuous-time Markov chain for the discrete diffusion process (see (Campbell et al., 2022)), which allows more flexible sampling on graphs (Siraudin et al., 2024; Xu et al., 2024).
can
Bayesian Flow Networks Graves et al. (2025) propose conceptually distinct approach discrete generative models: diffusion is applied to the parameters of distribution over samples rather than to the samples themselves. BFNs can be interpreted as an SDE, enabling more efficient sampling algorithms (Xue et al., 2024). This provides solid theoretical foundation for diffusion on discrete data while retaining the benefits smooth parameter changes, and it achieves competitive perfor- mance on protein and graph generation (Atkinson et al., 2025; Song et al., 2025; Tao & Abe, 2025). The flexible design of BFNs also permits joint generation of continuous and discrete quantities, for example the 3D positions, atom types, and charges in molecular generation (Song al., 2024).
Bayesian Sample Inference Lienen et al. (2025) extends BFNs by adding a prior over the distribu- tion parameters and offers simplified interpretation for the continuous-data case. Kollovieh et al. (2025) used the BSI framework to derive their generative model for hierarchies. However, they do not generalize the framework, i.e., do not derive SDE-based sampling algorithms, and do not optimize an ELBO as they specifically focus on hierarchy generation.
CONCLUSION
In this work, introduce GraphBSI, novel generative model for graphs based on Bayesian Sam- ple Inference with state-of-the-ar performance in large molecule generation benchmarks. Similar to Bayesian Flow Networks, GraphBSI iteratively refines a belief over the graph structure, modeled as categorical distribution over adjacency matrices, through Bayesian updates. We show that in the limit of infinitesimal time steps, GraphBSI converges to a Stochastic Differential Equation (SDE). Further, we employ the Fokker-Planck equation to derive generalized SDE with tunable noise pa- rameter, allowing us to interpolate between a deterministic probability flow ODE, the original SDE, and substantially more volatile sampler. We demonstrate that GraphBSI achieves state-of-the- of-tth-art art performance on the GuacaMol and Moses benchmarks for large molecule generation, outperform- ing existing models nearly all metries. Finally, in our ablations we empirically show that noise control critically influences performance.
Limitations and Future Work. GraphBSI, in its current implementation, suffers from the quadratic scaling of compute and memory requirements the number of nodes that comes with the application of graph transformer. Exploring a more memory-effiiint graph neural network ar- chitecture to generate larger graphs would be a promising avenue for future research. Further, while GraphBSI allows for variable-sized graphs, the number of nodes sampled beforehand instead of jointly generated with the graph features. Allowing for nodes to appear or disappear while gener- ating the graph. similar to jump diffusion (Campbell et al., 2023), might result more flexible generative process.
ACKNOWLEDGMENTS
This paper is supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Research, Technology and Space."
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A RELATIONSHIP TO BFNS AND DIFFUSION MODELS
A.1 RELATIONSHIP TO BFNS
There is close equivalence between Categorical Bayesian Sample Inference (BSI) and Categorical Bayesian Flow Networks (BFNs). In fact, Categorical BFNs can be seen as a special case of Cate- gorical BSI with specific choice of prior distribution and noise schedule. The dynamics of BFNs are recovered when choosing the sampler in Eq. (7) with y = 1 and ßo = O to parametrize 2o. i.e., making the prior logits deterministic. Note that require 30 > 0 to avoid numerical issues when approximating the score function. This generalized SDE allows BSI to vary stochasticity. Intuitively, increasing stochasticity allows the model to overwrite errors from previous predictions (see Sec. C.3 for a discussion on the extreme case), and empirically, increasing stochasticity proves crucial for performance Fig. To illustrate this, we will show the relationship between the components of both frameworks.
Input Distribution Both BFNs and categorical BSI parameterize the distribution over the data x using categorical distribution. The logits are denoted as z in BSI and as 0 in BFNs. In BSI, the parameters z are the logits
categorical distribution, i.e., p(x z) ~ Cat(softmax(()). In BFNs, the parameters 0 are the probabilities of each category, i.e., p(x x 0) ~ Cat(O). The two parameterizations are equivalent since 9 = softmax(z) and = lo) (up to an additive constant).
Output Distribution The output distribution in BFNs is intermediate distribution that is not needed in BSI.
Prior Distribution While Categorical BSI includes normal prior distribution over the logits of the categorical distribution (p(z t = 0) ~ M(po ( Categorical BFNs fix the parameters to 8, 1/K. Therefore, categorical BFNs can be seen special case of categorical BSI with no 0 and Bn = O.
Sender Distribution The sender distribution in categorical BFNs is an intermediate distribution that not required in categorical BSI.
Receiver Distribution The sender distribution in categorical BFNs is given as
pn x,
Σ V 1),
corresponds to the noisy measurement distribution in categorical BSI, ply xxaa)~ ~ ((t) Note that for + 0, it holds that:
pn(y | a) ~ (Na ((((Ksoltmax(((0) - (1)(())))).kK))) a).
The sender distribution for a + 0 is an affine transformation of the noisy observation function for BSI: we set y ~ p(y xx = (xx, 1/al) and compute y' aKy 1), then y' ~ Pe(y' x. a), where softmax (((axxx(((())) (f)) corresponds to the sample reconstruction x. Thus, in the small-a-limit two distributions have same-order approximation and therefore contain the same information. However, in the formulation of categorical BSI, we can directly see that is noisy observation of x and we do not require computing the distribution as a limit multinomial distribution as in BFNs.
