Lazy specification of transition matrix

[gdalle]

It would be cool to allow users to overload mul! for the transition matrix, especially when they have structured transitions that cannot be easily represented in a single matrix object.

Examples:

• discretization of markov process (matrix exponential)
• Link with HiddenMarkovModels.jl? MurrellGroup/CHMMera.jl#3 (diagonal + rank 1)

[murrellb]

FYI the way we handle this in MolecularEvolution.jl, where a phylogeny can be thought of as a branching HMM (instead of a lattice), is to have forward and backward functions called by the likelihood algorithms. For a simple transition probability matrix these just wrap mul!, but you can define any behavior you want (we mostly use various CTMCs). We also do something similar for more general state spaces, including continuous ones.

[gdalle]

Thanks! That's what I ended up doing in #121 but how do you handle Viterbi? Here the operation is a max-plus matrix multiplication (in log scale), not an ordinary one.

[murrellb]

We don't work in the log domain.