The likelihood function of (H, σ 2 ) conditional on observations of averaged energies from levels j 1 , . . . , j 2 is
L(H, σ 2 |y j 1 , . . . , y j 2 ) = j 2 i=j 1 g(y i ).
We use beta distribution and non-informative prior 1/σ 2 as independent priors on H and σ 2 , respectively,
π(H, σ 2 ) = H α−1 (1 − H) β−1 B(α, β) × 1 σ 2 .
The hyperparameters in beta distribution, α and β are calibrated by considering the impact of effective sample size (ESS) and the mean of the beta distribution, α α+β , which is linked to the Hurst exponent of an input signal. The ESS for the beta(α, β) prior is approximated with α + β and is closely related to the performance of the Bayesian estimation. For example, when ESS is large, the posterior distribution is dominated by the prior [13]. Based on simulations, we selected the ESS to be approximately 50% the original **data** size, but the ESS can be calibrated based on the level of certainty about H. The larger the ESS is, the more confident we are about the mean of a prior, that is, about the "true" value of H.