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Beyond Point Estimates: Likelihood-Based Full-Posterior Wireless Localization

Haozhe Lei, Hao Guo, Tommy Svensson, Sundeep Rangan

TL;DR

This work reframes wireless localization as posterior inference, producing uncertainty-aware spatial posteriors $p(\boldsymbol{x}^t|\boldsymbol{y})$ instead of point estimates. It introduces MC-CLE, a neural score-based approach that learns $g_\theta(\boldsymbol{x}^t,\boldsymbol{y})$ as an unnormalized log-likelihood and uses a Monte Carlo estimate of the partition function $Z_\theta(\boldsymbol{y})$ to train with a sampled cross-entropy loss. The method captures key LOS phenomena—mixture posteriors from false alarms, angular ambiguities from array geometry, and directionality from antenna gain patterns—and yields sharper, lower-entropy posteriors than Gaussian baselines. Demonstrations on realistic LOS simulations show improved likelihood estimates and geometry-consistent posterior heat maps, enabling calibrated uncertainty for planning, control, and radio resource management; future work will extend to NLOS and environment-aware scenarios.

Abstract

Modern wireless systems require not only position estimates, but also quantified uncertainty to support planning, control, and radio resource management. We formulate localization as posterior inference of an unknown transmitter location from receiver measurements. We propose Monte Carlo Candidate-Likelihood Estimation (MC-CLE), which trains a neural scoring network using Monte Carlo sampling to compare true and candidate transmitter locations. We show that in line-of-sight simulations with a multi-antenna receiver, MC-CLE learns critical properties including angular ambiguity and front-to-back antenna patterns. MC-CLE also achieves lower cross-entropy loss relative to a uniform baseline and Gaussian posteriors. alternatives under a uniform-loss metric.

Beyond Point Estimates: Likelihood-Based Full-Posterior Wireless Localization

TL;DR

This work reframes wireless localization as posterior inference, producing uncertainty-aware spatial posteriors instead of point estimates. It introduces MC-CLE, a neural score-based approach that learns as an unnormalized log-likelihood and uses a Monte Carlo estimate of the partition function to train with a sampled cross-entropy loss. The method captures key LOS phenomena—mixture posteriors from false alarms, angular ambiguities from array geometry, and directionality from antenna gain patterns—and yields sharper, lower-entropy posteriors than Gaussian baselines. Demonstrations on realistic LOS simulations show improved likelihood estimates and geometry-consistent posterior heat maps, enabling calibrated uncertainty for planning, control, and radio resource management; future work will extend to NLOS and environment-aware scenarios.

Abstract

Modern wireless systems require not only position estimates, but also quantified uncertainty to support planning, control, and radio resource management. We formulate localization as posterior inference of an unknown transmitter location from receiver measurements. We propose Monte Carlo Candidate-Likelihood Estimation (MC-CLE), which trains a neural scoring network using Monte Carlo sampling to compare true and candidate transmitter locations. We show that in line-of-sight simulations with a multi-antenna receiver, MC-CLE learns critical properties including angular ambiguity and front-to-back antenna patterns. MC-CLE also achieves lower cross-entropy loss relative to a uniform baseline and Gaussian posteriors. alternatives under a uniform-loss metric.

Paper Structure

This paper contains 12 sections, 23 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Posterior Density Estimation
  4. Challenges
  5. Proposed Solution
  6. Log Likelihood Estimation
  7. Sampled Cross Entropy Loss
  8. Position Estimation with Neural Networks
  9. Numerical Experiments
  10. CEL Performance
  11. Probability Mass Visualization
  12. Conclusion

Figures (4)

  • Figure 1: Example learned posterior $p(\boldsymbol{x}^t\!\mid\!\boldsymbol{y})$ for an unknown transmitter location $\boldsymbol{x}^t$ from receiver measurements $\boldsymbol{y}$ using an $8{\times}1$ uniform linear array. The posterior is visualized as a heat map.
  • Figure 2: Architecture of the MC-CLE model.
  • Figure 3: Relative log-probability maps for two RX–TX geometries (see \ref{['tab:rx-cases']}). Columns sweep the horizontal offset $\Delta x\!\in\!\{-80,-50,0,50,80\}$ m. Each panel spans a $100\times100$ m region discretized on a dense grid. The red star marks the TX, the blue circle the RX, and the cyan arrow its heading. Colour encodes the relative log-probability $L(\boldsymbol{x}_k)=\log\!(p_\theta(\boldsymbol{x}_k,\boldsymbol{y})\,K_{\sf grid})$; warmer colours indicate larger $L(\boldsymbol{x}_k)$. Each horizontal strip corresponds to one model: MC–CLE (top), Gauss–Cart (middle), and Gauss–Polar (bottom). MC–CLE produces sharper ridges aligned with the TX–RX geometry; Gauss–Cart yields smoother, elongated bands; Gauss–Polar concentrates mass in narrower angular sectors with less accurate peak localization. When the RX faces the TX, high-confidence ridges appear; side- or back-looking orientations spread the posterior into broader, lower-confidence regions.
  • Figure 4: Posterior entropy across candidates, averaged over episodes for each model with error bars indicating $\pm 1\sigma$. Lower entropy indicates more concentrated posteriors.