Bayesian Update Function The Bayesian update function categorical BFNs (Graves et al., 2025, Eq. 171) is the equivalent of Theorem 1 in categorical BSI. The update is simplified for BSI since the belief parameters are in logit space instead of probability space. Furthermore, the scaling of the receiver distribution leads to extra factor of in categorical BSI.
Bayesian Update Distribution This an intermediate that is not required in categorical BSL.
Accuracy Schedule The accuracy schedule can be chosen freely in categorical BSI. In categorical BFNs, the accuracy schedule is chosen as B(t) = 32)
Bayesian Flow Distribution The Bayesian flow distribution in categorical BFNs corresponds to Eq. (3) in categorical BSI. The two distributions are equivalent up to an affine transformation of the variable, explained above.
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Continuous Time Loss The continuous time loss in categorical BFNs (Graves et al., 2025, Eq. 205) corresponds to Eq. (5) in categorical BSI. Both are the L. loss between the reconstruction and the one-hot encoded data.
SDE formulation Both BSI and BFN sampling can be formulated as SDEs. Here, Theorem corresponds to (Xue al., 2024, Eq. 24). To do so, the authors also operate the logits of the categorical distribution instead of the probabilities.
Score function approximation The score function approximation for categorical BFNs (Xue al., 2024, Eq. 28) corresponds to Theorem 5 for 30 = 0 up to a constant. Note that a value of 30 > 0 avoids the division by zero in the score function approximation 0.
A.2 RELATIONSHIP TO DIFFUSION MODELS.
The logits evolve in way that closely resembles diffusion process in logit space. From Theo- rem we have our denoising dynamics
p(z++1
Ze, N(z: axx, a,I).
(11)
Moreover, the marginal of ze is given by
p(ze x) = NN + 8()x, (3o + B(O)
(12)
(see Eq. (3)). We define the corresponding "noising" process as the reverse-time conditional p(z: 1,+1, x). Using the standard Gaussian conditioning formula (Murphy, 2012, Eq. 4.125), we obtain
p(z: x))1, = 1++1,))=N(( -N( (Bo + =x(11) 8o + B(t) + az
((t))z++1 + Ot/o - 04
(3 + 8(1)) BB + B(t) aq 1)
(13)
Thus, the reverse transition is Gaussian, analogous to the posterior gxx:11 Xo) in standard diffusion models. While this not a typical diffusion process the sense that the derived forward dynamics over 2, are generally non-Markovian, related non-Markovian formulations have been pro- posed before (Song et al., 2020). Interestingly, a Markovian process is recovered when setting So = O, which coincides with the original BFN parameterization (Graves et al., 2025).
A.3 RELATIONSHIP WITH FLOW MATCHING MODELS
At noise level y = 0, Categorical BSI is closely related to Flow Matching. The sampling SDE Eq. (7) becomes an ODE where the right-hand side can be interpreted flow field to follow. However, we do not train to directly predict the flow field, but to reconstruct the clean sample. Similar to Dirichlet Flow Matching (DFM), Stark et al. (2024), Categorical BSI operates on distribution over the simplex. However, while Categorical BSI uses the logits of categorical distribution as a latent variable, DFM employs a mixture of Dirichlet distributions.
approximation of the
B BSI FOR SEQUENCE GENERATION
Categorical BSI can generate general categorical data it is not restricted to graphs. In this sec- tion, we demonstrate this capability empirically by training a categorical BSI model to generate sequences. We represent sequences with length I and vocabulary v in the one-hot encoded format as S E A C [0, 1 We call the resulting model SeqBSI.
Employing the same reconstruction model as Stark et al., 2024: Davis al., 2024, Convolu- tional Neural Network. We train on the toy dataset from Davis et al. (2024) with 1 4 and v E (5, 10, 20, 40, 60, 80, 100, 120, 140, 160) as well as a dataset of enhancer DNA sequences from fly brain cells Janssens et al. (2022) with ( = 500 and v 4 nucleotide bases.
Following Stark et al. (2024), we report the KL divergence for the toy task and the Fréchet Biological Distance (FBD) as a measure of distribution similarity. As demonstrated in Tab. 3, SeqBSI slightly outperforms Dirichlet Flow Matching (Stark et al., 2024) in the flybrain task. The comparison with Fisher Flow Matching
this metric is difficult, as their evaluation shows vastly different results for Dirichlet flow matching than the results reported in their own paper. On the toy dataset task. SeqBSI outperforms Dirichlet Flow Matching and is competitive with Fisher Flow Matching (see Fig. 9).
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Table 3: Results the enhancer DNA sequence dataset
flybrain FBD
Model
Steps
Random Sequence
876.0
Language Model
500
25.2
Linear FM
100
15.0
Dirichlet FM
100
15.2
SoqBSI (OU)
100
12.3
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C ANALYSIS OF SDE-BASED SAMPLING ALGORITHMS
In this section, we analyze the behavior of the SDE-based sampling methods Algs. 3 and 4.
C.I EQUIVALENCE OF THE TWO SAMPLING ALGORITHMS FOR INFINITE STEPS
is worth noting that for At O, the Ornsten-UUllenbed discretization and the Euler- Mally-Ma discretization of Eq. (7) converge to the same update step:
78'
At m + (Z: m.e RAt
(1 (--2211)-. 2 1))
(14)
2K
2m 1/8 (1- (1-1)AAAt) 1))
(Z: (1 kdt)
(15)
2K
+ x Al + 7 11
(16)
2: + + - (B + ///////))))))))))) /x)* - -2, + At
(17)
z: + + 7-
140 + 38 2₁ A NO.
