a vector , a subgroup and a section such that where , is a bounded, positive operator with bounded inverse and is a quasi-invariant measure on . It is not assumed that be invariant up to a phase under the action of and clearly, the best situation is when is a multiple of the identity. Although somewhat technical, this general construction is of enormous versatility for semi-direct product groups of the type , where is a closed subgroup of . Thus, it is useful for many physically important groups, such as the Poincaré group or the Euclidean group, which do not have square integrable representations in the sense of the earlier definition. In particular, the integral condition defining the operator ensures that any vector in can be written in terms of the generalized coherent states namely, which is the primary aim of any kind of coherent states. Coherent states: a Bayesian construction for the quantization of a measure set We now depart from the standard situation and present a general method of construction of coherent states, starting from a few observations on the structure of these objects as superpositions of eigenstates of some self-adjoint operator, as was the harmonic oscillator Hamiltonian for the standard CS. It is the essence of quantum mechanics that this superposition has a probabilistic flavor. As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction. There are, in a sort of duality, a Poisson distribution ruling the probability of detecting excitations when the quantum system is in a coherent state , and a gamma distribution on the set of complex parameters, more exactly on the range of the square of the radial variable. The generalization follows that duality scheme. Let be a set of parameters equipped with a measure and its associated Hilbert space of complex-valued functions, square integrable with respect to . Let us choose in a finite or countable orthonormal set : In case of infinite countability, this set must obey the (crucial) finiteness condition: Let be a separable complex Hilbert space with orthonormal basis in one-to-one correspondence with the elements of . The two conditions above imply that the family of normalized coherent states in , which are defined by resolves the identity in : Such a relation allows us to implement a coherent state or frame quantization of