fact, such a model is mathematically equivalent to the
well-known Kalman filter.
However, because exact inference is intractable in
our model, we instead adopt an approximate message passing scheme based on the Expectation Propagation (EP) algorithm (Minka 2001), which approximates each likelihood term by a Gaussian potential.
An important aspect of EP is that the approximations
are not made a priori, but in the context of all other
messages — conceptually, EP focuses on finding a
good approximation for the full posterior distribution instead of simply approximating the individual
non-Gaussian potentials. Full details of the message
passing scheme can be found in the conference paper
(Sheldon et al. 2013).
One additional detail of the message passing scheme
is worth mentioning because it reveals that the basic
computations performed by the overall approach
have a simple and intuitive interpretation from the
perspective of dealiasing.
As part of EP, we must solve the subproblem of
finding a Gaussian approximation of a particular
non-Gaussian distribution f(wi) that is the product of
the wrapped-normal likelihood ℓi(wi) with a prior
over wi from adjacent levels. We do this using
Laplace’s method — a variant of EP known as Laplace
propagation (Smola, Vishwanathan, and Eskin 2003)
— for which we need to find the mode of f(wi).
A very simple and intuitive local search procedure
converges to an approximate mode of f(wi) under a
one-term approximation to the infinite sum in the
wrapped normal density. The local search begins with
an initial value for wi, and then alternates between
two simple high-level steps. First, it uses the current
value of wi to dealias the radial velocity measurements by selecting the alias closest to the predicted
mean for each data point. Then, it refits wi by least
squares using the current dealiased values. The search
is very fast and usually converges in one or two steps,
and there is no measurable loss in overall performance compared with a much slower numerical
approach using many more terms to approximate the
wi+ 1 ℓi+ 1
Level i+ 1
; Smoothness prior
Level i– 1
Figure 9. Overall Graphical Model for Velocity Inference.
Round nodes represent variables: wi is the unknown target velocity for elevation level i. Square nodes represent factors.