Horseshoe Splatting: Handling Structural Sparsity for Uncertainty-Aware Gaussian-Splatting Radiance Field Rendering
Authors: Feng Wu
Tsai Hor Chan
Yihang Chen
Lingting Zhu
Guosheng Yin
Lequan Yu
Venue: ICLR 2026 Poster (2026)
Keywords: Bayesian Neural Network
Gaussian splatting
Horseshoe Prior
Structural Sparsity
Uncertainty
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We introduce Horseshoe Splatting, a Bayesian extension of 3D Gaussian Splatting (3DGS) that jointly addresses structured sparsity in per-splat covariances and delivers calibrated uncertainty.

Abstract: We introduce Horseshoe Splatting, a Bayesian extension of 3D Gaussian Splatting (3DGS) that jointly addresses structured sparsity in per-splat covariances and delivers calibrated uncertainty. While neural radiance fields achieve high-fidelity view synthesis and 3DGS attains real-time rendering with explicit anisotropic Gaussians, existing pipelines do not explicitly encode structural sparsity in the covariance—e.g., axis-wise variances or pairwise correlations—leaving noise-dominated components insufficiently regularized. Uncertainty is likewise essential for trustworthy and robust novel-view prediction, yet most 3DGS variants remain deterministic. We place a global-local Horseshoe prior on the covariance scales, whose spike-at-zero and heavy-tails adaptively shrink irrelevant directions while preserving the salient structure. We fit the model with a factorized variational inference scheme that mirrors the Horseshoe's inverse-Gamma augmentation, enabling Monte Carlo rendering and pixel-wise posterior uncertainty with minimal overhead. Theoretically, we establish posterior contraction rates for the scale parameters and transfer them to the rendered image via a local Lipschitz mapping, providing guarantees that estimation error and predictive uncertainty diminish with data. Empirically, Horseshoe Splatting produces high-quality uncertainty maps while matching state-of-the-art 3DGS visual fidelity and runtime, yielding a practical, uncertainty-aware renderer that is robust to structured sparsity in the radiance field.
The code is available at https://github.com/HKU-MedAI/Horseshoe-Splatting.

Published as a conference paper at ICLR 2026
HORSESHOE SPLATTING: HANDLING STRUCTURAL SPARSITY FOR UNCERTAINTY-AWAT GAUSSIAN- SPLATTING RADIANCE FIELD RENDERING
Feng Wu': Tsai Hor Chan"; Yihang Chen', Guosheng Yin', and Lequan Yu 'University of Hong Kong University of Pennsylvania
Lingting Zhu',
(fengwu96, yihange, 1tzhu99) @connect hwk.t....kkk Tsaihor. Chan@PennMed upenn edu, (gyin, 1qyu) @hku. hk
ABSTRACT
We introduce Horseshoe Splatting, Bayesian extension of 3D Gaussian Splatting (3DGS) that jointly addresses structured sparsity in per-splat covariances and delivers calibrated uncertainty. While neural radiance fields achieve high-fidelity view synthesis and 3DGS attains real-time rendering with explicit anisotropic Gaussians, existing pipelines do not explicitly encode structural sparsity the noise-
covariance-e.gg, axis-wise variances or pairwise correlations- dominated components insufficiently regularized. Uncertainty is likewise essential for trustworthy and robust novel-view prediction, yet most 3DGS variants remain deterministic. We place a global- Horseshoe prior on the covariance scales, whose spike-at-zero and heavy-tails adaptively shrink irrelevant directions while preserving the salient structure. We fit the model with a factorized variational inference scheme that mirrors the Horseshoe's 's inverse-Gamma augmentation, enabling Monte Carlo rendering and pixel-wise posterior uncertainty with minimal overhead. Theoretically, we establish posterior contraction rates for the scale parameters and transfer them to the rendered image via a local Lipschitz mapping, providing guarantees that estimation eттоr and predictive uncertainty diminish with data. Empirically, Horseshoe Splatting produces high-quality uncertainty maps while matching state-of-the-an 3DGS visual fidelity and runtime, yielding practical, uncertainty- renderer that is robust structured sparsity in the radiance field. The code is available ahtttss: /ttttt..thhgtggggggggggggg.ggh com/ MIKU-MedAI Horseshoe-Splatt.e
INTRODUCTION
Neural Radiance Fields (NeRFs) deliver figh-fidelity novel view synthesis by optimizing a continuous volumetric scene function (Mildenhall al. 202T). but their implicit networks are computation- ally intensive. The 3D Gaussian Splatting (3DGS) replaces per-ray network queries with explicit anisotropic Gaussians and differentiable rasterization pipeline, enabling real-time, high-resolution rendering while retaining strong visual quality (Kerbl et al.) 2023).
Despite its success, most 3DGS pipelines are deterministic, providing no notion confidence, which crucial under sparse views, occlusions, this setting can improve robustness by flagging unreliable regions and enabling active view selection mapping via prioritizing uncertain structural sparsity in the per-splat covariance (c.g.. axis-wise variances or pairwise correlations), which can leave noise-dominated directions insufficiently regularized (Figure Recent uncertainty- aware variants quantify aspects of scene ambiguity- e.g.. semantic/posterion map variance (Wilson) ctt al. 2025) or spatial depth uncertainty fields (Tan et al. 2025)-yet they do not target structured sparsity of the covariance itself, and hence cannot selectively suppress spurious variance or cross-axis
contents. Calibrated uncertainty in
Furthermore, these methods also do not explicitly encode
"Equal contribution.
1 "ooresponniig author.

---

Published as a conference paper at ICLR 2026
coupling in the Gaussian footprint. This gap limits both trustworthiness and robustness when signals are sparse or views are undersampled.
A principled way to address these issues is to bring Bayesian inference into 3DGS. While existing work has explored post-hoc pruning based on sensitivity (Hanson et al. 2025) or task-specific uncertainty modeling (Wilson et al. 2025 Tan al. 2025), there remains little on hierarchical priors that divectly regularize the per-splat covariance structure and yield coherent posterior uncertainty over rendered images.
To this end, we propose structured (global-local) priors on covariance scales. In particular, the Horseshoe prior provides high mass near zero (ag- gressive shrinkage of uninformative directions) and heavy tails (retention of salient structures)
(Carvalho et al. 2009 Pirronen and Vehtari 2017).
Placed on axis-wise scales-aae and when needed, on low-rank pairwise components-tth hierarchy conforms to how 3DGS forms elliptical screen- space footprints, allowing the renderer to suppress noise along irrelevant axes while preserving sharp, data-supported anisotropy and enabling pixel-wise uncertainty via posterior sampling.
Optimal Representation Unstructured Noise
Structured Noise
Figure 1: Unstructured and structured noise.
Building upon this, we introduce Horseshoe Splatting, a Bayesian 3DGS framework that imposes Horseshoe prior on per-splat covariance scales and fits factorized variational posterior. The variational family mirrors the Horseshoe's 's inverse-Gammn rendering from the learned posterior and pixel-wise uncertainty maps at time. To establish the theoretical soundness of our approach, we further prove that under a nonlinear observation model and a local Lipschitz renderer, the posterior over scales contracts at a near-minimax rate and transfers to image space, certifying that predictive uncertainty decreases with data.
augmentation, enabling Monte Carlo
Our contributions are: (1) A novel Bayesian formulation for 3DGS that imposes structural sparsity in the covariance via global-local Horseshoe hierarchy; (2) A tractable, factorized variational inference scheme that supports Monte Carlo (MC)-based rendering and pixel-wise posterior uncertainty with minimal overhead; (3) Uncertainty estimates that improve reliability and downstream tasks (e.g.. active view selection) without sacrificing speed; (4) Theoretically establishing posterior contrac- tion for scales and rendered images under standard smoothness and identifiability conditions; and (5) State-of-the visual fidelity on standard benchmarks, while providing calibrated uncertainty.
2 RELATED WORKS
Novel View Synthesis with NeRF and 3DGS. Novel view synthesis (NVS) has been significantly advanced by Neural Radiance Fields (NeRFs) (Mildenhall et al., 2021). which use continuous volumetric functions to achieve photorealistic rendering. Despite their success, NeRFs suffer from slow training and high computational cost (Müller al., 2022). Consequently, numerous variants
have emerged to improve rendering quality (Barron et al. (2021). accelerate training (Müller et al.)
2022), or model unbounded scenes (Zhang al.) 2020ap. More recently, 3D Gaussian Splatting (3DGS) (Kerbl et al. 2023) has become a highly efficient alternative, enabling real-time rendering and faster training by representing scenes with explicit Gaussian primitives. These Gaussians are typically initialized from sparse Structure from Motion (SIM) point cloud (Schonberger and Frahm 2016) and optimized with adaptive density control. Its superior performance has inspired rapid development, including extensions for dynamic scenes (Luiten et al. 2024 Sun al., 2024 Kim eet al., 2024), surface reconstruction (Guédon and Lepetit 2024
Lyu et al. 2024), and applications in
fields like autonomous driving (Bao al. 2025).
Uncertainty Estimation for Novel View Rendering. Quantifying prediction uncertainty is critical for trustworthy rendering in real-world applications (Amini-Naieni et al. 2024), yet remains a challenge for most NeRF and 3DGS methods. In the context of NeRF, uncertainty has been explored via several avenues: Bayesian approaches (Shen et al. 2021 2022). post-hoc methods (Goli et al. 2024), ensembles (Sünderhauf al. (023). and auxiliary networks (uue et al. 2024; Amini-Naieni et al. 2024). However, these techniques often introduce significant computational overhead, complex training, calibration difficulties. In contrast, uncertainty estimation for 3DGS is less explored
2

