  Bayesian approach for quantum parameter estimation has gained a renewed
interest from practical applications of quantum estimation theory. Recently, a
lower bound, called the Bayesian Nagaoka-Hayashi bound for the Bayes risk in
quantum domain was proposed, which is an extension of a new approach to point
estimation of quantum states by Conlon et al. (2021). The objective of this
paper is to explore this Bayesian Nagaoka-Hayashi bound further by obtaining
its lower bounds. We first obtain one-parameter family of lower bounds, which
is an analogue of the Holevo bound in point estimation. Thereby, we derive
one-parameter family of Bayesian logarithmic derivative type lower bounds in a
closed form for the parameter independent weight matrix setting. This new bound
includes previously known Bayesian lower bounds as special cases.