3
(18)
2
3+ 3o
7-1 3'V,, (ogg(()))/ VBBAII No 1)
24 + 3'olz, A +
(19)
2
C.2 STABILITY OF MARRUAMA SAMPLING
Let us explicitly write out the update step of the Euler-Maruyama discretization of Eq. (7):
At 2t + 3'AA1 +
1-1
3
M40
+
+38- Ze
A + - NN
(20)
2
B+80 30
=
(7-1)33 1)33'
(r 1)3² (HO + 38)
At
2: +
3'AA+
At + A -
2(3 + 30)
2(B + Bb)
(21)
As rule of thumb, the coefficient in front of z, should not be negative, i.c., the previous step should be over-corrected. This yields the condition
( - 1)33'
A 2 0
(22)
2(B + (0)
2(3 + Bb)
At 17- (7- 1) <
(23)
3'
For our precision schedule on moses (estart 3.0,, Fand 12.0, 90 1.0), find that
2(5(t)+ + 80)
min [E
0.48
(24)
3'(1)
The resulting maximum stable noise level y for different numbers of sampling steps in Tab. predicts the observed behavior in Fig. surprisingly well.
C.3 BEHAVIOR OF O UHLENBECK SAMPLING WITH INFINITE NOISE
Taking the limit 7 - oo Alg. yields an interesting sampling algorithm (see Alg. 5). In this limit, the update step becomes independent of the previous step 2,, replacing all previous information with the current prediction x. Empirically, we find that fixing the prior value after the initial sampling step, shown in Alg. works better in practice (see Tab. 5). This algorithm matches the Flowback algorithm from Song et al. (2025). We find that with budget of 50 sampling steps, this algorithm performs surprisingly well molecule generation. However, higher compute budget drastically
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Table 4: Maximum stable r for different numbers of sampling steps with the Buler-Maruyann discretization, following Eq. (24).
Number of Timesteps
At
Maximum Stable 7
25
0.040000
12.938480
0.020000
24.876960
100
0,010000
48. 48.753920
200
0.005000
96,507840
500
0.002000
239.769601
Misclasifeattin Rate for
Michosincatttn.o Rane
62
0.6
/hocker
Figure 5: Empirical misclassification rate of trained reconstructor on the moses dataset under the encoding distribution. Compared to simply sampling from the belief, returning a reconstruction far more likely to yield the correct train sample. Therefore, returning a quantization of the recon- struction instead of sampling from the belief significantly more efficient for molccule generation. However, deriving the ELBO under quantization is intractable to optimize. Therefore, we have the sampling-forman formulation to derive a tractable ELBO and the ciency after training.
to optimize effi-
reduces performance. We hypothesize that this is because an excessive amount of stochasticity is introduced. Song et al. (2025) address this by adaptively alternating between vanilla BFN steps and Flowback steps, effectively mixing Alg. with Alg. 5.
Algorithm 5 Sampling with -> 00
Algorithm 6 Fixed-prior sampling with y - co
Require: reconstructor fo. discretization At,
Require: reconstructor fo. discretization At,
precision schedule [0, 1] > R
precision schedule / [0. - R
Zo Nou,, Bo.)
2o NNu,.
4- Zo for 1 = 0... in steps of At do + f-)
z + Zo for = 0... in steps of At do
* < fo(z,t) t)
< S0 + B(t + At/2)
ffft+At/2)
y ~ N[p= &, Σ = 11aa T)
y ~ W(u= x, _ = 1/a 7)
Go from prior to in single step
Go from prior / in single step
z + 44 + a
t 20 + a
end for
end for
return Quantize (foiiizeeoole (fo 1))
return Quantize 1))
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Table 5: Results on the GuacaMol and Moses benchmarks for molecular generation with the Euler- (EM) and Censtein-UUhenbeee (OU) discretization, and with Alg. 5 ( - oc) and Alg. (7 oc, FP), as well as results for a linear scheduler (lin) with the same final precision as the exponential scheduler. Additionally, we include results obtained with the FlowBack (FB) sampler Song et al. (2025) using smaller value of Pu. as well the OU sampler with the same checkpoint. The EM sampler becomes unstable at 30- 0.05.
GuacaMol
Moses
Model
Steps
Val.
V.U. V.U.N.
