A Note on the Relation Between Balenzela's Algorithm for Two-Filter Formula for Smoothing and Information Filter

Recent paper by Balenzuela et al. presented an exact algorithm for computing the posterior distribution of current and future observations given the current state, $p(x_n\mathrm{\mid }{y}_n,\dots ,y_N)$, which is required when computing fixed-interval smoother of the state by a two-filter formula. In this note, it will be shown that their algorithm is equivalent to the backward filter obtained by applying an information filter to the reverse state-space model. Although their algorithm is proposed for complex Gaussian mixture distribution models, in this note, we consider the case of simple state-space models with respect to filter computation.