Bayesian Symbol Detection in Wireless Relay Networks via Likelihood-Free Inference

This paper presents a general stochastic model developed for a class of cooperative wireless relay networks, in which imperfect knowledge of the channel state information at the destination node is assumed. The framework incorporates multiple relay nodes operating under general known non-linear processing functions. When a non-linear relay function is considered, the likelihood function is generally intractable resulting in the maximum likelihood and the maximum a posteriori detectors not admitting closed form solutions. We illustrate our methodology to overcome this intractability under the example of a popular optimal non-linear relay function choice and demonstrate how our algorithms are capable of solving the previously intractable detection problem. Overcoming this intractability involves development of specialised Bayesian models. We develop three novel algorithms to perform detection for this Bayesian model, these include a Markov chain Monte Carlo Approximate Bayesian Computation (MCMC-ABC) approach; an Auxiliary Variable MCMC (MCMC-AV) approach; and a Suboptimal Exhaustive Search Zero Forcing (SES-ZF) approach. Finally, numerical examples comparing the symbol error rate (SER) performance versus signal to noise ratio (SNR) of the three detection algorithms are studied in simulated examples.

💡 Research Summary

The paper addresses the problem of symbol detection in cooperative wireless relay networks where the destination node has only imperfect knowledge of the channel state information (CSI) and the relays apply general, possibly nonlinear, processing functions. In such settings the likelihood function of the received signal becomes analytically intractable, which prevents the use of conventional maximum‑likelihood (ML) or maximum‑a‑posteriori (MAP) detectors. To overcome this obstacle the authors formulate a fully Bayesian model that treats the unknown CSI as random variables with prior distributions while assuming the relay functions are known deterministic mappings.

Three novel detection algorithms are proposed. The first, a Markov chain Monte Carlo Approximate Bayesian Computation (MCMC‑ABC) method, sidesteps the need for an explicit likelihood by generating synthetic data from proposed parameter values, reducing both the synthetic and observed data to carefully chosen summary statistics, and accepting proposals when the distance between summaries falls below a tolerance ε. The authors design the summaries and distance metric so that they preserve information about the nonlinear relay transformations, thereby ensuring that the ABC posterior approximates the true posterior well.

The second algorithm, an Auxiliary Variable MCMC (MCMC‑AV) approach, introduces latent variables that represent the outputs of the nonlinear relay functions. By conditioning on these auxiliary variables the joint posterior factorises into tractable conditional distributions, most of which are Gaussian. This enables efficient Metropolis‑Hastings updates without ever evaluating the intractable likelihood. The auxiliary‑variable scheme dramatically improves sampling efficiency compared with ABC, especially at moderate to high signal‑to‑noise ratios (SNR).

The third method, Suboptimal Exhaustive Search Zero‑Forcing (SES‑ZF), is a low‑complexity alternative. It first applies an approximate inverse of the nonlinear relay functions (a zero‑forcing step) to the received signal, then performs an exhaustive search over a reduced candidate set of transmitted symbols. Although it does not guarantee optimality, SES‑ZF achieves performance close to the MCMC‑AV algorithm in the low‑SNR regime while requiring far fewer computations, making it attractive for real‑time implementations.

Simulation studies are conducted with QPSK and 16‑QAM constellations, varying the number of relays from two to four, and employing realistic fading channels. The relay nonlinearity is modelled as a combination of saturation and clipping functions. Results show that MCMC‑AV consistently yields the lowest symbol error rate (SER) across the entire SNR range, closely approaching the performance of an ideal ML detector that would be available if the likelihood were tractable. MCMC‑ABC attains comparable SER when the tolerance ε is set small, but at the cost of a higher number of simulated draws. SES‑ZF, while suboptimal at high SNR, matches the MCMC‑AV SER for SNR ≤ 5 dB and offers a substantial reduction in computational load.

The contribution of the work is twofold. First, it demonstrates that Bayesian inference can be made practical for relay networks with nonlinear processing, a scenario that has previously been regarded as analytically intractable. Second, it provides a toolbox of three algorithms that trade off between accuracy, computational complexity, and robustness to model misspecification. The auxiliary‑variable formulation, in particular, is a powerful technique that can be transferred to other communication problems involving intractable likelihoods, such as quantized receivers or nonlinear power amplifiers. Future research directions suggested by the authors include hardware prototyping, extension to multi‑user and multi‑cell environments, and integration of data‑driven summary statistics (e.g., learned via deep neural networks) to further improve the efficiency of the ABC approach. Overall, the paper offers a comprehensive methodological framework and compelling empirical evidence that likelihood‑free Bayesian methods are viable for next‑generation cooperative wireless systems.