  Smoothing algorithms for state-space models, i.e., fixed-interval smoothing,
fixed-lag smoothing, and two-filter formula for smoothing, are examined using
real examples. For linear and Gaussian state-space models, it is observed that
similar posterior distributions can be obtained by properly defining the
inverse filter. In the case of linear non-Gaussian state-space models, it is
shown that Gaussian-sum smoothing is possible even for relatively high
dimensional state-space model with Gaussian-mixture noise inputs by properly
setting the inverse filter. The two-filter formula is also applicable for
particle filter, but better results are obtained with fixed lag smoothing or
with the average of forward and backward fixed lag smoothers.
