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Variational Bayesian Inference Based Progressive Multi-Target MIMO Sensing with Nuisance Parameters
Jiayi Yao, Shuowen Zhang
TL;DR
Multi-stage Bayesian sensing depends on characterizing posterior PDFs for multiple targets, including when reflection coefficients are nuisance parameters. The paper uses a variational Bayesian surrogate in a progressive MIMO framework, reporting polynomial complexity and refined sensing performance. The model assumes reflection coefficients remain unchanged across stages.
Problem
Progressive Bayesian beamforming and sensing rely on prior-stage posterior PDFs, whose characterization for multiple targets is difficult and can require grid-based calculations with exponential complexity.
Source: §1 · p. 1, §3.3 · p. 2Method
The framework derives a closed-form surrogate posterior PDF using variational Bayesian inference and uses it for subsequent-stage Bayesian beamforming and sensing.
Source: §1 · p. 1, §4.1 · p. 3Results
The proposed framework progressively refines sensing performance and outperforms benchmark schemes, while approaching numerical-posterior MMSE and PCRB performance at moderate-to-high receive SNR.
Source: §5 · p. 4Takeaways & Limitations
Surrogate posterior PDFs concentrate around the actual target angles over stages, and beamforming progressively concentrates power around highly probable angles.
Source: §5 · p. 4The reflection coefficients are modeled as unknown nuisance parameters that remain unchanged across sensing stages.
Source: §2 · p. 2
Abstract
from arXiv · showhide
This paper proposes a progressive Bayesian multiple-input multiple-output (MIMO) sensing framework for multiple targets over multiple stages. We consider a practical yet challenging scenario where the targets' angles are unknown and random parameters to be estimated, while the targets' reflection coefficients are unknown nuisance parameters. With an initial prior probability density function (PDF) for the targets' angles, we progressively update the prior PDF for each sensing stage as the posterior PDF obtained from the previous stage, based on which Bayesian transmit beamforming optimization is performed to minimize the sum posterior Cram\'{e}r-Rao bound (PCRB) in estimating the targets' angles and Bayesian sensing is performed with the help of new observations in this stage. To analytically characterize the intractable and high-dimensional posterior PDF with low complexity, we propose a variational Bayesian inference based approach which derives a surrogate posterior PDF in closed form with only polynomial complexity over the number of targets, in sharp contrast to existing numerical calculation approaches with exponential complexity. Numerical results validate the efficacy of our proposed framework in progressively refining sensing performance.
1. INTRODUCTION
Bayesian beamforming uses statistical priors and PCRB, while progressive sensing reuses posterior PDFs to adapt strategies over multiple stages. The paper addresses the difficulty of characterizing those PDFs in multi-target MIMO sensing with nuisance reflection coefficients.
1. INTRODUCTION
- Bayesian beamforming uses full prior distributions and PCRB, a lower bound on MSE for estimators independent of the true parameter values [2–9].Earlier beamforming studies generally used predictions of unknown parameters to characterize CRB.Source: §1 · p. 1
- Progressive sensing uses each stage’s posterior PDF as the next stage’s prior, enabling different beamforming and sensing strategies across stages.The posterior PDF is also the central challenge because both operations depend on it.Source: §1 · p. 1, Figure 1 · p. 1
- Grid-based numerical posterior calculations grow exponentially with target count, while prior VALSE-based work addressed SIMO or opportunity-illuminator settings without transmit beamforming [15–17].Those works jointly estimated angles and reflection coefficients under specified priors, leaving general MIMO systems with nuisance parameters without a low-complexity characterization approach.Source: Figure 1 · p. 1
- The framework derives a closed-form surrogate posterior PDF with polynomial complexity, then uses it in later-stage Bayesian beamforming and sensing.The paper reports progressive refinement and superiority over benchmark schemes.Source: §1 · p. 1
2. SYSTEM MODEL
The system models multi-target MIMO sensing across stages, with unknown target angles and unknown reflection coefficients treated as nuisance parameters. Observations update the angle prior between stages, while target and prior assumptions define the model.
2. SYSTEM MODEL
- The model considers static point targets whose random azimuth angles are sensed over multiple stages using echoes and prior PDFs; reflection coefficients are unknown nuisance parameters unchanged across stages.The system assumes a line-of-sight channel between the base station and each target.Source: §2 · p. 2
2.1. Observation Model
- At each stage, the transmitter sends probing signals and receives target echoes under a shared-range-bin assumption, with circularly symmetric complex Gaussian receiver noise.The transmit and receive array steering vectors depend on each target’s angle.Source: §2.1 · p. 2
2.2. Prior Model
- The nuisance-parameter priors are known and targets are independent; the initial angle prior is assumed known, and each stage’s posterior becomes the next stage’s prior.The initial angle prior may come from historic data or a target appearance pattern.Source: §2.2 · p. 2
3. PROPOSED PROGRESSIVE BAYESIAN MULTI-TARGET MIMO SENSING FRAMEWORK
The framework updates angle priors with each stage’s posterior, uses Bayesian beamforming to minimize summed conditional PCRB, and performs Bayesian sensing from new observations. Its variational posterior characterization addresses the exponential complexity of numerical calculation with a polynomial-complexity alternative.
