Equivalence and Divergence of Bayesian Log-Odds and Dempster's Combination Rule for 2D Occupancy Grids

Tatiana Berlenko; Kirill Krinkin

cs.ROFeb 21, 2026

Equivalence and Divergence of Bayesian Log-Odds and Dempster's Combination Rule for 2D Occupancy Grids

Authors: Tatiana Berlenko, Kirill Krinkin

Organizations: Constructor University, Bremen, Germany · JetBrains LTD

Abstract

We introduce a pignistic-transform-based methodology for fair comparison of Bayesian log-odds and Dempster's combination rule in occupancy grid mapping, matching per-observation decision probabilities to isolate the fusion rule from sensor parameterization. Under BetP matching across simulation, two real lidar datasets, and downstream path planning, Bayesian fusion is consistently favored (15/15 directional consistency, p = 3.1e-5) with small absolute differences (0.001-0.022). Under normalized plausibility matching, the direction reverses, confirming the result is matching-criterion-specific. The methodology is reusable for any future Bayesian/belief function comparison.

Figures & tables

Experiment	Metric	$\delta$	$p$	90% CI	Equiv.
H-002 (single)	Cell Accuracy	0.02	$<10^{-10}$	$[+0.0013,\;+0.0014]$	✓
	Boundary Sharpness	0.03	$<10^{-10}$	$[+0.0096,\;+0.0108]$	✓
	Brier Score	0.01	$<10^{-10}$	$[-0.0057,\;-0.0054]$	✓
H-003 (dyn.)	Cell Accuracy	0.02	$<10^{-10}$	$[+0.0097,\;+0.0113]$	✓
	Boundary Sharpness	0.03	1.0000	$[+0.0647,\;+0.0754]$	×
	Map Entropy	0.02	1.0000	$[-0.0278,\;-0.0245]$	×

Table 1: TOST equivalence testing results. “Equiv.” indicates whether the difference falls within the equivalence margin

\delta

Experiment	Metric	$d$	95% CI	Direction
H-002 (single)	Cell Accuracy	$+13.51$	$[+8.01,\;+19.02]$	B $\geq$ D
	Boundary Sharpness	$+7.78$	$[+4.58,\;+10.98]$	B $\geq$ D
	Brier Score	$-14.93$	$[-21.00,\;-8.85]$	B $\leq$ D
H-003 (dyn.)	Cell Accuracy	$+6.14$	$[+3.59,\;+8.69]$	B $\geq$ D
	Boundary Sharpness	$+5.96$	$[+3.48,\;+8.44]$	B $\geq$ D
	Map Entropy	$-7.11$	$[-10.04,\;-4.17]$	B $\leq$ D

Table 2: Effect sizes (Cohen’s

d

) with 95% confidence intervals for Bayesian vs. Dempster’s rule.

d=\text{mean}(\text{B}-\text{D})/\text{SD}

. Sign interpretation depends on metric polarity; see Direction column.

Experiment	Metric	$\bar{x}_{\mathrm{B}}$	$\bar{x}_{\mathrm{D}}$	$\Delta$	Equiv.
H-002	Cell Accuracy	$0.9925$	$0.9912$	$+0.0013$	✓
	Boundary Sharpness	$0.1474$	$0.1372$	$+0.0102$	✓
	Brier Score	$0.0239$	$0.0295$	$-0.0056$	✓
H-003 dyn.	Cell Accuracy	$0.9848$	$0.9743$	$+0.0105$	✓
	Boundary Sharpness	$0.3135$	$0.2435$	$+0.0701$	×
	Map Entropy	$0.0940$	$0.1202$	$-0.0261$	×

Table 3: Cross-condition summary: mean difference (Bayesian

-

Dempster) and TOST verdict. Higher is better for Cell Accuracy and Boundary Sharpness; lower is better for Brier Score and Map Entropy.

Setup	Metric	Bayesian	Dempster	$\Delta$	95% CI
1-source	Cell Accuracy	$0.8971$	$0.8729$	$+0.0176$	$[+0.0137,\;+0.0217]$
	Boundary Sharpness	$0.2067$	$0.2061$	$+0.0006$	$[-0.0068,\;+0.0074]$
	Brier Score	$0.0949$	$0.1110$	$-0.0128$	$[-0.0158,\;-0.0098]$
2-way split	Cell Accuracy	$0.9015$	$0.8729$	$+0.0212$	$[+0.0174,\;+0.0252]$
	Boundary Sharpness	$0.2027$	$0.2061$	$-0.0034$	$[-0.0103,\;+0.0032]$
	Brier Score	$0.0909$	$0.1110$	$-0.0160$	$[-0.0189,\;-0.0131]$

Table 4: Intel Research Lab dataset validation. Bayesian vs. Dempster’s rule on real 2D LiDAR data.

