Estimation Performance for the Bayesian Hierarchical Linear Model

Mohammed Nabil El Korso (1) , Remy Boyer (2) , Pascal Larzabal (3) , Bernard-Henri Fleury (4)

1 LEME - Laboratoire Energétique Mécanique Electromagnétisme
2 L2S - Laboratoire des signaux et systèmes
3 SATIE - Systèmes et Applications des Technologies de l'Information et de l'Energie
4 AAU - Aalborg University [Denmark]

Bayesian hierarchical modelling is a well-established branch of Bayesian inference. In this letter, we derive and study the estimation performance for the Bayesian hierarchical linear model (BHLM). Specifically, we consider a linear model with hierarchical priors for the involved amplitude and noise vectors. We provide closed-form expressions of the Bayesian Cramér-Rao bound (BCRB) for the following settings: (i) an arbitrary prior and hyperprior and (ii) a Gaussian-Y prior for the amplitudes, while the prior of noise is a Gaussian-X in both cases. Gaussian-X means that the conditional prior given the hyperparameter is Gaussian and X is the hyperprior. For the hierarchical distribution associated with spherical invariant random variables, the BCRB has a compact closed-form expression and enjoys several interesting properties that are discussed. Finally, we provide a theoretical analysis of the statistical efficiency of the linear minimum mean square error (MMSE) estimator in the low-and high-noise variance regimes when the hyperparameters are stochastically dominant. Index Terms—Bayesian Cramér-Rao bound, hierarchical linear model, performance analysis.

Domaines: Traitement du signal et de l'image [eess.SP]

Mohammed Nabil El Korso, Remy Boyer, Pascal Larzabal, Bernard-Henri Fleury. Estimation Performance for the Bayesian Hierarchical Linear Model. IEEE Signal Processing Letters, 2016, 23 (4), pp.488-492. ⟨10.1109/LSP.2016.2528579⟩. ⟨hal-01264666⟩