  We propose a novel approach to evaluating the ionic Seebeck coefficient in
electrolytes from relatively short equilibrium molecular dynamics simulations,
based on the Green-Kubo theory of linear response and Bayesian regression
analysis. By exploiting the probability distribution of the off-diagonal
elements of a Wishart matrix, we develop a consistent and unbiased estimator
for the Seebeck coefficient whose statistical uncertainty can be arbitrarily
reduced in the long-time limit. To validate the effectiveness of our method, we
benchmark it against extensive equilibrium molecular dynamics simulations
conducted on molten $\mathrm{CsF}$ using empirical force fields. We then employ
this procedure to calculate the Seebeck coefficient of molten $\mathrm{NaCl}$,
$\mathrm{KCl}$ and $\mathrm{LiCl}$ using neural-network force fields trained on
ab initio data over a range of pressure-temperature conditions.