---

Published as a conference paper at ICLR 2026
Variational Inference
Horseshoe Pfor
-lvGGamma
ojiky InvGammaaaaaaa
Py Mean
PH
sir
c Variance
Pu
Global Parameter
Local Parameter
Gousslan Aucrruree
N(B f
RSSTR
Positon:: x
Nagatiue Log- liklihood
Uncertainty- aware Sampling
Computing
Opecta:: a
coelficients
Structured Gaussians
Single Gaussian
Lires Live
Adaptive
hiss Seniaee
Differentialee
Camera
Projection
Points
Render Output
Ground Truth
Uncertainty Nap
Figure 2: The framework of our proposed Horseshoe Splatting.
area. Recent works include FisherRF (Jiang et al., 2024). which uses Fisher information estimate uncertainty, and several methods that apply variational inference (Savant et al. 2024 Li and Cheung 2024) or uncertainty-awan regularization (Kim al. 2024). Despite these advances, effectively modeling the anisotropic variance of each Gaussian primitive remains key challenge, motivating our hierarchical Bayesian approach for more expressive uncertainty 3DGS. Beyond radiance fields, non-RF 3D reconstruction routinely uses confidence uncertainty (e.g., probability-volume variance or learned confidence in the multi-view stereo) to filter unreliable depth and guide fusion; our goals align with this tradition, but in the explicit 3DGS setting.
Shrinkage Priors for Sparsification. Shrinkage priors are Bayesian regularizers that pull noisy parameters toward zero while preserving strong signals. A prominent family of such priors is the global-local shrinkage class (Zhang al. 20206 Cadonna al. 2020 Bhadra al. 2017). which can be expressed as scale mixtures of Gaussians: each parameter 0 is drawn from N (0, T =x;) with global scale 7 and individual local scales A₃. This hierarchy enables adaptive sparsity: the global scale sets the overall shrinkage strength, while each local scale finely tunes how aggressively an individual coefficient driven toward zero. Effective shrinkage priors exhibit two key properties: sharp peak at zero to suppress noises, and heavy tails to avoid over-shrinking important signals. The Horseshoe prior (Carvalho et al. (2009) is prime example, achieving these properties through specific hierarchical construction. Despite their success in statisties and machine learning, their capability in tackling the sparsity issues 3DGS remains underexplored.
METHODOLOGGY
We present Horseshoe splatting, which imposes structural sparsity in the covariance via global-local Horseshoe hierarchy to suppress the noise along irrelevant axes while preserving sharp, data-supported anisotropy. Figure presents the overall framework with the detailed algorithm in Appendix E
3.1 SPARSITY-INDUCED PRIOR ON 3D GAUSSIAN
3D Gaussian Splatting. In 3D Gaussian Splatting (Kerbl et al. 2023), a scene is modeled explicitly as a set anisotropic Gaussians ("splats"), each parameterized by a center position μı; R3, an opacity a; E [0, 1], color e; [0, 1 derived from Spherical Harmonies (SH), and full covariance, Σ, = R, ST R. where R; E SO(3) is a rotation matrix and S = diag($;1 (111 $12, s2a) > 0 is a optimized to minimize the reconstruction error of set of posed input images, using a depth-aware rasterization that projects each 3D Gaussian 2D elliptical splat the image plane. The rendered color pixel is computed front-to-back
diagonal scale matrix. During training, these parameters

---

Published as a conference paper at ICLR 2026
compositing of overlapping splats,
N
f(u) = (11- Co T, = exp
aa)). J&l
This explicit, non-grid representation enables real-time, the discretization artifacts common in voxel or mesh-based methods.
novel view synthesis without
Horseshoe Prior on the Scaling Matrix. For each splat and axis j (1,2,3). we place global-local Horseshoe prior on the diagonal scale Sij of S diag(8:1 812, 813). Let 0j > 0 be the (axis-wise) global shrinkage and Aaj= > the local shrinkage. We model
Sij Pijz fl
MBB 02
with learnable mean B.y and scale oij softplus(p:s) The half Cauchy (C-) Horseshoe priors Aij ~ C+ C+ 1) and 0z C+ (0, b) are expressed via the IG-IG (InverseGaama) augmentation,
X Pb (L/2,, 1/Mg), vi IG(1/2, 1), 03 0115 IG(1/2, 1/E), & IG 1/6²).
This yields the joint density
N II (M o3 1/2, 1/26) ==1j=1
,D,, (DI (8))))
x fic(rr 1/22, 1) II fio 1/2, 1/6) fi(E)) 1/2,, 1/b?),
with Σ; = R,S.RR This parameterization keeps the Gaussian prior on Sy (with variance modulated by 22) while leveraging the Horseshoe's spike-and and-nnavy-taen behavior for adaptive shrinkage.
Stochastic Variational Inference. As exact Bayesian inference under the Horseshoe hierarchy is intractable, we adopt a mean-field variational family that mirrors the IG-IG augmentation. Writing the stacked variables
=(sigh X2 D.y {uy), 0 [0])
1/ use
(8. uu, 00000
aNl siy; Big, oG) qi aij, bis) gn cej. diy)
II gI aj, 3,) qi(E) 7j. 8,),
with independent factors. The evidence lower bound (ELBO) is
C(g) Egl l( [8.3)] +
Σ By log fa(sss) Big, of x9)2))
1.j
+ B log fiol fio(A?) 1/2, 1/M,)) + log fia(Us) 1/2, 1)]
+
Σ Eg log lic(0) 1/2, 1/E,) + log Sic(Es 1/2, 1/6?)] - loglog log X2, 02, E)].
The Gaussian prior term has closed form,
Eg log fa(s1) Bes, 03.00303335) = 1 log(2n) (2g( - log o
E,p log log - 1 Edd log X3))
[, - =]
1
E
E.
20
0
where for X ~ IG(a, 3) (shape-rate), we use E(log X) log 3 - v((a)) (where (-)) denotes the digamma function) and E(1/X) = a/3. and B(1/X) = a/3. We optimize C(g) by stochastic gradient ascent: Sty is sampled via the reparameterization 8.j B., oiy E, (0, 1), while the IG factors admit low- variance pathwise gradients via implicit/rranspor reparameterizztiins for Gamma/IG families; the KI. terms between IG factors remain analytic.
4

