  Multiplying a likelihood function with a positive number makes no difference
in Bayesian statistical inference, therefore after normalization the likelihood
function in many cases can be considered as probability distribution. This idea
led Aitchison to define a vector space structure on the probability simplex in
1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product
on this space in 2001, endowing the probability simplex with a Hilbert space
structure. In this paper we present the noncommutative counterpart of this
geometry. We introduce a real Hilbert space structure on the quantum mechanical
finite dimensional state space. We show that the scalar product in quantum
setting respects the tensor product structure and can be expressed in terms of
modular operators and Hamilton operators. Using the quantum analogue of the
log-ratio transformation it turns out that all the newly introduced operations
emerge naturally in the language of Gibbs states. We show an orthonormal basis
in the state space and study the introduced geometry on the space of qubits in
details.
