Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tatiana Berlenko, Kirill Krinkin

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.

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

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.
Paper Structure (80 sections, 5 theorems, 23 equations, 6 figures, 6 tables)

This paper contains 80 sections, 5 theorems, 23 equations, 6 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Bayesian Occupancy Grid Mapping
  4. Belief Function Occupancy Grid Mapping
  5. Alternative combination rules
  6. Comparison Studies and the Sensor Model Gap
  7. The methodological gap
  8. Theoretical Foundations
  9. Bayesian Occupancy Grid Fusion
  10. Dempster--Shafer Belief Function Fusion
  11. Equivalence Theorem
  12. Monotonic Decay of $m_{OF}$
  13. Fair Comparison Methodology
  14. The Sensor Model Equivalence Problem
  15. Pignistic Transform Matching
  16. ...and 65 more sections

Key Result

Theorem 3.2

Define the mapping $\varphi\colon \mathbb{R} \to \mathcal{M}$, where $\mathcal{M}$ denotes the set of consonant BBAs on $\{O, F\}$ with $m_{OF} > 0$, from log-odds values by: where $\sigma(L) = (1 + e^{-L})^{-1}$. Then:

Figures (6)

  • 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.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 3.1: Consonant mass functions
  • Theorem 3.2: Single-observation equivalence
  • proof
  • Corollary 3.3: Single-observation equivalence
  • proof
  • Lemma 3.4: Closed-form accumulation under consonant observations
  • Theorem 3.5: Monotonic decay of $m_{OF}$
  • proof
  • Corollary 3.6: Exponential decay under consonant observations
  • Remark 3.7: Scope of \ref{['thm:mof-decay']}