How to estimate discrete probability distribution from a dataset of pairwise frequencies? My problem is best explained with an example..
Suppose every day I count the cars driving down my street, but because I'm a slow counter, I only count two different types of cars per day. On the first day, I count 30 trucks and 65 cars. On the second day, I count 25 trucks and 12 motorcycles. On the third day I count 19 motorcycles and 72 cars. On the fourth day I count 11 vans and 14 motorcycles. And on the fifth day I count 12 vans and 44 trucks.
While the total volume of traffic changes day to day, I expect the relative proportion of vehicle types to remain near constant.
How do I use this information to develop a discrete probability distribution for the vehicles on my road? Put another way, how can I determine the probability that the next car I observe will be type X?
I suspect there's a nice conditional probability tree solution to this, but I can't quite put my finger on it.
UPDATE
Upon more thought, I think this may be a good candidate for maximum likelihood estimation. If I assume $P(\text{car})= p_1$, $P(\text{truck})= p_2$, $P(\text{motorcycle})= p_3$, and $P(\text{van})= p_4$, then I can calculate the likelihood of my observed pairs of data given some values for $p_1$, $p_2$, $p_3$, and $p_4$.  Now I just need to define and optimize my likelihood function.. Hmm..

Since you have a good idea, let's formalize it.
Begin with notation.  We need to suppose you know all the possible types of vehicle.  Let's index them with whole numbers $1,2,\ldots,k.$  You assume a constant chance $p_i$ that vehicle $i$ will appear at any time, so (obviously) these chances are non-negative and sum to unity.
When on a particular day $t$ you count only vehicles of types $i$ and $j\ne i,$ let $x_{ti}$ and $x_{tj}$ be those counts.  I will assume you decide, independently of any of the preceding data or other related information, which vehicles to count.  (Thus, for instance, you don't just count the first two types of vehicles you encounter: that would provide more information.)
If the appearances of the vehicles are independent, then the chance of observing $x_{ti}$ and $x_{tj}$ must be proportional to the powers of the probabilities,
$$\Pr(x_{ti}, x_{tj}) \,\propto\, (1-p_i-p_j)^{-(x_{ti}+x_{tj})}\,p_i^{x_{ti}}\,p_j^{x_{tj}}.$$
Proceeding from day to day, again assuming independence, these probabilities continue to multiply.  Thus, for each pair $\{i,j\},$ we may once and for all collect our counts $x_{ti}$ and $x_{tj}$ over all the days $t$ when we observed both these vehicle types.  Let $X_{ij}$ be the sum of all such $x_{tj}$ (and, therefore, $X_{ji}$ is the sum of all such $x_{ti}$).  Set $X_{ii}=0$ for all $i.$
For example, indexing Truck, Car, Motorcycle, and Van with the numbers $1$ through $4,$ respectively, the data in the question yield the matrix
$$X = \begin{array}{l|rrrr}
& \text{Truck} & \text{Car} & \text{Cycle} & \text{Van} \\
\hline\text{Truck} & 0 & 65 & 12 & 12 \\
\text{Car}   & 30 & 0 & 19 & 0 \\
\text{Cycle} & 25 & 72 & 0 & 11 \\
\text{Van}   & 44 & 0 & 14 & 0
\end{array}$$
For instance, on days when you were observing trucks ($1$) and cars ($2$), you saw a total of $X_{12}=65$ cars and $X_{21} = 30$ trucks.
Let $\mathbf{p} = (p_i)$ be the vector of probabilities and let $\mathbf{x} = (x_j) = \left(\sum_{i=1}^k X_{ij}\right)$ represent the total counts of each type of vehicle (the column sums of $X$).
Upon taking logarithms of the total probability, we obtain a log likelihood
$$\Lambda = C + 2 \mathbf{x}\cdot \log \mathbf{p} - \sum_{i\ne j} (X_{ij}+X_{ji})\log(p_i+p_j)$$
where $C$ is what became of all the implicit constants of proportionality.  Maximize this to obtain the maximum likelihood estimate of $\mathbf p.$  I obtain the value $\hat {\mathbb{p}} = (0.26, 0.53, 0.13, 0.081).$

Generally, with more than two types of vehicles, the equations for the critical points of the likelihood are polynomials of degree two or higher: you won't obtain closed formulas.  This requires numerical optimization.
