Mixed Discrete-Continuous Bayesian Inference: Censored Measurements of Sparse Signals

This paper addresses Bayesian inference of sparse signals by censored measurements. To reduce the sensor's duty-cycle, signals below a threshold are censored, i.e., are set to zero. Sparse signals are random vectors that, or whose elements, are zero with given probabilities. The corresponding probabilistic model induces random measurement and signal vectors of mixed absolute-continuous, discrete, and singular-continuous nature. Therefore, mixed probability densities, the expectation regarding these densities, and a generalized Bayes' rule are constructively derived. For the inference, proper a-posteriori expected loss functions are defined. Their derivative-free minimizations gives Bayesian inferrers similar to traditional minimum-mean-square-error (MMSE), maximum a-posteriori (MAP), and median estimators and detectors. The result provides a unified Bayesian-inference framework. Eventually, this leads to closed-form solutions for the inference problem, numerical results, and the analysis of the probability of censorship.