---

Published as a conference paper at ICLR 2026
3.2 INTEGRATING WITH RECONSTRUCTION LOSS
We now combine our variational objective with the 3D Gaussian Splatting (3DGS) reconstruction loss form the final training criterion. Let D {Iu]u=1 be the set of ground pixel colors. and ((S.)) the corresponding rendered colors (for single sample of {S.]). We define the reconstruction term as the expected negative log-likelihood under the variational posterior,
Free
1 M
Σ
Σ
In fN fimi) oi)
where f(m)) rendered with x)(") sampled from g, and x² is learnable variance parameter. We introduce surrogate parameter p such that log(1 + e²) to control the scale of prior variance.
The variational regularizer the negative ELBO's KL component,
LKL KT. KT [[( X2 0²) p(s, X² 0²)]
Putting these together, the total loss minimized by stochastic gradient descent is
Leotal Free + CKL
[S:)] + [L[g(s, A², 0²) 11 X2, 03)
The prior term p(-) of the KL divergence acts as an automatic scaling factor that balances the regularization power of the Horseshoc assumption. In practice, approximate both terms by Monte Carlo sampling from q and use their analytic KI. expressions under the inverse- Gamma factors. Minimizing Ltotal thus jointly drives accurate image reconstruction and enforces the Horseshoe- induced sparsity and uncertainty regularization on the Gaussian scales.
3.3 POSTERIOR INFERENCE AND UNCERTAINTY ESTIMATION
After training, we generate posterior predictions by sampling ancestrally through the Horseshoe hierarchy, avoiding any auxiliary Gaussian variance parameters. For each splat i and axis j. let the learned mean and base scale be (3kj, oiy), and let the variational factors over the global-local shrinkage variables be qIG (8:;; az B3) and qIG (A²z; axj. bij) (with optional IG factors for the aug-
mentation variables Ej, Pay when used). For m = 1,..., M. we draw (2(m) qIG (IG(0)), A
qiG () (O 1), and set the scale sample s/() = Big + Tie ((m)) ((m) (m)
Eij which is
exactly the conditional Gaussian p(s:j | 68, , X2 integrated against the variational posteriors of the Horseshoe scales (using the standard IG-IG
1G augmentation of the half Cauchy) (Makalic and Schmidt
2015; Ghosal et al. 2000). We then form >.(mm) R& diag( 811 (m) (m) [m])
$/2 .93 Rd render each draw
with the differentrable 3DGS rasterizer, and approximate the posterior predictive for each pixel u as
M
p(īn D)
p(u
R,, a,
R
M YA
Per-pixel predictive mean, variance, and credible intervals are computed from the rendered samples [ī(m) 1 M 1= yielding calibrated uncertainty maps while preserving the real-time rendering pipeline of 3DGS (Kerbl et al. 2023).
THEORETICAL ANALYSIS
We develop the theoretical properties, in particular the posterior concentration rates of the scaling matrices and the resulting rendered scenes. Under scene sparsity, only small subset of diagonal scale entries in the Gaussian covariances meaningfully contribute to rendering fidelity. Global-local shrinkage priors such as the Horseshoe concentrate posterior mass by aggressively shrinking noise- dominated coordinates toward zero while leaving large, hignal-bearing coordinates relatively unshrunk. Consequently, the posterior over the scale vector contracts around the sparse ground truth at minimax rate. Because the 3DGS renderer is smooth map these scales to images, this contraction propagates the rendered scenes with (at most) Lipschitz deformation of the rate.
5

---

Published as a conference paper at ICLR 2026
We model the effective measurements used to infer the per-splat scales noisy nonlinear map y = g(s") ) + E, where summarizes how multi-view image evidence responds to changes in the per-splat scales (e.g.. via differentiable rasterization statisties), and y E R denotes the observed data vector we use to infer the per-splat scale 8. We carry contraction from the scale space & to image space via local Lipschitz renderer R. The Horseshoe prior delivers adaptive sparsification and heavy-tailed protection, and its hear-minimat contraction in sparse normal means extends to our nonlinear setting by local linearization and standard testing/prior-mat arguments.
Assumption 1 (Nonlinear observation model). Let s* E IR be k-sparse. We observe y g(8") + e, where MO, I,,,,) (where I,, is the Py Py identity matrix), g RAN IR is twice continuously differentiable on a neighborhood (s*)..
Assumption 2 (Local identifiability and curvature). Let J. = Vg((8)) There exist 0 < Kmiin S Fimax Oc and Hg oo such that for all E (s*),
Kmin | s" l2 S [|J((--8")
s2)122 S Kmax | 1
( S lgg
where || -loop denotes the operator norm. Consequently, 9 admits the expansion g(s) = g(s") (s s*) + r(s) with [|(()]]22 S _ 1
- s*
Assumption (Horseshoe prior). Each coordinate has the global-local Horseshoe hierarchy (half-Cauchy scales) in an Toverse-Gammne mixture form:
s8 0y ~ (0 uE), ve de IG 2: h²): de 1e TG 2: 21)
0² ~ IG 2:086 )
Assumption (Growth regime). As N, P - oc, o(3N) and log caN
k = PPP)
Assumption 5 (Local Lipschitz renderer). Fix R2 C,, 0², The renderer R RSN + RM, I R(s), continuously differentiable (C*)) on neighborhood N(s*) and thus locally Lipschitz:
[R(s) - RR(s)) l12 & LR|ss s- s'1l2, ss,s'E NNs).
with LR. SuPpsEN(>> [| co.
Lemma 1 (Horseshoe contraction for the linearized model). Consider the surrogate linear model y g(s") + (s s*) E, with NO0, . Jp,). under Assumptions 2 and Then there exists
a"
C > 0 such that the posterior for s contracts at rate
02 k log
E3N
N. = C
P
min
and the posterior mean attains the same rate up to constants.
Under Assumptions 1 we linearize at s* and apply Lemma to obtain Horseshoe posterior contraction for the scale vector after the variance rescaling o² -²7P and with ambient dimension = 3N; the constants Kmin and H, control, respectively, local identifiability and the Taylor remainder. The rendered-iman image rate then follows from Assumption [ by Lipschitz transfer. incurring at most a multiplicative factor LR. Theorems I
J formalize these two steps precisely. Theorem (Contraction for sparse scales under nonlinear observations). Under Assumptions there exist constants C, C,M 0 such that if the target radius satisfies Hg ENP o(Fmin) (ensuring the sccondoorder remainder is negligible at that scale), then
2x 02 le loog
33N
s0112 M M v) 0. E [I(
R2
where II(: y)) denotes the posterior distribution. Consequently, E.- ||E[s y) s0113 = 2 Theorem (Contraction for the rendered image). Let I R(s) and I' R(s"). Under Assump- tion 5 with E LR&EN, from Theorem]
| there exists M' >> 0 such that
Eg- TT
- I" I2
M' EN.P v) + 0.
Remark 2.1 (Adaptive sparsification of splats). Theorem 7 shows Horseshoe shrinkage automatically drives near-zero scales to (near) zero at a near minimax rate depending on k and log(3N). This justifies pruning redundant splats and stabilizes optimization in sparse regions.

---

Published as a conference paper at ICLR 2026
Remark 2.2 (Preservation of large structures). Contraction together with the Horseshoe's 's heavy tails implies negligible bias for large coordinates; salient geometry (large Sxy) is preserved while noise is shrunk (van der Pas et al. 2014).
Remark 2.3 (Renderer sensitivity). Image-space contraction holds with a Lipschitz loss of at most LR. which is finite under the differentiable rasterizationno-ccoppoiiiit used by 3DGS on bounded neighborhoods.
EXPERIMENTS
5.1 DATASETS AND EXPERIMENTAL SETTING
Datasets. Our experiments are conducted on two datasets: the Light Field LF dataset (Yücer et al. 2016) and LLFF dataset [Mildenhall et al. 2019). For the LF dataset which contains 8 scenes with dense 360 views, we adopt the setup from CF-NeRF (Shen et al.) (022) and evaluate on the torch, basket, africa, and statue scenes. For the LLFF dataset, we use all 8 forward-facing scenes, including fern, flower, fort ress, horns, leaves, orchids, room, and trex, following established protocols (Shen et al. 2022 Li and Cheung 2024).
Task & Evaluation Metries. We focus on the novel view synthesis (NVS) task, where the objective is to render photorealistic images from new camera poses given only a sparse set of input views.
We assess model performance using standard metrics for both image reconstruction quality and uncertainty estimation. To evaluate the fidelity of synthesized views, we employ three metrics: Peak Signal-to-Noise Ratio (PSNR) for reconstruction accuracy, the Structural Similarity Index Measure (SSIM) for perceptual similarity, and the Learned Perceptual Image Patch Similarity (LPIPS) for perceptual distance. We evaluate the quality of our uncertainty predictions using two primary metrics. First, the Area Under the Sparsification Error (AUSE) curve, which measures the correlation between the estimated uncertainty and the true prediction error (MAE). Second, the Negative Log-I LikeLiiel (NLL), which measures the likelihood of ground truth in the predictive distribution.
Comparable Methods. 1) CF-NeRF (Shen et al. 2022 A NeRF-based method that captures uncertainty by learning conditional distribution
radiance values via a normalizing flow model. 2) S-NeRF (Shen et al. 202T): A Bayesian variant of NeRF that performs variational inference on the model's weights derive uncertainty for both rendered color and depth. 3) Bayes' Ray (Goli et al.) 2024): Introduces a spatial uncertainty field that perturbs input ray coordinates and propagates this geometric uncertainty through the radiance field using Laplace approximation. 4) FisherRF (Jang et al. 2024): A 3DGS method that uses Fisher information to estimate uncertainty; 5) Variational 3DGS (Li and Cheung 2024): variational inference framework for 3DGS that introduces multi- scale representation to explicitly model uncertainty. 6) Ensemble GS: Following Li and Cheung (2024). 10 3DGS models are trained with different subsets of initialization points from Structure from Motion (SIM) (Schonberger and Frahm 2016) and different random seeds. The variance of the predictions between all models is regarded as the predictive uncertainty.
Implementation Details. Our method extends the official 3DGS Py framework (Kerbl et al. 2023). We use standard StM initialization and the original training schedules for all base 3DGS parameters. The learning rates for our variational local and global scale parameters are to 1 x 10-4 and 1 x 10 10-5, respectively. At inference, uncertainty is estimated via 10 Monte Carlo samples. All experiments were run on a single NVIDIA RTX 3090 GPU.
5.2 EXPERIMENTAL RESULTS
Quantitative Results. We evaluate our method's performance in both uncertainty estimation and novel view synthesis, with results detailed in Table and Table
Uncertainty Estimation. For depth uncertainty on the LF dataset, our method achieves state-of- the-art average AUSE of 0.18, outperforming all baselines. This is highlighted in the basket scene, where our 0.10 AUSE marks 23% improvement over the next best method. For RGB image uncertainty, our method also excels. On the LF dataset, it delivers the best performance with a leading AUSE of 0.25 and significantly lower NIL of -0.74. On the more challenging LLFF dataset,
7

