The paper analyzes the probability distribution of the occupancy
numbers and the entropy of a system at the equilibrium composed by
an arbitrary number of non-interacting bosons. The probability
distribution is derived both by tracing out the environment from a
bosonic eigenstate of the union of environment and system of
interest (the empirical approach) and by tracing out the
environment from the mixed state of the union of environment and
system of interest (the Bayesian approach). In the thermodynamic
limit, the two coincide and are equal to the multinomial
distribution. Furthermore, the paper proposes to identify the
physical entropy of the bosonic system with the Shannon entropy of
the occupancy numbers, fixing certain contradictions that arise in
the classical analysis of thermodynamic entropy. Finally, by
leveraging an information-theoretic inequality between the entropy
of the multinomial distribution and the entropy of the
multivariate hypergeometric distribution, Bayesianism of
information theory and empiricism of statistical mechanics are
integrated into a common ''infomechanical'' framework.