We describe probability distributions, dubbed compressible priors, whose independent and identically distributed (iid) realizations result in compressible signals. A signal is compressible when sorted magnitudes of its coefficients exhibit a power-law decay so that the signal can be well-approximated by a sparse signal. Since compressible signals live close to sparse signals, their intrinsic information can be stably embedded via simple non-adaptive linear projections into a much lower dimensional space whose dimension grows logarithmically with the ambient signal dimension. By using order statistics, we show that N-sample iid realizations of generalized Pareto, Studentâ€™s t, log-normal, Frechet, and log-logistic distributions are compressible, i.e., they have a constant expected decay rate, which is independent of N. In contrast, we show that generalized Gaussian distribution with shape parameter q is compressible only in restricted cases since the expected decay rate of its N-sample iid realizations decreases with N as 1/[q log(N/q)]. We use compressible priors as a scaffold to build new iterative sparse signal recovery algorithms based on Bayesian inference arguments. We show how tuning of these algorithms explicitly depends on the parameters of the compressible prior of the signal, and how to learn the parameters of the signalâ€™s compressible prior on the fly during recovery.

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