Efficient Bayesian estimation of permutation entropy with Dirichlet priors

Estimation of permutation entropy (PE) using Bayesian statistical methods is presented for systems where the ordinal pattern sampling follows an independent, multinomial distribution. It is demonstrated that the PE posterior distribution is closely approximated by a standard Beta distribution, whose hyperparameters can be estimated directly from moments computed analytically from observed ordinal pattern counts. Equivalence with expressions derived previously using frequentist methods is also demonstrated. Because Bayesian estimation of PE naturally incorporates uncertainty and prior information, the orthodox requirement that $N \gg D!$ is effectively circumvented, allowing PE to be estimated even for very short time series. Self-similarity tests on PE posterior distributions computed for a semiconductor laser with optical feedback (SLWOF) system show its PE to vary periodically over time.