---

Published as a conference paper at ICLR 2026
Table I:
Depth uncertainty estimation (AUSE-MAE) performance the LF dataset. The best result in boldface, and the second-best is underlined.
LF Dataset
africa
basket
statue
torch
Average
2022 S-NeRF Shen
CF-NeRF (Shen al.
0.35
0.31
0.46
0.97
0.52
2021
0.66
0.38
0.67
0.74
0.61
Bayes' Ray (Gol
2024
0.27
0.28
0.17
0.22
0.23
FisherRF Diang et al.
2024
0.21
0.17
0.25
0.24
0.22
Variational 3DG (Li and Cheung 2024
0.19
0.13
0.21
0.23
0.19
Ensemble GS (x10)
0.16
0.22
0.17
0.26
0.20
Horseshoe Splatting (Ours)
0.19
0.10
0.20
0.24
0.18
Table 2: NVS and uncertainty estimation results on the LF and LLFF datasets. The best result is in boldface, and the second-best is underlined.
Synthesized View Quality
Uncertainty Quality
Dataset
Method
PSNR
SSIM
LPIPS
AUSE 4
NLL
CF-NeRF (hen al 2022
24.32
0.835
0.202
0.49
-0.37
S-NeKF (Shen et al.
20.21
0.761
0.248
0.62
1.32
FisherRI Uiang cl al 2124
29.13
0.927
0.076
0.54
7.02
LF
Variational 3DGS
Cheung 2024
27.39
0,914
0.26
-0.30
Ensemble GS (x10)
64
0.902
0.088
0.29
-0.34
Horseshoe Splatting (Ours)
30,05
0.947
0.064
0,25
-0.74
CF-NeRF (Shen et al. 2022
21.74
0.782
0.190
0,48
0.58
S-NeRII (Shen el al. 2021
20.10
0.744
0.221
0.59
0.91
FisherRI (tang et al 2024
25.34
0.849
0.125
0,51
7.05
LLFF
Variational 3DGS (L Cheung 2024
23.97
0.806
0.172
0.32
0.23
Ensemble GS (x10)
24.54
0.810
0.157
0,30
0.26
Horseshoc Splatting (Ours)
25.86
0.864
0.110
0.31
0.14
it achieves the best NLL of 0.14, indicating more accurate predictive distribution. These results confirm our model's ability to produce well-calibrated uncertainty across modalities and datasets.
Novel View Synthesis. As shown in Table 2 our method also outperforms others for rendering quality. On the LF dataset, achieves a PSNR of 30.05, surpassing the closest competitor by a significant margin of over 0.9, while also leading in SSIM and LPIPS. On the more challenging forward facing LLFF dataset, our method continues to excel, leading in image fidelity metrics. This demonstrates that approach not only provides superior uncertainty but also produces state-of state-of-the-art high- fidelity renderings.
Qualitative Results. Figure 3 visualizes rendering results on the fern scene. Our method achieves the best reconstruction quality. correctly rendering fine details like complex leaf structures. The error map shows that our errors are minimal and confined inherently challenging areas, such as thin leaf edges and dark boundary lines. The main advantage of our method is shown in the uncertainty maps. Our uncertainty estimate exhibits the strongest correlation with the actual error map, especially in difficult regions like the plant's 's stem. In contrast, while FisherRF achieves good reconstruction, its uncertainty estimation poor. This is because its Hessian-based approximation relies on depth information, which does not translate well to the RGB space, resulting in noisy and uninformative uncertainty. In summary, the visualization confirms that our method not only delivers superior reconstruction quality but also provides more accurate and reliable uncertainty estimates.
Active View Selection. A key application of reliable uncertainty is guiding data acquisition in active learning. We conduct an experiment on the LLFF dataset, starting with 10% of views and iteratively adding one view every 500 training steps based on uncertainty, up to 30% of total views. Specifically, we select the view with the highest integrated pixel-wise variance. We strictly enforce this budget alignment for all baselines, and the model is further trained for 7K steps after all images are chosen.
As shown in Table 4 our method significantly outperforms all baselines. This result confirms that our well-calibrated uncertainty is highly effective in identifying the most informative views, leading to greater data efficiency in reconstruction.
Inference Time Analysis. We report the inference speed of 3DGS basedd uncertainty methods in T5 The inference time is measured as the average time to render 1000 frames on the torch scene of the LF dataset. At just 0.03 seconds per view, our method is by far the fastest, running
8

---

Published as a conference paper at ICLR 2026
Table 3: Horseshoe Prior compared with Laplace Prior and Gaussian Prior.
Synthesized View Quality
Uncertainty Quality
Dataset
Method
PSNR
SSIM
LPIPS
AUSE +
NLL 4
Depth AUSE !
Laplace Prior
30.04
0.942
0.065
0.37
10.58
0.19
L.F
Gaussian Prior
30.01
0.941
0.067
0.38
9.15
0.18
Horseshne Prior
30.05
0.947
0.064
0.25
-0.74
0.18
Luplace Prior
25.74
0.860
0.112
0,42
8.22
LLFF
Gaussian Prior
25.61
0.859
0.116
0.42
6.98
Horseshoe Prior
25.86
0.864
0.110
0.31
0.14
Novel View
Unncertaitty
Hiror
Ensemble
S NeRRRF
CF-NeRF
FisherRF
Var 3DGS
Ours
Figure 3: Visualization of predicted uncertainty maps of novel view renderings.
approximately 9x faster than the costly Ensemble GS approach and significantly outpacing both FisherRF and Variational 3DGS. This result demonstrates that our approach provides state-of-the-an performance while simultaneously achieving the fastest rendering speed.
Table 4: The experiment on active learning.
Table 5: Inference time for 3DGS-base methods.
PSNR
SSIM 1
LPIPS
Inference Time (s)
Random
20.97
0.65
0.234
Ensemble GS (x10)
0.27 :: 0.050
FisherRF
23.37
0.81
0.144
FisherRF
+ 0.003
Variational 3DGS
21.35
0.69
0.212
Variational 3DGS
0.06 1 0.020
Horseshoe Splatting (Ours)
26.23
0.87
0.104
Horseshoe Splatting (Ouss)
0.03 4:: 0.005
5.3
ABLATION STUDY.
Effectiveness the Horseshoe Prior.
Table 6: Ablation study on hyperparameters p and AKL-
To validate our core contribution, we re- place the hierarchical Horseshoe prior with standard Laplace prior and Gaussian prior. As shown in Table 3 while render- ing quality remains comparable, the Horse- shoe prior yields dramatically better uncer-
Synthesized View Quality
Uncertainty Quality
PSNR
SSIM t
LPIPS
AUSE
NLL
P=
25.86
0.864
0.110
0.31
0.14
25.82
0.863
0.111
0.33
0.15
P=
25.78
0.863
0.110
0.33
0.14
25.96
0,866
0.109
0.33
0.15
tainty estimates. The most striking result the NLLI score, which plummets on both datasets (e.g., from 9.15 to -0.74 LF), indicating far more accurate predictive distribution. This confirms the Horseshoe prior's heavy-tailed nature is critical for modeling structural sparsity and producing well-calibrated predictive distribution.
p=
25.49
0.855
0.118
0.34
0.18
AKL O.I
25.61
0.861
0.113
0.32
0.16
AKL 0.01
25.70
0.860
0.114
0.33
0.15
AKE 0.001
25.86
0.864
0.110
0.31
0.14
AKL 0.0001
25.75
0.862
0.113
0.33
0.16
Sensitivity to Hyperparameter Prior Variance p and KL Weight AKI. We study the impact of two key hyperparameters: the log-scale prior variance p and the KL-divergence weight λKL. with results on the LLFF dataset shown Table | For p. which controls the global shrinkage strength, aim to find balance between rendering quality and uncertainty estimation. As shown in the table, p = 5 achieves the best uncertainty scores while maintaining strong rendering performance. Although 2 yields slightly higher PSNR, its uncertainty metrics worse. Therefore,