Val. 1
Uniq. 1
FCD
SNN
Scaf
Train
100.0
100.0
100.0
100.0
GraphRSI (EM)
86.6
85.4
3.74
GraphRSI (OU)
91.9
91.9
24.2
3.88
GraphBSI (EM)
97.5
97.5
40.7
97.5
100.0
97.9
93.6
1.83
0.47
15.7
GraphBSI (OU)
97.1
97.1
96.8
89.3
49.7
98.2
100.0
97.8
94.5
1.92
0.48
14.4
DeFoG
50
91.7
91.7
91.2
92.3
$7.9
83.9
99.9
96.9
96.5
1.87
0.50
23.5
GraphBSI (EM)
50
97.5
97.5
97.2
90.7
65.6
99.3
100.0
96.5
96.9
1.06
0,50
15.2
GraphBSSI (OU)
50
99.2
99.2
98.7
93.7
71.3
99.7
100.0
94.6
98.2
1.19
0.52
15.1
GraphBSI (y 00)
50
99.6
99.6
98.3
95.1
61.4
99.9
99.9
89.9
99.2
1.58
0,56
11.7
GraphBSI (y FP)
50
99.6
99.6
98.3
97.4
75.1
99.9
99.9
89.7
99.1
1.06
0,56
13.1
GraphBSI (FB)
50
99.6
100.0
95.9
97.5
1.15
0,51
15.0
DiGress (CADD)
500
92.2
82.3
74,2
76.2
37.19
0.24
0.0
DiGress
500
85.2
85,2
85.1
92.9
68.0
85.7
100.0
95.0
97.1
1.19
0.52
14.8
DisCo
500
86.6
86.6
86.5
92.6
59.7
88.3
100.0
97.7
95.6
1.44
0,50
15.1
Cometh
500
98.9
98.9
97.6
96.7
72.7
90.5
92.6
99.1
1,27
0,54
16.0
DeFoG
500
99.0
99.0
97.9
97.7
73.8
92.8
92.1
98.9
1,95
0.55
14.4
GraphBFN
500
89.0
98.3
1.07
0.59
10.0
GraphBSI (EM)
500
98.8
98.8
94.6
82.6
100.0
92.5
99.1
0.72
0.54
14.3
GraphBSI (OU)
90.7
GraphRSI (EM,lin)
0.85
GmphBBI (OU,lin)
99.2
GraphBSI (FB.9 = 1)
100.0
99.6
80.7
99.6
2.84
GraphBSI ( 0.05)
500
100.0
99.9
85.9
99.4
1.32
0.57
11.7
GraphBSI (OU,A, 0.03)
500
99.2
100.0
90.4
99.4
1.00
0.55
12.4
MA
0.8
0.5
0.4
(ODE)
0.05
(Vonilla)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
(a) Exemplary trajectories
0.6
0.4
0.05
10
0.0
0.0
0.4
0.8
(b) Quantiles (1%, 10%, 50%, 75%, 90%, 99%) over 10000 trajectories
Figure 6: Illustration of the trajectories the categorical sampler with two categories with a fixed reconstruction f(z,t) t) é₁ for different noise levels While higher values of y result in more volatile trajectories (see Fig. 6a), the marginal distribution is preserved if the score function is known exactly (see Fig. 6b). Since we approximate the score function in practice, the noise level is crucial hyperparameter to fin-tune during inference.
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2
SNNT
Normalized Score
FCD4
1
Filterst
Valid?
0
10
11
12
13
14
15
Final Precision
Figure 7: Key metrics on the Moses benchmark with a linear scheduler, ending at different final precisions. The model was trained with final precision of 15, and to generate this plot, sampling was stopped early instead of training a new model for each precision value. While too small final precision values yield noisy samples, too large final precision values waste sampling steps.
FB
FCD (1)
SNN Score (1)
100% Filters (1)
Change vs = 0 (%)
-25
-50
-75
-100
20
250
1000
100
1000
100 250
1000
Noise Level (t)
Noise Level (?)
Noise Level (t)
vs. noise level y for different numbers of function evaluations (FE) and discretization schemes. Our custom Ornstein-Uhlenben denoted OU, while the standard Euler-Maruyama scheme is written Fuler. Some values for the Fuler scheme are missing since the sampler becomes unstable if 7 A/ becomes too large (see Sec. C.2).
Figure 8: Change in metrics relative
discretization scheme
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109
10
Minimal KLL divergence
10
10-3
Diricblet FM
Fisher Flow (No OT, Sphere)
Fisher-Flow (OT, Sphere)
Lincar FM
10
Mulinomiil
D3PM
SeqBSI
10
40
100
120
140
160
Number of categories, K
Figure 9: KL divergence on the toy sequences benchmark by (Davis et al., 2024), reporting the lowest KL divergence for each vocabulary size over five random seeds. The model is trained on 100, 000 samples with sequence of length four and varying vocabulary size. Find the details of the dataset generation in the original paper.
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D PROOFS
Theorem 1. Given prior belief p(x z) = Cat(x softmax(z)), after observing y ~ Ny μ
x, 22² 1/al) at precision a, the posterior belief p(x 2,y, Cat(x softmax
with
Zpost = 2 y.
(25)
Proof. We need to compute the Bayesian update of the belief parameters. Each dimension can be considered independently since the noise is isotropic. Let start with single- variable prior belief (at(soltmax with z E R and a noisy observation y X, a ~ NN = _ 1/a 7) of the Let now consider any class 1 ,....,.. We write ét for the one-hot encoding of class I. Since we are only interested in the ratio of the posterior probabilities, we can ignore any factors that do not depend and normalize at the end. We have:
true sample E A c-1 at precision
p(x ét
softmax(z)) (z), x exp(zi)
(26)
ply x ē,, a.) =N é, E² 1/aa
(27)
p(x = è1 2. =x ply x a) -p(x = ê: z)
(28)
= =N é1, E² = -1/a 1) softmax(2);
(29)
lly - 6/12
exp
exp(2z)
(30)
2- 1/a
||y|| -2 (y, êx) + 1|ē:112 + z
exp
(31)
2. 1/ox
exp (a: Y1 + z))
(32)
Let us now normalize the results to obtain the posterior probabilities:
exp (a Yt + 2))
p(x = êt a. Er= exp (a yv zp) = softmax(z + a y)
(33)
Putting everything together, find that the posterior belief is p(x 2,y,a) softmax (oltmax(Zpost)) )) with
Cat(x
7post 2 + a.
