Implement MMAP

[GiggleLiu]

It requires a mixed use of floating point numbers and Tropical numbers:

MMAP : computing the most likely assignment to the query variables, $X_M \subset X'$ after marginalizing out the remaining variables $X_S = X' \setminus X_M$.

$$\begin{align} MMAP(X_i|E=e) = \arg \max_{X_M} \sum_{X_S} \prod_{F} f(x_M, x_S, e) \end{align}$$

[GiggleLiu]

Proposal

We first marginalize $X_S$ with real numbers, the contraction result is a set of tensors. Then we contract the remaining tensor network with tropical algebra.

1. find clusters connected by hyper-edges in $X_S$,
2. optimize the contraction order for each cluster,
3. contract each cluster with real numbers,
4. optimize the clusters contraction order,
5. contract the clusters with logged tropical numbers.