---

Published as a conference paper at ICLR 2026
prioritize well-calibrate uncertainty without significantly compromising rendering quality, select -5 for our main experiments.
For ÄKL. the model achieves the best overall performance at λKL 0.001, leading in both rendering quality and uncertainty metrics. This indicates that while the model is robust, a smaller KL weight is beneficial. We use λKL 0.001 in experiments.
Verification of Structural Sparsity. Table 7 quantitatively confirms the induced structural sparsity. Across all thresholds c {5.3. 1, 0.5), the Horseshoe prior consistently yields the high- est proportion of near-zero scales (e.g.. 64.62% vs. 61.01% for Laplace at e = 5). This quan- titative evidence is further corroborated by the visualization of the posterior densities in Fig- ure 4 Compared to the Gaussian and Laplace baselines, the Horseshoe posterior over the scale parameters $ty exhibits distinct "spike-ate maikee-t-zzeo"
Table 7: Comparison of Structural Sparsity Induc- tion on LF torch.
Sparsity Ratio (% of scales < c)
Method
<= 5
<=
3
<=
0.5
Prior
60.73
31.57
6.59
2.98
Gaussian Prior
60.85
31.64
6.64
3.00
Laplace Prior
61.01
31.95
6.78
3.07
Horseshoe Prior
64.62
34.09
7.17
3.21
distribution combined with heavy tails. This be- havior indicates aggressive shrinkage of noise-dominated scales while simultaneously preserving large, signal-bearing soefficients-aligninn perfectly with the theoretical properties of Horseshoe priors (Carvalho et al., 2009) Pironenn and Vehtari 2017;; van der Pas et al. 2017). In contrast, the Laplace prior's lighter tails risk over-shrinking significant features, while the Gaussian prior fails to induce meaningful sparsity. These results confirm that the performance gains are specific to the
Horseshoe hierarchy, rather than simply the result of applying "any sparsity prior" (Park and Casella 2008
Bhadra et al. 2019).
10000
10.00
800
Figure 4: Posterior density of covariance scale (s:y) the LF torch scene under different priors (Left: Gaussian; Middle: Laplace; Right: Horseshoe). The Horseshoe prior exhibits a pronounced spike at along with heavy tails, inducing structural sparsity by suppressing noise while preserving salient scales, whereas Gaussian and Laplace priors fail to achieve this optimal balance.
CONCLUSION
We propose Horseshoe Splatting, a Bayesian extension of 3D Gaussian Splatting that places global-local Horseshoe prior on per-splat covariance scales to encode structural sparsity and deliver calibrated uncertainty. This prior's spike-at-zero and heavy tails aggressively shrink noise-dominated directions while preserving data-supported anisotropy. directly tackling artifacts that arise in deter- ministic 3DGS pipelines and enabling pixel-wise uncertainty via factorized variational inference and Monte-Carke Carlo rendering. Theoretically, we establish posterior contraction for the scale vector and transfer it to image space under locally Lipschitz renderer, while empirically we maintain the strong real-time fidelity of 3DGS and obtain reliable uncertainty maps that highlight under-constrained regions. Together, these results show that principled prior design turns explicit splatting into robust, uncertainty-awarc renderer. Looking ahead, we plan to explore alternative structured priors (e.g.. low-rank/ Wishart constraints) and adaptive splat birth-pruning to further couple sparsity, fidelity, and confidence.
ACKNOWLEDGEMENTS
This work supported in part by the Research Grants Council of Hong Kong (27206123, 17200125, C5055-24G, and T45-401/22-N), and the Hong Kong Innovation and Technology Fund
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Niki Amini-Naieni, Tomas Jakab, Andrea Vedaldi, and Ronald Clark. Instant uncertainty calibration of NeRFs using meta-calibrator. In European Conference on Computer Vision, pages 309-324. Springer, 2024.
Yanqi Bao, Tianyu Ding, Jing Huo, Yaoli Liu, Yuxin Li, Wenbin Li, Yang Gao, and Jiebo Luo. 3D Gaussian Splatting: Survey, technologies, challenges, and opportunities. IEEE Transactions on Circuits and Systems for Video Technology, 2025.
Jonathan T Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin-Brualla, and Pratul P Srinivasan. Mip-NeRF: A multiscale representation for anti- anti-aliasing neural radiance fields. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5855-5864, 2021.
Anindya Bhadra, Jyotishka Datta, Nicholas G Polson, and Brandon Willard. The Horseshoe+ estimator of ultra-sparse signals. Bayesian Analysis, 1044::1105-11111 2017.
Anindya Bhadra, Jyotishka Datta, Nicholas Polson, and Brandon Willard. Lasso meets Horseshoe. Statistical Science, 34(3):005-422 2019.
Annalisa Cadonna, Sylvia Frühhwitth-Schnater and Peter Knaus. Triple the gamma-A A unify- ing shrinkage prior for variance and variable selection in sparse state space and TVP models. Econometrics, 8(2):20, 2020.
Carlos M Carvalho, Nicholas G Polson, and James G Scott. Handling sparsity via the Horseshoe, In Artificial Intelligence and Statistics, pages 73-80. PMLR, 2009.
Ghosal, J. K. Ghosh, and W. van der Vaart. Convergence rates of posterior distributions. Annals of Statistics, (8(2):5005531 2000.
Lily Goli, Cody Reading, Silvia Sellán, Alec Jacobson, and Andrea Tagliasacchi. Bayes' rays: Uncertainty quantification for neural radiance fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20061-20070, 2024.
Antoine Guédon and Vincent Lepetit. Sugar: Surface-aligned Gaussian Splatting for efficient 3D mesh reconstruction and nigh-quality mesh rendering. In Proceedings of the IEEÉ/CVF Conference on Computer Vision and Pattern Recognition, pages $354-5363. 2024.
Alex Hanson, Allen Tu, Vasu Singla, Mayuka Jayawardhana, Matthias Zwicker, and Tom Goldstein. Pup 3D-GS: Principled uncertainty pruning for 3D Gaussian Splatting. In Proceedings of the Computer Vision and Pattern Recognition Conference, pages 5949-5958, 2025.
Wen Jiang, Boshu Lei, and Kostas Danilidis. FisherRF: Active view selection and mapping with radiance fields using fisher information. In European Conference on Computer Vision, pages 422-440. Springer, 2024.
Bernhard Kerbl, Georgios Kopanas, Thomas Leimkühler, and George Drettakis. 3D Gaussian Splatting for real-time radiance field rendering. ACM Trans. Graph. (SIGGRAPH), 2023.
Mijeong Kim, Jongwoo Lim, and Bohyung Han. 4D Gaussian Splatting in the wild with uncertainty- aware regularization. Advances in Neural Information Processing Systems, 37: 17209999-122 2024.
Ruiqi Li and Yiu-ming Cheung. Variational multi-scale representation for estimating uncertainty in 3D Gaussian Splatting. Advances in Neural Information Processing Systems, 37:87934-8799 2024.
Jonathon Luiten, Georgios Kopanas, Bastian Leibe, and Deva Ramanan. Dynamic 3D Gaussians: Tracking by persistent dynamic view synthesis. In 2024 International Conference on 3D Vision (3DV), pages 800-809. IEEE, 2024.
Xiaoyang Lyu, Yang- Tian-Tianoo Sun, Yi-Hua Huang, Xiuzhe Wu, Ziyi Yang, Yilun Chen, Jiangmiao Pang, and Xiaojuan Qi. 3DGSR: Implicit surface reconstruction with 3D Gaussian Splatting. ACM Transactions Graphics (TOG), 13 2024.