\Delta

= Bayesian

-

Dempster. Higher is better for Cell Accuracy and Boundary Sharpness; lower is better for Brier Score.

Setup	Metric	Bayesian	Dempster	$\Delta$	95% CI
1-source	Cell Accuracy	$0.9599$	$0.9512$	$+0.0071$	$[+0.0051,\;+0.0092]$
	Boundary Sharpness	$0.3421$	$0.3341$	$+0.0114$	$[+0.0054,\;+0.0176]$
	Brier Score	$0.0382$	$0.0463$	$-0.0072$	$[-0.0092,\;-0.0054]$
2-way split	Cell Accuracy	$0.9618$	$0.9518$	$+0.0071$	$[+0.0052,\;+0.0092]$
	Boundary Sharpness	$0.3396$	$0.3331$	$+0.0105$	$[+0.0046,\;+0.0164]$
	Brier Score	$0.0363$	$0.0457$	$-0.0073$	$[-0.0092,\;-0.0056]$

Table 5: Freiburg Building 079 dataset validation. Bayesian vs. Dempster’s rule on real 2D LiDAR data (SICK LMS200, 360 beams).

\Delta

= Bayesian

-

Dempster. Higher is better for Cell Accuracy and Boundary Sharpness; lower is better for Brier Score.

$R$	Bayes. sharpness	DS sharpness	$\Delta_{\mathrm{sharp}}$	$\Delta_{\mathrm{acc}}$
1	$0.724\pm 0.032$	$0.600\pm 0.036$	$+0.124$	$+0.007$
2	$0.725\pm 0.117$	$0.599\pm 0.098$	$+0.126$	$+0.008$
3	$0.714\pm 0.025$	$0.573\pm 0.014$	$+0.141$	$+0.009$
5	$0.711\pm 0.040$	$0.580\pm 0.041$	$+0.131$	$+0.008$

Table 6: Mechanism test: boundary sharpness and cell accuracy across robot counts (

R

). Each robot executes

500/R

steps in the single-agent environment. Values are mean

\pm

SD over 15 runs.

\Delta

is the paired difference (Bayesian

-

DS ).

Figure 1: Single-agent comparison: violin plots of Bayesian vs. belief function fusion across 15 independent runs for cell accuracy, boundary sharpness, and Brier score. Bayesian is consistently better on all metrics (15/15 directional consistency).

Figure 2: Multi-robot comparison: violin plots for dynamic-baseline and noisy-sensor conditions across 15 independent runs. Bayesian outperforms belief functions on all metrics with 15/15 directional consistency. Larger separation is visible for boundary sharpness and map entropy.

Figure 3: Forest plot of Cohen’s

d

(Bayesian

-

belief functions) with 95% Hedges–Olkin CIs across all nine simulation comparisons. All CIs exclude zero and favor Bayesian. The direction is consistent across all experiments and metrics.

Figure 4: Mechanism test: boundary sharpness and cell accuracy as a function of robot count (1, 2, 3, 5) under fair comparison. Shaded bands show

\pm 1

SD across 15 runs. The Bayesian advantage is stable across robot counts with no crossover, confirming the sensor model confound hypothesis.

Figure 5: Intel Research Lab real-data validation: Bayesian vs. Dempster’s rule for 1-source, 2-way split, and 4-way split configurations. The direction is consistent with simulation results across all metrics and configurations.

Figure 6: Freiburg Building 079 real-data validation: Bayesian vs. Dempster’s rule for 1-source, 2-way split, and 4-way split configurations. Results confirm the Intel Lab findings on an independent dataset.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Experiment	Metric	$d$	HO 95% CI	NCT 95% CI	Disc.
Single-agent	Cell Accuracy	$+13.51$	$[+8.01,\;+19.02]$	$[+8.54,\;+18.48]$	0.54
	Boundary Sharpness	$+7.78$	$[+4.58,\;+10.98]$	$[+4.88,\;+10.67]$	0.31
	Brier Score	$-14.93$	$[-21.00,\;-8.85]$	$[-20.41,\;-9.44]$	0.59
Multi-robot (dyn.)	Cell Accuracy	$+6.14$	$[+3.59,\;+8.69]$	$[+3.83,\;+8.44]$	0.25
	Boundary Sharpness	$+5.96$	$[+3.48,\;+8.44]$	$[+3.71,\;+8.19]$	0.24
	Map Entropy	$-7.11$	$[-10.04,\;-4.17]$	$[-9.75,\;-4.45]$	0.29

Appendix

Table S1: Cohen’s

d

CI verification: Hedges–Olkin (1985) approximation vs. exact non-central

t

inversion (Steiger, 2004). Maximum discrepancy is 0.59, confirming qualitative conclusions are unaffected.

