HMMSP rejection upper bound

[axch]

Can we do better than 0?

The definition is the maximum over all possible initial vectors and transmission and emission matrices of the marginal probability of the observations, integrating out the state sequence. That may be computable directly; an upper bound would be the likelihood of the maximum likelihood state sequence, which might be easier to compute.

[lenaqr]

Seems like if the dimensionality of the hidden state space isn't fixed, the log-likelihood can be arbitrary close to 0.

[axch]

FWIW, holding any one of the arguments fixed suffices to determine the dimension, so perhaps an improvement over 0 is possible in that case. This example also suggests that there is value in being able to separate, e.g., the dimension from the actual values in the matrices, so that a client could reject on the latter while holding the former constant and gain the benefit (potentially) of a tighter bound.