Forecast reconciliation of multivariate time series is the process of mapping a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints . Commonly used projection matrix based approaches for point forecast reconciliation are OLS (ordinary least squares), WLS (weighted least squares), and MinT (minimum trace). Even though point forecast reconciliation is a well-established field of research, the literature on generating probabilistic forecasts subject to linear constraints is somewhat limited . Available methods follow a two-step procedure . Firstly, it draws future sample paths from the univariate models fitted to each series in the collection (which are incoherent). Secondly, it uses a projection matrix based approach or empirical copula based reordering approach to account for contemporaneous correlations and linear constraints . The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score, or simply using a projection matrix derived for point forecast reconciliation . This paper proves that (a) if the incoherent predictive distribution is Gaussian then MinT minimizes the logarithmic scoring rule; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS . We show these theoretical results using a set of simulation studies . We also evaluate them using the Australian domestic tourism data set.