---

Published as a conference paper at ICLR 2026
Enes Makalic and Daniel F Schmidt. A simple sampler for the Horseshoe estimator. IEEE Signal Processing Letters, 23(1):179 2015.
Ben Mildenhall, Pratul P Srinivasan, Rodrigo Ortiz-Cayon Nima Khademi Kalantari, Ravi Ra- mamoorthi, Ren Ng, and Abhishek Kar. Local light field fusion: Practical view synthesis with prescriptive sampling guidelines. ACM Transactions on Graphics (ToG), 38(4):1-14, 2019.
Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. NeRF: Representing scenes as neural radiance fields for view synthesis. Commumications of the ACM, 65(1)999-106, 2021.
Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. Instant neural graphics primitives with multiresolution hash encoding. ACM Trans. Graph. (SIGGRAPH), 2022.
Richard Nickl. Bayesian Non-linear Statistical Inverse Problems. Cambridge University Press, 2022.
Trevor Park and George Casella. The Bayesian Lasso. Journal of the American Statistical Association, 103((82):681 1686, 2008.
Juho Piironen and Aki Vehtari. Sparsity information and regularization in the Horseshoe and other shrinkage priors. Electronic Journal of Statistics, 11:5018-5051 2017.
Luca Savant, Diego Valsesia, and Enrico Magli. Modeling uncertainty for Gaussian Splatting. arXiv preprint arXiv:2403. 1otte22443.877.,,. 18476, 2024.
Johannes L Schonberger and Jan-Michael Frahm. Stracture-from-mottio revisited. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 4104-4113, 2016.
Jianxiong Shen, Adria Ruiz, Antonio Agudo, and Francese Moreno-Noguet Stochastic neural radiance fields: Quantifying uncertainty in implicit 3D representations. In 2021 International Conference 3D) Vision (3DV), pages 972-981. IEEE, 2021.
Jianxiong Shen, Antonio Agudo, Francesc Moreno-Noguer, and Adria Ruiz. Conditional-flow NeRF: Accurate 3D modelling with reliable uncertainty quantification. In European Conference Computer Vision, pages 540-557. Springer, 2022.
Jiakai Sun. Han Jiao, Guangyuan Li, Zhanjie Zhang, Lei Zhao, and Wei Xing. 3DGStream: On-the- fly training 3D Gaussians for efficient streaming of photo-realistic free-viewpoint videos. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 20675-20685, 2024.
Niko Sünderhauf, Jad Abou-Chakra, and Dimity Miller. Density-aware NeRF ensembles: Quantifying predictive uncertainty in neural radiance fields. 2023 IEEE International Conference on Robotics and Automation (ICRA), pages 9370-9376. IEEE, 2023.
Zhen Tan, Xieyuanli Chen, Jinpu Zhang. Lei Feng, and Dewen Hu. Uncertainty normal- guided Gaussian Splatting for surface reconstruction from sparse image sequences. arXiv preprint arXiv:2503. / 2025.
van der Pas, B. Szabó, and A. van der Vaart. Adaptive posterior contraction rates for the Horseshoe. Electronic Journal of Statistics, 11(2):3196-333 12255 2017.
S. L. van der Pas, J. K. Kleijn, and W. van der Vaart. The Horseshoe estimator: Posterior concentration around nearly black vectors. Electronic Journal of Statistics, 8(2):2585-22) 2618, 2014.
Joey Wilson, Marcelino Almeida, Min Sun, Sachit Mahajan, Maani Ghaffari, Parker Ewen, Omid Ghasemalizadeh, Cheng-Hao Kuo, and Arnie Sen. Modeling uncertainty in 3D Gaussian Splatting through continuous semantic splatting. In 2025 IEEE International Conference on Robotics and Automation (ICRA), pages 3284-3290. IEEE, 2025.
Shangjie Xue, Jesse Dill, Pranay Mathur, Frank Dellaert, Panagiotis Tsiotra, and Danfei Xu. Neural visibility field for uncertainty-drie driven active mapping. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 18122-18132, 2024.
12

---

Published as a conference paper at ICLR 2026
Kaan Yücer, Alexander Corkine-Hornune Oliver Wang, and Olga Sorkine- Hornnng.. Efficient 3D object segmentation from densely sampled light fields with applications to 3D reconstruction. ACM Transactions Graphics (TOG), 35(3):1-15, 2016.
Kai Zhang, Germot Riegler, Noah Snavely, and Vladlen Koltun. NeRF++: Analyzing and improving neural radiance fields. arXiv preprint urXiv:2010.07492, 2020a.
Yan Dora Zhang, Brian P Naughton, Howard D Bondell, and Brian Reich. Bayesian regression using prior on the model fit: The R2-D2 shrinkage prior. Journal of the American Statistical Association, pages 1-13, 2020b.
13

---

Published as a conference paper at ICLR 2026
APPENDIX SUMMARY
Section A describes our use of Large Language Models (LLMs). Section outlines the resources provided to ensure the reproducibility of work. Section contains proofs of our main theorems and derivations for the KL divergences, Section presents additional qualitative results. Section provides the pseudo-code for our method. Section provides the additional experiment results.
A THE USE OF LARGE LANGUAGE MODELS (LLMs)
Large Language Models (LLMs) were used in this work solely for language polishing and improving the clarity of writing. No LLMs contributed to research ideation, experimental design, analysis, or the development of core scientific content. All conceptual and technical contributions are original to the authors.
B REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our work, we provide a comprehensive set of resources. Our full implementation available as anonymous code repository, linked in the abstract, which includes all necessary code and usage instructions to replicate our experiments.
C TECHNICAL DETAILS
C.I PROOFS OF MAIN THEOREMS
Proof of Lemmal 7 Set z = J.(8 8*). Since Gmin (J.) 2 Kmins the Moore-Penrose bound gives |s s" 2 I || || = K k |=/2|
where J! denotes the Moore Penrose pseudoinverse of J. Under the surrogate linear model, the likelihood is Gaussian,
E~A(O o Pp.)
y = g2+))++++
Equipping with the Horseshoe prior (specified by Assumption 3) induces global-local shrinkage prior on z through the linear map J.. By the testing-pluss pars--puuu-oo---iii program for posterior con- traction (Ghosal al. (000)) together with near-minimax Horseshoe rates in sparse normal means (van der Pas et al. 2014 2017).
e3N k log
|| C
v) - O.
P
k
Pulling back through J scales Euclidean error by at most Kmii hence
s0113 - K m || = Onl 0² klog(e3N//
Kinin
which is the desired rate; the posterior-mean bound follows from Jensen's 's inequality using the posterior tail control (Goosa et al. 2000).
Proof of Theorem
By a second-order Taylor expansion at 8*,
||(s))]22 < H ||s s'113.
g(s") + J.(s s²) (s)) +E,
For the linearized likelihood with mean g(s*) ) + (s s*), Lemmal (invoking (Ghosa et al. 2000 van der Pas et al. 2014 2017)) yields contraction at radius
2x = C 2² & 3og(e33/K))
P
min
14

---

Published as a conference paper at ICLR 2026
On the complement of the contraction ball {||s s*112 >> EN,P), the remainder satisfies ||r())]la I + ||s =11 o(Eminn 8 s" (12) under the side condition Hg EN.P o(Kmin). Therefore, the nonlinear and linearized log-likelihood ratios differ by o(1) uniformly on the alternative, so exponentially powerful tests and KL-ball prior mass transfer by Le Cam-type contiguity/bracketnng arguments (Ghosal et al., 20000 Nickl 2022). Consequently,
II Is
s s112 M M U) 00
and Eg [|| y] s01 -s-
(
Proof of Theorem
By local Lipschitzness of the renderer (Assumption 5).
| - 12 S LR || -s s²122-
Hence for any M' > 0,
I M'LRENP) { s'112 > M'EN.r)-
Taking posterior probabilities and expectations under 8" and applying Theorem D gives contraction in image space at rate E = LRRE P; this is the standard Lipschitz pushforward used for nonlinear Bayesian inverse problems (Nickl 2022).
C.2 CLOSED FORMS KL DIVERGENCES
KL Divergence Between Two Inverse Gamma Distributions Below is the KL divergence between two inverse-Gamma distributions,
((>)= Inv-T(a;; Ogg Bg), p(x) Inv-r(a;; Ops 9p).
where
ga --1 exp -
Inv-I'(x; a, T(a))
The KL divergence KL(qllp) is
KI(glP) / ((x) In ((x) p(z)
dr
Bg
Bp
= (a. ap) v(a)) In [(ag)) + InT(ap) + ap In + ag
ag:
Bp Ba
where (-) is the digamma function.
D ADDITIONAL VISUALIZATION RESULTS
D.1 DEPTHI AND UNCERTAINTY ON THE LF DATASET
Figure 5 shows additional qualitative results for depth rendering and uncertainty estimation on the LF dataset. Our method accurately captures the geometry and produces uncertainty maps that correlate well with the error maps.
D.2 RGB AND UNCERTAINTY ON THE LLFF DATASET
pressens additional qualitative results for novel view synthesis and uncertainty estimation on the LLFF dataset. Across all scenes, our method generates high-fidelity renderings, and the corresponding uncertainty maps correlate well with the reconstruction errors.
E ALGORITHM DETAILS
The pseudo-code for our Horseshoe Splatting algorithm is provided below.
15

