Sensibly combine pairwise distributions Let's say that I have $P(X, Y)$, $P(X, Z)$, and $P(Y, Z)$. Are there a set of reasonable assumptions I can make that will allow me to combine them into a $P(X, Y, Z)$?
A practical use-case might be that I have software that can fit a bivariate distribution but my data has more than two variables.
 A: 
Are there a set of reasonable assumptions I can make that will allow me to combine them into a $P(X,Y,Z)$?

Short answer: There are a few ways, but software should not be an excuse for using them. If your data has $N$ variables, you really should fit an $N$-dimensional distribution to your whole dataset instead of patching a bunch of pairwise densities, which is costly and error-prone.
Long answer: Under certain conditions, there are a few reasonable things to do.


*

*The Maximum Entropy Principle (MaxEnt) is a statistical/information-theoretical principle that basically says that in the face of uncertainty you should choose the most uninformative distribution. 
Following your question, let's say you can accurately estimate the marginal pairwise densities $P(X,Y)$, $P(Y,Z)$ and $P(X,Z)$. Then, your MaxEnt estimator $\hat{P}(X,Y,Z)$ for the full joint would be the distribution over $X$, $Y$ and $Z$ that matches all your pairwise marginals and has the highest possible entropy. You can formalise this as a constrained optimisation problem:
\begin{align}
& \operatorname{maximise} & {-}\sum_{x,y,z} \hat{P}(x,y,z) \log \hat{P}(x,y,z) \\ ~ \\
& \text{subject to} & \sum_x \hat{P}(x,y,z) = P(y,z) \\
&                   & \sum_y \hat{P}(x,y,z) = P(x,z) \\
&                   & \sum_z \hat{P}(x,y,z) = P(x,y) \\
\end{align}
Using MaxEnt would be a good idea if, for example, your data measurement process is fundamentally restricted to pairs of variables and you don't have access to full $(x,y,z)$ tuples.

*You can also make some assumptions on the shape of your joint distribution and impose some factorisation on it. For example, if you assume that the full joint distribution can be written as
$$P(X,Y,Z) = P(X,Y|Z) P(Z)$$
then you can get away with estimating those two densities, $P(X,Y|Z)$ and $P(Z)$ -- assuming you can fit a bivariate conditional density.
Conclusion: both using MaxEnt and factorising your joint distribution are reasonable things to do, but either of them is going to take significant effort and will likely result in suboptimal results compared to fitting $P(X,Y,Z)$ directly. If it's just a matter of software then get a better fitting programme.