(34)
Theorem 2. For categoricul BSI, the lok-liiol of x under Alg. is lower-bounded by
k
logp(x):
E
[logp(x p(x
[log
[x)]]
E
- -x12], (35)
where g(z) x, N(x M4 3()xx +
Proof. For any distribution p(x) and any latent variable i.e. any choice of prior p(z), encoding distribution p(z x), and likelihood p(x 2), we have the variational lower bound
log p(x( 2
EE [logp(x z)] KL(p(z) x)llp(z))
(36)
on log p(x) Kingma & Welling (2013). We choose the beliefs 2o.-.., 25 as latent variables at the discretized time steps to...., tx. We choose the encoding distribution to be the distribution of the beliefs under Alg. with the reconstruction network fa replaced by the true sample x:
k-1
p(zo ..., Zz x) N2o
Mo. Bol)
II
p(z 21,x,tt)
(37)
#=0
The transition distribution p(zi+1
Zi, x, (4) can be computed from Theorem 1:
- 2, Oi Yz 05 Niy ب ===xx x, 1/o.))))) N Na 2, + a; x, ozT)
(38)
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The distribution of p(z) following Alg. factorizes similarly:
A-1
pizo. 3x)) Nzo po, (o,8o1) BoT)
II
p(z++1 22 0)
(39)
with the transition distribution
p(zi+1 12, N(zi+1 2 + as to)
(40)
Let us now compute the KL divergence:
(T
2k x) (p(zo,.... 21,))
(41)
log p(z0..... 2x x)
(42)
Mo mo p P(Ko.... (oc)
p(zo... pizo...., 2x))
log N(zo 140 (0 II 1i=0 p( 24, 1,,x,44 (4)
(43)
Nzzo Mo, Bol) II: p(z[+1
t2. 0) Zzy 1z,,t,
p P( Rid [x]
p(2++1
Z; x, (:) 2i,4;,0 0)
log
(44)
p(zi+1
p 2e (x)
(KL(pzz)+1
(..tt:0))
(45)
2)mp((2)))) (x)
E
22 + x, a. a, I z, + aa fo(2,44) o;; [)]
(46)
1 xsmp(zx. xx
|2 + +a.
(z. + as fe(z,,t)))1B
(47)
2rrpp(2,(x)) [x)
20x
aa;
|x fo (z,, t.)1
(48)
2
E
[(8(t++1) - B(t:))/21|x fo(z,,
(49)
x xx)
k (8((+t))-
E (0, 2,00((,,1 (x)
- -3(4))lxx So(2i, 4:11
(50)
2
Since p(x 20.-.., 2x) = p(x 2x)) = Cat(xx softmax(z:)), we can plug in Eq. (3) to obtain the final result:
logp(x) 2
E [logp(x 1x)]]
E
((())1))-((4))166(22. - - (51)
xy =x)0(x)
2
(0.--1111 2, [x/x1)
where q(z x,t) N(z p4 + B(t)x 8, + 3(t)1).
Theorem 3. As Al + 0, the update equation Theorem converges to the following SDE:
dz, = 3 (t))fo(z,, t)dt + (B'(t) where dW, is a Wiener process and zo N N 30 T).
(52)
Proof. Take the update equation Theorem with an infinitesimal time step At -> 0, it holds that
a (3(1 A1) ((1) + (1)
(53)
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Published as a conference paper at ICLR 2026
Therefore, have:
24-21 24 + σy
(54)
~ "s - aN 2²2 1/al)
(55)
2. + Maax, E² al)
(56)
Z + (1))&S BB
(57)
We identify this as the Euler-Maruyane discretization of the SDE above.
Theorem 4. The SDE in Theorem is generalized by the following family of SDEs with equal marginal densities pz(z:)
dz. = 80 (z), t)dt + 7-1
2 Vt) log po + (1)
(58)
where dW, Wiener process and zo p(z t= =) 0).
Proof. We need to show that the evolution of the probability density p. (z,) of Eq. (6) matches that of Eq. (7). The evolution characterized by the Fokker-Planck equation:
(pc(2e)
Σ Ve, (Plesfalen)) t) + 7-1 (t)) logpi(z)) p((ze) + 2 2
Ot
Σ Vay (()))(((2,)00))z
7-1
(1) (pr(z)) Vn, logpi(z))) + 2 2 2 px
2
1
Σ Ve, ((t))f(zzzttpplll - 1
8 (t) (z,) + +2 7 (t)V p((i))
2
Σ vn (8(()) (p: + (1) 8 2x
Which equals the Fokker-Planck equation the SDE Eq. (6).
Theorem 5. The BSI loss Eq. (5) also is score matching loss with the score model se t) param- eterized as
s0(2, M40 ((,))5(2,())) 1 Vx log pe(z)
(59)
B(t) 3o
Proof. Score matching Song et al. (2021) is generative model that learns to approximate the score function V, Pr (z) distribution (z) by minimizing the score matching loss:
Lecore = E4-1((011) E (2lx)) [2)) V pz x)]]
(60)
where X [0, 1] R is a positive weighting function. The distribution pa (z x) is the distribution of the latent variable at time & given the true sample x. For categorical BSI, we have from Eq. (3);
(z x) = (/ μo + B(t)x, (So + {(t))z)
(61)
The score function of an isotropic Gaussian can be computed in closed form:
Vg log μ. (2,021) Vz (_ ||z 11/12
(62)
202
(63)
Plugging in the parameters of Pe(z x), we find: (no + 3(1)x)
140 + +((1)x z
Va lo + + Btt)
(64)
Bo B(t)
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With the proposed score model parameterization sø(z, t), we find:
Lacore EE- (A))).,,))(((E)Bxx (z/x) [11s0(2, V~ log x)]
(65)
Mo ((t) 1, t)
Mo + B(t)x [
(66)
B(t)) + 500
8o 3(t)
2
((t)(fo(z,t) -x)
[c]
(67)
3(t) + 90
B(t)² E (3(1) + 80)2
Eptx E px(())) [[[[fo2211) x)][
(68)
(69)
Choosing the weighting
A(t) = (t)) (3(t) + +b)2
(70)
23(t)²
we find that the score matching loss equals the BSI loss Eq. (5). Therefore, the BSI loss Eq. (5) is score-matching loss with the weighting Sec. D and the score function s0(2, parameterized as Eq. (59).