---

Published as a conference paper at ICLR 2026
Africa
Statue
Torch
Ground Truth
Rendered Depth
Error Mup
Uncertainty Mup
Figure 5: The visualization of depth rendering and uncertainty on LF dataset.
Algorithm 1 Horseshoe Splatting (training and posterior predictive rendering)
Input: images D, camera poses, number of Gaussians N, initial {14, R4, Ci Gr, ) N base scale
params {Bej. oij Jim.,,,, j=1.33
Prior (global-local Horseshoe): st Oy N( 03,03223).
X | Vis
IG( 2 1/vsg), Ii 2: 1), 03 Ey E 1/E,);
1/EE), ( 1/b²)
Init VI factors: for all i set ()g ai big). quuluii Cijy diyd;;); for each axis set qIG 3,), (E) 7g. 55)
1: for epoch = to Repoch do
2: Sample shrinkage scales: X2 ~ QI (A2)), Ujj~ (Vg), 03~91a(00 ng(())) Ey (EG)
3: Sample noise: Eig ~ NO0 1) and set 8új Big + Jig 0j Aij Eij
4: Build S = diag(S11, 822, 8:3) and Σ, = S R.
5: Render predicted pixels {f.} with 3DGS rasterization
6: Lrec f - E,, p(I, I.)
7: CKL E.,, KL (A2)) IG 1/m;)]] (L)a(()))) [ 1)]
+ +3 [KL (3) [G 1/&)] (L[oa (E;) IG 1/6*)] 9: Update {Mis Ri, 0i,
8:
02 Big, )i and all VI parameters by gradients of Cree LKI.
10: end for
Posterior predictive & uncertainty (MC rendering):
11: for m = to M do
12:
Draw X m)
qi (A2) (?(m)
(na(09) (m)
Sij (0,1
%i (m) Bij + aig glm) (m) (m)
Xis Eij s(m) R2 diag(s
(m) (m) (m)) R.
13:
8822 88
14: Render image f(mm); store pir-pixel samples
end for
16: return posterior mean i Em f(m) and per-pixel variance [(r[
F ADDITIONAL EXPERIMENTAL RESULTS
F. EVALUATION ON CHALLENGING LARGE-SCALE DATASETS
To validate the robustness of our method in diverse and challenging scenarios, we conducted compre- hensive additional experiments the Tanks & Temples and Mip-NeRF 360 datasets, which feature
16

---

Published as a conference paper at ICLR 2026
Flower
Fortress
Horms
Orchids
Ground Truth
Rendered Image
Ervo Map
Uncertainty Mup
Figure 6: The visualization of RGB images rendering and uncertainty on the LLFF dataset.
large-scale, unbounded outdoor scenes with complex geometries and varying lighting conditions. The quantitative results, presented in Table demonstrate our method's 's exceptional generalization capabilities. Specifically, we achieve substantial improvement in uncertainty calibration compared to baselines: on Tanks & Temples, our method reduces the Negative Log Likeliiod (NLL) from 2.46 (Variational 3DGS) 0.58, and Mip-NeRF 360 from 2.88 to 0.72. This confirms that
17

---

Published as a conference paper at ICLR 2026
Horseshoe prior effectively induces meaningful structural sparsity in complex wild set- tings. Furthermore, we maintain or exceed the rendering quality (PSNR/SSIM//PIIP)) of competing uncertainty-awart methods, proving that our improved uncertainty modeling does not compromise visual fidelity.
Table Novel View Synthesis (NVS) and Uncertainty Estimation on large-scale datasets (Tanks Temples and Mip-NeRF360), Our method achieves significantly better uncertainty metrics (AUSE, NLL) while maintaining superior visual quality. Best results are in boldface.
Synthesized View Quality
Uncertainty Quality
Dataset
Method
PSNR t
SSIM f
LPIPS
AUSE
NLL +
FisherRF
23.41
0.837
0.198
0.57
5,94
Tanks & Temples
Variational 3DGS
23.45
0.835
0.199
0,47
2.46
Ours
23.67
0.845
0.186
0.35
0.58
FisherRF
27.36
0.803
0.238
0.56
6.11
Mip-NeRF360
Variational 3DGS
27.28
0.797
0.250
0.51
2.88
Ours
27.68
0.810
0.207
0.35
0.72
F.2 COMPUTATIONAL COST ANALYSIS
To demonstrate accessibility and efficiency, we benchmarked our method against Vanilla 3DGS and other uncertainty-ant methods single NVIDIA RTX 3090 GPU (Table ( It is crucial to note that Vanilla 3DGS is deterministic and inherently incapable of estimating uncertainty; obtaining uncertainty via standard Ensemble strategy (e.g., the 10 baseline) would linearly scale the training time and resource usage by factor of 10, making it prohibitively expensive. While our Bayesian formulation incurs moderate overhead compared
single deterministic 3DGS (training: 42s + 87s), this represents highly efficient trade-off compared to ensembles. Moreover, compared to other single-model uncertainty methods, our approach is the only one to maintain real-time rendering speeds (33 FPS), significantly outperforming FisherRF (8 FPS) and Variational 3DGS (17 FPS). With peak memory usage of only 1.43 GB and a compact model size of 59 MB, our method remains lightweight enough for deployment commodity hardware, demonstrating its practicality for real-world applications.
Table Computational Cost on the LF torch scene, All experiments were conducted on a single
NVIDIA RTX 3090 GPU.
Method
Training Time
FPS
Peak Memory
Model Size
3DGS(x
42.17s
494
0.47GB
49MB
FisherRF
55.22s
1.20GB
128MB
Variational 3DGS
81.08s
17
0.87GB
99MB
Ours
87.33s
33
1.43GB
59MB
F.3 DOWNSTREAM UTILITY OF UNCERTAINTY
To further demonstrate the practical value of our uncertainty estimates, we evaluated our method on two downstream tasks: Active View Selection on the large-scale Tanks & Temples dataset, and Out- of-Distribution (OOD) View Detection on the LLFF dataset. The results are presented Tables and Regarding active view selection, our uncertainty-guiid
11.
selection on Tanks & Temples achieves PSNR 20.05, significantly outperforming Random selection (16.87) and Variational 3DGS (19.86), confirming that our uncertainty estimates successfully identify the most informative geometric regions under sparse data conditions. In the OOD view detection experiment on LLFF, our method achieves a remarkable AUROC of 0.8732 in identifying novel views, far surpassing Variational 3DGS (0.7345) and FisherRF (0.5696). This demonstrates that our uncertainty is well-calibrated enough to serve as a reliable signal for anomaly detection in safety-critical scenarios.
18

---

Published as a conference paper at ICLR 2026
Table 10: Active Learning on Tanks & Temples.
Table 11: OOD View Detection on LLFF
Method
PSNR
SSIM
LPIPS
Method
AUROC
Random
16.87
0.61
0.345
FisherRF
0.5696
FisherRF
17.00
0.61
0.346
Variational 3DGS
19.86
0.67
0,330
Variational 3DGS
0.7345
Ours
20.05
0.71
0.322
Ours
0.8732
19


