  Optimisation problems in science and engineering typically involve finding
the ground state (i.e. the minimum energy configuration) of a cost function
with respect to many variables. If the variables are corrupted by noise then
this approach maximises the likelihood that the solution found is correct. An
alternative approach is to make use of prior statistical information about the
noise in conjunction with Bayes's theorem. The maximum entropy solution to the
problem then takes the form of a Boltzmann distribution over the ground and
excited states of the cost function. Here we use a programmable Josephson
junction array for the information decoding problem which we simulate as a
random Ising model in a field. We show experimentally that maximum entropy
decoding at finite temperature can in certain cases give competitive and even
slightly better bit-error-rates than the maximum likelihood approach at zero
temperature, confirming that useful information can be extracted from the
excited states of the annealing device. Furthermore we introduce a microscopic
bit-by-bit analytical method which is agnostic to the specific application and
use it to show that the annealing device samples from a highly Boltzmann-like
distribution. Machines of this kind are therefore candidates for use in a wide
variety of machine learning applications which exploit maximum entropy
inference, including natural language processing and image recognition. We
further show that the limiting factor for performance in our experiments is
likely to be control errors rather than failure to reach equilibrium. Our work
also provides a method for determining if a system is in equilibrium which can
be easily generalized. We discuss possible applications of this method to spin
glasses and probing the performance of the quantum annealing algorithm.