Theorem 6. Fixing the prediction & = [a(zi, 1) and the values 3 = B (1/2), B' = -1+ in Eq. (7) in time interval [t, + A1] yields an Ornstein Menttiin-Uhlenbeck process with the exact marginal
33' (1--- NN I),
Ze4 At ~ m71 + (Z: -m)e xbt + V
(71)
2x
where K.
(7-1)88' 203++8p
= p40 + (B + B /x)
Proof. The SDE in Eq. (7) with fixed parameters 3, & is given as
dz. = 3'tt xdt 7 B Va, log)r(z:)dl + VBddW,
(72)
where dW, is Wiener process and Z, p(z t).
Let us insert Theorem 5 to obtain
dz: B'xdl + Y-1
3r 140 + 3 f
di + v/3dd,
(73)
8 + 30
(7 - 1)/3'
2(30 + (3)
+ (++
dt + VyB'dw, x- 21
(74)
2(Bo + B)
7
Setting K
(x-1),8'
and m μm + (B + 3' /x), we find
2[
dz, k(m z dt V/dWW
(75)
which is an
process. The exact marginal distribution of an Ornttein-Ullenbed Uhlenbeck process is given as Uhlenbeck & Ornstein (1930):
7/9
Zet + (z: mme kAt
(1 E- 2mA)) (
(76)
2x
E ADDITIONAL RESULTS
Tab. shows our method is competitive the QM9 dataset with removed hydrogen atoms, achiev- ing state-of-the-an results on validity and FCD. We explicitly model charges on the nodes, enabling high validity scores.
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Table 6: Results the QM9 dataset.
QM9 (without H)
Q (with H)
Model
Steps
Val.
Uniq.
FCD
Val.
Uniq.
FCD
Train Set
99.3
100.0
0.05
99.3
100.0 97.6 0.4
0.05
DiGress
500
99.0
96.2
95.4 1.1
DiGress (CADD)
500
96.3
83.4
5.25
DisCo
500
99.3 0.6
Fisher FM
500
Cometh
500
99.6
96.8
0.01
DeFoGi
98.9
96.2
0.00
DeFoCi
0.00
Ours
50
93.7
0.30
Ours
500
99.9
96.2
0.09
99.8
96.6
Table Hyperparameters used for the results in Tabs. 1 and The precision schedule is parame- terized as 3(t) Astart (exp(t- log( Bon( (staar)) 1).
Belief Parameters
Sampler 10% steps
Sampler 100% steps)
Dataset
B
Bend
g(0)
7 (OU)
7 (Euler)
(OU)
(Euler)
GuacaMol
20.0
10.0
200.0
200.0
12.0
Moscs
10.0
20.0
90.0
120.0
Planar
3.0
1.0
200.0
SBM
20.0
200.0
Tree
100.0
Table 8: Datasets with training samples and maximum number of nodes. For Moses, we use the test_sca do split for benchmarking, which is the standard test split.
Dataset
Train samples
Max. Nodes
GuacaMol (Brown al., 2019)
1.3M
88
Moses (Polykovskiy et al., 2020)
1.6M
30
Planar (Martinkus al., 2022)
128
64
SBM (Martinkus et al., 2022)
128
187
Tree (Bergmeister al., 2024)
128
64
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Table 9: Molecular metrics
Metric
Short
Description
Validity
Val.
The fraction of generated molccules that are chemically valid according to RDKit.
Uniqueness
Uniq.
The number of unique molccules generated (counting permutations as same molecule) divided by the total number generated molecules when generating 10,000 molccules.
Novelty
Nov.
The fraction of generated molecules that present in the training set.
Valid & Unique
V.U.
The fraction of generated molecules that are both valid and unique.
Valid, Unique & Novel
V.U.N.
The fraction of generated molecules that are valid, unique, and novel.
KL Divergence
KL.
The normalized KL-Divergence between the distributions of various physicochemical deseriptors between the generated set and the training set.
Fréchet ChemNet Distance (Moses)
FCD
Distance between the distributions of learned features of the generated molecules and those of the validation sct, computed by pretrained ChemNet model.
Fréchet ChemNet Distance (GuacaMol)
FCD
Same as for Moses, but normalized with the transform x + exp(-0.2z)
Similarity to Nearest Neighbor
SNN
The average Tanimoto similarity between each generated molccule and nearest neighbor in the test set
Scaffold Similarity
Scaf.
Cosine similarity between the firquuecces of scaffold substructures in the generated set and the test
Table 10: Synthetic graph metrics metrics
Metric
Short
Description
Valid & Unique
V.U.
The fraction of generated graphs that are both valid and unique among 40 generated graphs. For the planar and tree datasets, we check if the generated graphs are planar/trce graphs. The SBM datasct does not have straightforward validity criterion, therefore a test with Bayesian inference used with likelihood threshold.
Average Ratio
Ratio
For each of several metries, ratio defined as the Maximum Mean Diserepancy (MMD) between the generated and training set divided by the MMD between the training and the test set. The average ratio is the ratio metric averaged over all metrics. The metries are degree, clustering coefficient, orbit counts, spectral-, and wavelet metrics.
