In [4], the posterior probability of the unknown variables
was approximated as Gaussian distribution with diagonal covariance
matrix. This means that the unknown variables were approximated to be
independent given the observations. Notice that this assumption is
false even if the unknown variables are assumed to be independent a
priori. For instance, the factors are assumed to be independent a
priori but observations induce dependencies between them. These
dependencies are strongest for factors at the same time instant, that
is, *s*_{i}(*t*) and *s*_{j}(*t*) have a posterior dependence but *s*_{i}(*t*_{1})and *s*_{j}(*t*_{2}) can be nearly independent when *t*_{1} is far from *t*_{2}.

The NLFA model has an indeterminacy of the rotation of the factors and the model can utilise this by choosing the rotation which makes the factors independent not only a priori but also a posteriori. However, the inclusion of the dynamic model causes posterior dependencies between factors at different time steps and these dependencies do not vanish for any rotation or any other mapping of the factor space. The smaller the process noise the stronger the dependence will be.

Taking into account the full posterior covariance between
**s**(*t*-1) and
**s**(*t*) is computationally costly if the
dimension of the factor space is large. In practice, the most
significant posterior correlations are the posterior autocorrelations
of the factors, that is, the correlations between *s*_{i}(*t*-1) and
*s*_{i}(*t*). They can be taken into account without increasing the
computational complexity significantly.

In [4], the approximation of the posterior probability of
the factors had the factorial form

(5) |

where

(6) |

where

Here denotes a Gaussian distribution over with mean and variance .

Given *s*_{i}(*t*-1), the posterior variance of *s*_{i}(*t*) is
and the posterior mean is
.
The approximate posterior
*q*(*s*_{i}(*t*) | *s*_{i}(*t*-1),
**X**) is thus parametrised by the mean
,
linear
dependence
and variance
as defined by
(7), whereas in NLFA the posterior of the factor was
parametrised by mean and variance alone.

It is easy to see by induction that if the past values of the factors
are marginalised out, the posterior mean of *s*_{i}(*t*) is
and posterior variance
is

Notice that the marginalised variances are computed recursively in a sweep forward in time.