predicts that the measured radial velocity a(ϕ) will
come from a Normal distribution centered at μ(ϕ).
But whenever the predictive probability density falls
outside the unambiguous interval, the model assigns
probability to an event that cannot occur, because all
measurements will always fall inside the unambiguous interval. In the example shown, the model places
essentially all probability mass on an impossible
How can we fix the model? We still believe it is a
good model for the true radial velocity — the only
problem is aliasing. The obvious solution is to augment the model to also model the process of aliasing
by mapping the probability density for any value a
outside the unambiguous interval to its alias ā
within the unambiguous interval. This leads to a model
for the aliased radial velocity ā(ϕ) where only values
inside the unambiguous interval have positive probability density, and each point ā accumulates the
density of each of its aliases in the original model.
Mathematically, this is described by the wrapped normal density function (Breitenberger 1963):
N w a µ,; 2() = N a +2kVmax µ,; 2 ( )
a ; ;Vmax ,Vmax ] (
With this definition in place, our only change to the
original model is to replace the normal likelihood in
the uniform velocity model by the following
wrapped normal likelihood:
The resulting density is shown in the right panel of
figure 7: it looks like the original normal density but
is positive only on the unambiguous interval and
wraps around at ±Vmax. 4
Our high-level approach is to use the wrapped normal density in place of the normal density in the uniform velocity model and then fit the parameters.
However, the likelihood surface is considerably more
complex than that of the simple linear model, so fitting the parameters is more difficult. This is again
consistent with the general principle that joint inference is preferable, but it is usually more demanding
Problem: Multimodal Likelihood
Our goal is to find the parameters u and v that maxi-
mize the log-likelihood of the wrapped normal mod-
el. In the original model, the log-likelihood was
a ; ( ):Nw µ; ( ),; 2 ( )
Predicted mean Predictive density
Figure 7. Illustration of Wrapped Normal Model.
See text for explanation.