27


Reviews:
[{"review_id": "EJAxM7aPEB", "reviewer": "Reviewer_w6nS", "summary": "The paper extends Bayesian Sample Inference (BSI) to categorical/graph generation, introduces an SDE view with a controllable noise family ($\\gamma$), and studies two discretizations (EM/OU). On molecular benchmarks, the method achieves competitive performance with relatively few function evaluations (e.g., 50/500 NFE). The contribution is integration-oriented: it adapts BFN/BSI ideas to discrete graphs with a clearer Bayesian update and an SDE formulation, offering a tunable trade-off between speed and quality. The writing is generally clear, but several theoretical statements and implementation details would benefit from refinement.", "questions": "1) Could you add a minimal appendix probe comparing means/covariances of $z_t$ across a few $t$ and $\\gamma$ values as an intuitive check for the “same marginal” claim? This can be very lightweight.  \n2) For Theorem 5, do you plan to include the missing $\\tfrac{1}{2}$ factor in a revision/erratum and explicitly state the condition under which it is strictly aligned with Eq.(5)?  \n3) For the final decoding, which default path do you recommend—**categorical sampling** or **argmax quantization**—and why (stability, diversity, alignment with prior work)?  \n4) If adding figures is difficult, would you consider providing a reference configuration in the repo for a “**same wall-clock budget**” comparison (fixed batch/GPU/time) so others can verify fairness?  \n5) For the “affine equivalence” phrasing, would you consider noting in the appendix that “same-order approximation” is more accurate when covariance scalings differ, to avoid readers inferring strict equivalence?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- **Originality**: Systematically ports BSI to **discrete/categorical graphs** and unifies deterministic ODE and stochastic sampling paths via **SDE + controllable noise**. This “bridging” removes some practical limitations of prior BFN-style derivations and constitutes originality via domain adaptation and simplification.  \n- **Quality**: A clear training objective (ELBO) and a family of samplers (controlled by $\\gamma$) are provided, with strong **inference efficiency** (competitive at low NFE). Ablations on $\\gamma$ and time grids are helpful; engineering appears solid.  \n- **Clarity**: Key intuitions (“belief contraction,” “same marginal family”) are conveyed through diagrams and concise algorithms; the separation of main text and appendix reads well.  \n- **Significance**: For **discrete structure generation** (notably molecular graphs), the **low-step, efficient** generation is practically relevant. Methodologically, it offers a clean, implementable link between BFN/BSI and diffusion-SDE paradigms, with potential impact on broader discrete domains.", "weaknesses": "1) **ELBO Reconstruction Term** In the main theorem, the reconstruction term is written as $\\mathbb{E}[p(x|z)]$ instead of the standard $\\mathbb{E}[\\log p(x|z)]$. While the conclusion that this term is independent of $\\theta$ still holds, the ELBO should use the expected log-likelihood. Please unify the notation in the main text and appendix.\n\n2) **Weighting in Theorem 5** To be strictly consistent with the loss in the main text, a factor $\\tfrac{1}{2}$ is needed, i.e.,  \n$$\n\\lambda(t) = \\frac{\\beta'(t)}{2} \\cdot \\frac{(\\beta(t) + \\beta_0)^2}{\\beta(t)^2}.\n$$  \nThe current statement may double the scaling. A short clarification aligning Theorem 5 with Eq. (5) would help.\n\n3) **Over-Strong “Affine Equivalence” Wording** The appendix claims an affine equivalence between the receiver distribution and BSI observations, but the covariance scaling does not match exactly. A safer phrasing is “same-order approximation in the small- $\\alpha$ limit,” avoiding over-commitment.\n\n4) **Final Decoding Inconsistency** The text mentions sampling from $\\mathrm{Cat}(\\mathrm{softmax}(z))$ while algorithms return `Quantize(f_\\theta(\\cdot,1))`. Since sampling vs. quantization affects diversity and faithfulness differently, please unify the terminology and state the default choice with a brief rationale.\n\n5) **Empirical Support for “Same Marginal” is Indirect** The claim that different $\\gamma$ share the same marginal is mainly supported through downstream metrics. A lightweight check in $z$ -space (e.g., comparing first and second moments at a few $t$, $\\gamma$) would make the claim more tangible—no large-scale new experiments needed.\n\n6) **Comparative/Robustness Details Could Be Lightly Strengthened** If space permits, adding small variance bands over a few seeds on key plots or clarifying a “same wall-clock budget” comparison protocol would improve perceived fairness and reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "N4IQUmaMIR", "reviewer": "Reviewer_3cUV", "summary": "This paper presents GraphBSI, a new one-shot generative model for discrete graphs based on Bayesian Sample Inference (BSI).\nUnlike conventional diffusion or Bayesian Flow Network (BFN) models, GraphBSI performs generation by refining a belief distribution in parameter space rather than evolving discrete samples directly. The authors derive a categorical BSI formulation and show that in the continuous-time limit, it becomes a stochastic differential equation (SDE). They further generalize this to a family of SDEs with a noise control parameter γ, allowing smooth interpolation between deterministic probability-flow ODEs and stochastic samplers. On standard molecule generation benchmarks (GuacaMol and MOSES), GraphBSI achieves  superior results across most metrics.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Extends the Bayesian Sample Inference framework to discrete graphs.\n\nIntroduces both Euler–Maruyama and Ornstein–Uhlenbeck discretization schemes.\n\nOutperforms baselines on key metrics with fewer sample steps.", "weaknesses": "How sensitive is GraphBSI to the choice of the precision schedule β(t)? The paper only mentioned a monotonically increasing schedule.\n\nSince GraphBSI is similar to BFN, a direct comparison with GraphBFN regarding training time would better highlight its practical advantages.\n\nIn addition, GraphBFN is not included in Table 2. Including GraphBFN results would allow for a clearer assessment of GraphBSI’s improvements over prior BFN-based approaches. The sampling_step=50 for GraphBSI is also missed in Table 2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1JaF4KkUJk", "reviewer": "Reviewer_skZ3", "summary": "The paper proposed a method based on BSI for graph generation, achieving outstanding performance.", "questions": "1. Which preprocessing procedure (which version of the implementation) the authors apply for the MOSES dataset?