Dataset	Setup	Metric	$B{=}5$ (0.5 m)	$B{=}10$ (1.0 m)	$B{=}20$ (2.0 m)
Intel Lab	1-source	Cell Accuracy	$[+0.014,\;+0.021]$	$[+0.014,\;+0.022]$	$[+0.013,\;+0.023]$
		Boundary Sharpness	$[-0.006,\;+0.007]$	$[-0.007,\;+0.007]$	$[-0.007,\;+0.007]$
		Brier Score	$[-0.016,\;-0.010]$	$[-0.016,\;-0.010]$	$[-0.016,\;-0.009]$
	2-way split	Cell Accuracy	$[+0.018,\;+0.024]$	$[+0.017,\;+0.025]$	$[+0.017,\;+0.026]$
		Boundary Sharpness	$[-0.010,\;+0.003]$	$[-0.010,\;+0.003]$	$[-0.011,\;+0.004]$
		Brier Score	$[-0.018,\;-0.014]$	$[-0.019,\;-0.013]$	$[-0.020,\;-0.013]$

Appendix

Table S2: Block size sensitivity for spatial block bootstrap CIs. Each cell shows the 95% CI for the metric delta (Bayesian

-

Dempster). All significance verdicts are stable across block sizes.

$N$	$L$	$p(O)$	$m_{O}$	$m_{OF}$	$\mathrm{BetP}(O)$	$\mathrm{BetP}-p$
1	2.0	0.8808	0.7616	0.2384	0.8808	$+$ 0.0000
2	4.0	0.9820	0.9432	0.0568	0.9716	$-$ 0.0104
5	10.0	0.999955	0.9992	0.0008	0.9996	$-$ 0.0003
10	10.0	0.999955	$\approx 1$	$<10^{-6}$	$\approx 1$	$+$ 0.00005
20	10.0	0.999955	$\approx 1$	$<10^{-12}$	$\approx 1$	$+$ 0.00005

Appendix

Table S3: Convergence comparison for

N

identical occupied observations (

l_{\mathrm{occ}}=2.0

L_{\max}=10

a=0.7616

). At

N=1

the methods are identical by construction (pignistic matching). The Bayesian clamp saturates at

N=5

; Dempster–Shafer overtakes at

N\approx 7

Metric	$L_{\max}=5$	$L_{\max}=10$	$L_{\max}=20$	$L_{\max}=\infty$	Consist.
Cell accuracy	$+0.0013$	$+0.0013$	$+0.0013$	$+0.0013$	15/15
Boundary sharpness	$+0.007$	$+0.010$	$+0.010$	$+0.010$	15/15
Brier score	$-0.006$	$-0.006$	$-0.006$	$-0.006$	15/15

Appendix

Table S4:

L_{\max}

ablation: Bayesian+count minus Dempster delta (15 runs each). Higher is better for cell accuracy and boundary sharpness; lower (more negative) is better for Brier score.

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Large-Scale Continuous Occupancy Mapping via Variance-Weighted Submap Joining

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Open paper AskPDF arXiv

Continuous occupancy mapping methods that learn spatial representations from sensor observations support predictions at arbitrary locations, but training a single global model can become computationally expensive as the amount of training data grows during large-scale mapping. Constructing local continuous occupancy submaps reduces the size of individual learning problems, but relative pose errors can cause inconsistent predictions in overlapping regions. We address these challenges by formulating large-scale continuous occupancy mapping as a submap joining problem. Each local submap is represented by a relevance vector machine regressing a continuous average log-odds field. Repeated observations within each cell are summarized by their average log-odds and observation count, yielding a Gaussian likelihood with count-dependent precision and enabling Gaussian-conjugate inference. The resulting local fields provide predictive means, variances, and analytical spatial derivatives. We then formulate submap joining as a joint pose--field optimization problem, with consistency constraints weighted by the predictive uncertainty of overlapping local submaps. With uncertainty weights held fixed within each Gauss--Newton iteration, our joint pose--field system can be reduced to an equivalent pose-only system. After the optimized submap poses are obtained, the global field value at any covered location can be recovered in closed form through variance-weighted fusion of predictions from the corresponding local submaps, without training an additional global model. Experiments on simulated and real-world 2D LiDAR datasets demonstrate improved pose and occupancy-map accuracy over the evaluated baselines, reduced map storage and joining time, and improved map-construction scalability.

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