  We use the spectral kinetic theory of soliton gas to investigate the
likelihood of extreme events in integrable turbulence described by the
one-dimensional focusing nonlinear Schr\"odinger equation (fNLSE). This is done
by invoking a stochastic interpretation of the inverse scattering transform for
fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear
wave field in terms of the spectral density of states of the corresponding
soliton gas. We then apply the general result to two fundamental scenarios of
the generation of integrable turbulence: (i) the asymptotic development of the
spontaneous (noise induced) modulational instability of a plane wave, and (ii)
the long-time evolution of strongly nonlinear, partially coherent waves. In
both cases, involving the bound state soliton gas dynamics, the analytically
obtained values of the kurtosis are in perfect agreement with those inferred
from direct numerical simulations of the the fNLSE, providing the long-awaited
theoretical explanation of the respective rogue wave statistics. Additionally,
the evolution of a particular non-bound state gas is considered providing
important insights related to the validity of the so-called virial theorem.