\n2. Do you have results with 5 or 10 steps, since the method already reaches very high performance with only 50 steps?\n3. How do the authors evaluate the robustness of the framework, separately in terms of hyperparameters during training, and sampling?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper clearly presents the method being used. It is easy to follow.\n\nIt has a thorough structure, including theoretical guarantees and experimental analysis.\n\nThe experimental results are impressive, especially with only 50 steps.", "weaknesses": "Regarding the soundness of the paper, it is not clear how this work relates to graphs, and it seems mainly adapted for discrete data, and the transformers and graph features mainly follow the previous work. I would like to ask the authors to clarify whether they are the first to adapt BSI to discrete data modalities. If yes, is the main approach is adding softmax operations on the original feature z used in common BSI?\n\nBeyond that, I kindly ask the authors to clarify better the relationship between BSI and BFN. The paper states that BFN is a more generalizable version of BSI, but a more thorough discussion would help the reader understand the connection better. In terms of application, how does this help in your implementation or analysis, and what can BFN not achieve?\n\nAdditionally, mathematically, what is the difference between this method and a continuous diffusion on the simplex space, or a continuous diffusion on the logits space with softmax applied? If different, what makes BFN more advantageous than these normal diffusion formulations?\n\nClarifying these points would help me better understand the contribution of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VRzUkUqQyQ", "reviewer": "Reviewer_WtYP", "summary": "This paper introduces Discrete Bayesian Sampling Inference (BSI), a novel framework for graph generation. The core idea is to model the generative process not through discrete state transitions, but by evolving a continuous belief state, which is argued to better capture the dynamics of discrete variable evolution. The authors provide a theoretically grounded framework to derive the training and sampling methodologies. GraphBSI is evaluated on molecular graph generation benchmarks, where it demonstrates significant performance improvements over current state-of-the-art models.", "questions": "- Q1: Though the title states that the model is designated for graph generation, my main question is whether the model is specifically working for graph generation, or it can bring similar improvements to other discrete generation tasks. If this is designed for graph generation, I would like to see more on how can BSI improve the graph generation? If not and BSI can generally improve the discrete generation, it might be ideal to conduct generative tasks to modalities other than graphs, such as text modeling, protein co-design, etc[1][2]. \n\n- Q2: I look into the appendix and understand that the loss in eq. 5 is derived through the lower bound of log-likelihood. Honestly this is a bit surprised as from the assumption, the x is sampled from a softmax version of z, it seems cross-entropy loss to be more reasonable. Could the authors elaborate more on advantage of choosing the loss to be squared distance ? I am convinced with the derivation but It just seems a bit unnatural. \n\n- Q3: In the abstract, the authors claim that the advantage of BSI comes from introducing a continuous belief z that can better capture the evolution of dynamics. Would the authors provide more evidence/justification/intuition on why this works? \n\n\n[1] A. Campbell et al. A Continuous Time Framework for Discrete Denoising Models. NeurIPS 2023\n\n[2] A. Campbell et al. Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design. ICML 2024", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The proposed Discrete BSI presents a compelling and theoretically interesting mechanism for discrete data generation. The idea of operating on a continuous belief state to model the evolution of discrete variables addresses a key challenge in generative modeling.\n\n- The application of BSI to graph generation yields significant performance improvements over state-of-the-art methods on established benchmarks like Moses and GuacaMol. This highlights the practical efficacy of the approach.\n\n- The theoretical foundations are presented rigorously, with (mostly) well-stated theorems and proofs that are easy to follow.", "weaknesses": "- My primary concern is the precise positioning of the paper's contribution. It is somewhat ambiguous whether the main novelty lies in the general framework for discrete generation (placing it alongside with discrete diffusion, flow models, etc.) or specifically in its application to graph generation. Clarifying this would significantly strengthen the paper's impact by helping the reader understand its broader context and significance. Honestly I would be happy to improve the score if this question is properly addressed. \n\n- Insufficient Ablation of Performance Gains: While the empirical gains are impressive, the source of these improvements is not fully elucidated. The provided ablation studies on noise levels and time distortion are helpful, but a deeper analysis is needed to isolate which specific components of the GraphBSI model are most critical to its success.\n\n- The theoretical framework is solid but I would appreciate if the authors can provide some easy-to-follow explanations alongside. Some of the results are slightly against intuition (e.g. Theorem 2 where the lower bound is a squared error for categorical variables) and can be difficult to understand at the first glimpse. Also, I would appreciate if the authors can compare their methods with a few existing continuous state discrete diffusion/flow models, both theoretical and empirical. Such as [1, 2, 3]. \n\n[1] H. Stark. et al. \"Dirichlet Flow Matching with Applications to DNA Sequence Design\". ICML 2024\n\n[2] O. Davis. et al. \"Fisher flow matching for generative modeling over discrete data\". NeurIPS 2024\n\n[3] H. Zheng. et al. \"Continuously augmented discrete diffusion model for categorical generative modeling\". Arxiv.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)