Comparing the distribution fits of a bivariate and a univariate model Suppose I've done an experiment and I have a distribution of observations $x$ that vary between $-\pi$ and $\pi$.
Now suppose each $x$ is associated with a second observation $y$ that may or may not influence the value $x$ takes.  This second variable $y$ also varies between $-\pi$ and $\pi$.
I have two models for how $x$ should be distributed.  One model ignores $y$; the other does not.  I want to see which fits my data better.
To make this concrete, let's say that Model 1 is a von Mises PDF.  For each $x$ value, it returns a probability density.  Its shape is determined by a single parameter $\kappa$, the concentration.  So we have Model 1 = $VM(x; \kappa)$.
Let's say Model 2 has its basis in the von Mises PDF for $x$, but is more complicated.  For each $x$, the mean of the von Mises is biased in the direction of the corresponding $y$. The conditional density of $x$ given $y$ here is $VM(x|y; g(y), \kappa)$, where $\mu=g(y)$ is the mean of the von Mises on each trial. ($g$ is some function of y with some number of parameters -- say two parameters $h$ and $w$.)
Finally, say I want to compare the models using the AIC.  Maximizing the likelihood of Model 1 is straightforward. I simply find the $\kappa$ that maximizes the summed probabilities of my $x$ values. For Model 2, it's not as clear to me what my likelihood function should be.
My question is, what should my likelihood function be for Model 2?  It's clear I need to find the $\kappa$, $h$, and $w$ that maximize the likelihood, but . . . is the likelihood function the joint density for all $x$ and $y$?  The marginal density of $x$?  Something else?  Why?  Which model parameters would I count in the computation of the AIC? Is it important to note that my data sample the space of all possible $x$ and $y$ pairs less thoroughly than the data sample the space of all possible $x$ values (ignoring $y$)?
 A: In attempting this comparison, the marginal density for $x$ would be the likelihood function to use for Model 2.  Like the likelihood function for Model 1, it computes a single probability for each $x$ value.  However, for Model 2, the marginal density is
\begin{align}
p(x)=\frac{1}{2\pi}\int^\pi_{-\pi}VM\bigg(x|y; g(y; h, w), \kappa\bigg)dy,
\end{align}
assuming $y$ is drawn from a uniform distribution.  This is a bit tricky, if not impossible, to compute analytically.
What you can do is approximate it numerically using one of the methods described in this answer.
Another option is to fix $y$ in your experimental design at a value expected to reveal differences between Model 1 and Model 2.  (This is possible only if $y$ is an independent variable.)  In such a design, the differences between the models would reduce to the absence/presence of a constant bias in the mean of the von Mises.
As a final note, characterizing $y$'s influence on $x$ doesn't require fitting a PDF to observations.  This influence could be characterized in many different ways, with other sorts of fits to other representations of the data.
