In this communication, we propose a new Bayesian framework to characterize the concentration parameter of the von Mises distribution. To achieve this, we equip this parameter with a Lie group structure. We design a Lie group (LG) estimator by incorporating prior information modeled by a Gaussian distribution on$\mathbb{R}^{+}$. This estimator is determined using a dedicated optimization algorithm on$\mathbb{R}^{+}$. The performance of this estimator is then evaluated by computing a new expression of the Bayesian Cramér-Rao bound on the Lie group (LG-BCRB)$\mathbb{R}^{+}$. The consistency between the proposed estimator and the LG-BCRB is validated through numerical simulations by comparing it with the Bayesian Mean Squared Error.

The Modified Cramér-Rao Bound (MCRB) proves to be of significant importance in non-standard estimation scenarios, when in addition to unknown deterministic parameters to be estimated, observations also depend on random nuisance parameters. Given the interest of applications that involve estimation on Lie Groups (LGs), as well as the relevance of nonstandard estimation problems in many practical scenarios, the main concern in this communication is to derive an intrinsic MCRB on LGs (LG-MCRB). For this purpose, a modified unbiasedness constraint must be defined, yielding a modified Barankin Bound. A closed-form formula of the LG-MCRB is then provided for a LG Gaussian model on $S O(2)$, representing $2 D$ rotation matrices, while considering non-Gaussian random nuisance parameters. The validity of this expression is then assessed through numerical simulations, and compared with the intrinsic CRB on LGs for a simplified illustrative scenario, involving a concentrated Gaussian prior distribution on the random nuisance parameters.