  The inverse Ising problem and its generalizations to Potts and continuous
spin models have recently attracted much attention thanks to their successful
applications in the statistical modeling of biological data. In the standard
setting, the parameters of an Ising model (couplings and fields) are inferred
using a sample of equilibrium configurations drawn from the Boltzmann
distribution. However, in the context of biological applications, quantitative
information for a limited number of microscopic spins configurations has
recently become available. In this paper, we extend the usual setting of the
inverse Ising model by developing an integrative approach combining the
equilibrium sample with (possibly noisy) measurements of the energy performed
for a number of arbitrary configurations. Using simulated data, we show that
our integrative approach outperforms standard inference based only on the
equilibrium sample or the energy measurements, including error correction of
noisy energy measurements. As a biological proof-of-concept application, we
show that mutational fitness landscapes in proteins can be better described
when combining evolutionary sequence data with complementary structural
information about mutant sequences.