Reviews:
[{"review_id": "wu5GOLh2hk", "reviewer": "Reviewer_WvZK", "summary": "This paper introduces Horseshoe Splatting, a Bayesian extension of 3D Gaussian Splatting (3DGS) that addresses structural sparsity in per-splat covariances while providing calibrated uncertainty estimates. The key innovation is the application of a global-local Horseshoe prior on covariance scales, which adaptively shrinks noise-dominated directions while preserving salient anisotropic structure. The authors develop a factorized variational inference scheme that enables Monte Carlo rendering with pixel-wise posterior uncertainty estimation. Theoretically, they establish posterior contraction rates for scale parameters and propagate these guarantees to rendered images via local Lipschitz mapping. Empirically, the method achieves state-of-the-art visual fidelity on standard benchmarks while producing high-quality uncertainty maps with minimal computational overhead.", "questions": "- Only covariance scales are stochastic; opacity, color (SH), and positions appear deterministic. Are there failure modes where scale uncertainty alone is insufficient?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper addresses a genuine limitation of existing 3DGS methods: the lack of explicit structural sparsity encoding in per-splat covariances and the absence of principled uncertainty quantification\n- The observation that noise-dominated components in covariance structures remain insufficiently regularized is well-articulated and demonstrated through visualizations (Figure 1)\n- The choice of Horseshoe prior is well-justified by its spike-at-zero and heavy-tail properties, which are precisely what is needed for adaptive sparsification\n- The method achieves state-of-the-art uncertainty estimation (Table 1: average AUSE of 0.18 on depth, NLL of -0.74 on RGB for LF dataset)\n- Novel view synthesis quality is not sacrificed for uncertainty: PSNR of 30.05 on LF dataset, outperforming baselines by significant margins", "weaknesses": "- The paper addresses a genuine limitation of existing 3DGS methods: the lack of explicit structural sparsity encoding in per-splat covariances and the absence of principled uncertainty quantification\n- The observation that noise-dominated components in covariance structures remain insufficiently regularized is well-articulated and demonstrated through visualizations (Figure 1)\n- The choice of Horseshoe prior is well-justified by its spike-at-zero and heavy-tail properties, which are precisely what is needed for adaptive sparsification\n- The method achieves state-of-the-art uncertainty estimation (Table 1: average AUSE of 0.18 on depth, NLL of -0.74 on RGB for LF dataset)\n- Novel view synthesis quality is not sacrificed for uncertainty: PSNR of 30.05 on LF dataset, outperforming baselines by significant margins", "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": "w2Yl2dsMHG", "reviewer": "Reviewer_xYVt", "summary": "This paper proposes Horseshoe Splatting, which applies a global–local Horseshoe shrinkage prior to the covariance scales of each Gaussian in 3D Gaussian Splatting (3DGS) to encode structured sparsity. Using an inverse-gamma augmentation equivalent to the half-Cauchy prior, the method builds a factorized variational family for Monte Carlo rendering and pixel-wise uncertainty estimation. The authors also provide theoretical guarantees on posterior contraction under Lipschitz assumptions and validate the method on LF/LLFF datasets with improved view and uncertainty quality.", "questions": "It would be helpful if the authors could validate the method on more wild or challenging scenes, such as MipNeRF or Tanks and Temples, to demonstrate broader applicability.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper’s exploration of uncertainty modeling in 3DGS is valuable. Incorporating both uncertainty and structured sparsity is novel, as most existing 3DGS uncertainty modeling focuses on geometry, semantic fields, or Fisher information approximations, with little direct regularization on covariance structures.\n2. The experiments are comprehensive, jointly evaluating RGB and depth uncertainty (e.g., NLL, AUSE) and active view selection scenarios, with ablations (Gaussian vs. Horseshoe) demonstrating the benefit of the proposed prior.", "weaknesses": "1. The local Lipschitz assumption may be fragile since 3DGS rendering involves depth sorting and α-blending, which can cause discontinuities around occlusion or visibility changes. More clarification or evidence on how local neighborhoods avoid these discrete transitions would strengthen the theory section.\n2. The active-view selection strategy lacks details — the paper mentions adding one view every 500 steps until 30%, but the convergence criterion, acquisition function, and budget alignment across methods are not clearly defined.", "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": "TZ2Nhafteq", "reviewer": "Reviewer_QC4h", "summary": "This paper introduces Horseshoe Splatting, a Bayesian variant of 3D Gaussian Splatting (3DGS) that explicitly models structural sparsity and uncertainty in radiance field rendering. The method places a global–local Horseshoe prior on per-splat covariance scales to suppress noisy or redundant directions while retaining key anisotropic structures. A variational inference framework enables Monte Carlo rendering and pixel-wise uncertainty estimation with little computational cost. Experiments on Light Field and LLFF datasets show that the approach achieves state-of-the-art image quality and well-calibrated uncertainty, while maintaining real-time rendering performance.", "questions": "1. Generalization and Robustness – How does Horseshoe Splatting perform on dynamic or large-scale outdoor scenes where scene sparsity and motion are more complex?\n  2. Computational Cost – What is the training time and memory overhead compared to vanilla 3DGS? Could this method realistically run on commodity GPUs?\n  3. Ablation on Horseshoe Hierarchy – How sensitive is performance to the hierarchical prior choice? Would simpler priors (e.g., Laplace or Gaussian scale mixtures) achieve similar uncertainty quality?\n  4. Downstream Use of Uncertainty – Beyond visualization, have the authors tested whether the uncertainty maps improve robustness in active learning or out-of-distribution detection?\n  5. Scalability and Model Release – Are there plans to release pretrained models or a lighter implementation for reproducibility?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Innovative Bayesian extension of 3DGS that meaningfully incorporates structural sparsity and uncertainty estimation.\n  2. Solid theoretical grounding, with posterior contraction guarantees linking model uncertainty to data consistency.\n  3. Strong empirical validation, outperforming prior methods on fidelity and uncertainty metrics without slowing inference.\n  4. Clear writing and presentation, with convincing visual and quantitative results.", "weaknesses": "1. Limited Conceptual Novelty Beyond Integration. While the Horseshoe prior is applied creatively to 3DGS, the overall contribution is more of a principled integration of known Bayesian shrinkage techniques rather than a fundamentally new rendering paradigm. The work’s novelty thus lies mainly in adaptation, not invention.\n  2. Dependence on Heavy Computational Infrastructure. The approach relies on large-scale Gaussian Splatting models and variational inference, which may limit reproducibility and accessibility. It’s unclear how well the method scales down to smaller datasets or lightweight hardware settings.\n  3. Lack of Diversity in Experimental Scenarios. Experiments are limited to static and relatively clean datasets (LF, LLFF). The paper does not demonstrate how the model behaves under dynamic scenes, noisy inputs, or severe view sparsity — conditions where uncertainty modeling would be most critical.\n  4. Insufficient Analysis of Practical Utility of Uncertainty. While uncertainty maps are visually convincing, the paper provides little quantitative evidence of how uncertainty benefits downstream tasks, beyond one active view selection experiment. More demonstration of practical value would strengthen the impact.", "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": "Q4813uQMIs", "reviewer": "Reviewer_W9qb", "summary": "This paper proposes Horseshoe Splatting, a Bayesian extension of 3DGS for radiance field rendering. The key idea is to introduce a global–local Horseshoe prior over per-splat covariance scales, allowing the model to encode structured sparsity in the Gaussian footprints while also providing pixel-wise uncertainty estimates.  Empirically, the proposed method achieves state-of-the-art image quality and well-calibrated uncertainty on the LF and LLFF benchmarks, outperforming both prior NeRF- and 3DGS-based uncertainty modeling methods, while maintaining real-time rendering speed.", "questions": "See the weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper introduces a conceptually clear global–local Horseshoe for 3DGS uncertainty estimation while keeping sufficient efficiency.\n\n- Theoretical derivations are provided to support the method. \n\n- Strong performance is shown on LF and LLFF.", "weaknesses": "- It is unclear whether the observed benefits are specific to the Horseshoe, or simply due to introducing any sparsity prior. Also, the ablation only reports improved metrics on the datasets, not how much structural sparsity was actually induced. Thus, the paper lacks quantitative evidence that the prior truly enforces the claimed structural sparsity. It should be further verified whether Horseshoe is necessary and essential.\n\n- The assumption of scaling sparsity places priors on covariance scales, which has not been well verified whether this is sufficiently generalizable for various scenes. Also, as it does not rely on color or opacity, the effectiveness may be limited in scenes dominated by photometric ambiguity rather than geometric uncertainty, such as non-Lambertian surfaces.\n\n- The experiments regarding active view selection may be too ideal. Considering there initially does not contain too many views in the LF dataset, starting with 10% of views may be too sparse and seems to be a toy study that can hardly match the real-world application. Would like to see if the method can still work in more scenarios.", "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)