3.1. Bayesian Beamforming at the Transmitter
- Transmit beamforming minimizes the sum conditional PCRB across targets; the optimization is convex, and progressively updated priors concentrate power around the targets’ actual angles.The conditional PCRB is used as a lower bound on MSE for Bayesian estimators and observation histories.Source: §3.1 · p. 2
3.2. Bayesian Sensing at the Receiver
- At each stage’s end, the receiver calculates the angle posterior and uses it to obtain MMSE or marginal MAP angle estimates.Source: §3.2 · p. 2
3.3. Complexity with Numerical Posterior PDF Calculation
- The proposed posterior characterization has polynomial complexity in target count, unlike numerical posterior calculation, whose complexity grows exponentially and becomes prohibitive for many targets.The posterior is needed for both Bayesian beamforming and Bayesian sensing.Source: §3.3 · p. 2
4. PROPOSED VARIATIONAL BAYESIAN INFERENCE BASED POSTERIOR PDF CHARACTERIZATION
The posterior characterization reparameterizes angles, restricts observations to a selectable stage window, and uses variational inference to build tractable per-angle surrogate PDFs. Iterative updates yield a posterior for subsequent stages with polynomial rather than exponential complexity in the number of targets.
4.1. Equivalent Transformation and Tractable Approximation
- Pseudo angles provide a one-to-one reparameterization of physical angles that simplifies the array steering vectors and observation model.Source: §4.1 · p. 3
- The stage-s observation window includes observations from ρ(s) through s, allowing a trade-off between conditional-PDF calculation complexity and sensing performance.Source: §4.1 · p. 3
- Mean-field factorization turns the high-dimensional posterior optimization into parallel low-dimensional optimizations of per-angle surrogate PDFs; the independence approximation is exact when M = 1.The approach seeks a tractable surrogate posterior by maximizing the evidence lower bound.Source: §4.1 · p. 3, §4.2 · p. 3
4.2. Optimizing Surrogate Posterior PDF for Each Angle
- The method averages the reflection-coefficient auxiliary distribution over the current angle surrogate, with only one-dimensional integration for each angle under mean-field factorization.Source: §4.2 · p. 3
- Iterative per-angle updates produce a posterior for next-stage beamforming and sensing, with polynomial complexity in target count rather than the exponential complexity of numerical calculation.Source: §4.2 · p. 3, §4.3 · p. 4
5. NUMERICAL RESULTS
The numerical results report competitive angle estimation and lower computation than numerical posterior methods, while showing progressive refinement and an observation-window accuracy–complexity trade-off.
5. NUMERICAL RESULTS
- Posterior-PDF characterization converges in 15 iterations under a 10^-6 convergence threshold.The convergence evaluation uses S = 1 and different numbers of targets.Source: §5 · p. 4
- The proposed framework outperforms the listed estimation benchmarks while matching numerical marginal MAP and MMSE performance with significantly less computation.The benchmarks use the same Bayesian beamforming but different estimation methods.Source: §5 · p. 4
- At M = 2, the framework outperforms all schemes except numerical MMSE and approaches numerical MMSE and the PCRB at moderate-to-high receive SNR.Fig. 4 presents root sum MSE/PCRB under M = 2 and S = 1.Source: §5 · p. 4, Figure 4 · p. 4
- As sensing stages progress, surrogate posterior PDFs approximate the posterior PDFs and concentrate around the true angles, while beamforming concentrates power around probable angles.The framework also improves inaccurate ESPRIT initial estimates by exploiting surrogate posterior PDFs.Source: §5 · p. 4, Figure 5 · p. 4
- Smaller ρ(s) enlarges the observation window and improves sensing accuracy at the cost of higher complexity.Fig. 6 evaluates observation windows at receive SNR 13 dB, S = 5, and M = 2.Source: §5 · p. 4, Figure 6 · p. 4
6. CONCLUSIONS
The conclusion describes a staged Bayesian MIMO sensing framework that uses variational Bayesian inference to produce a closed-form surrogate posterior for angle estimation with nuisance reflection coefficients.
6. CONCLUSIONS
- At each sensing stage, beamformed probing signals are designed to minimize the sum conditional PCRB for target angles using the prior PDF.The resulting observations support Bayesian sensing and beamforming at subsequent stages.Source: §6 · p. 4
- Variational Bayesian inference derives a closed-form surrogate posterior PDF for use in Bayesian sensing and subsequent-stage beamforming.Source: §6 · p. 4
- The framework has polynomial complexity in the number of targets, and numerical results demonstrate complexity and performance advantages alongside progressive sensing refinement.Source: §6 · p. 4