I am interested in modelling human activities using sensor data with HMMs and would like to incorporate prior knowledge during inference. The normal procedure is to model K different activities with K separate HMMs. To test an unknown sequence, compute its likelihood from each of the HMMs and the HMM with maximum value is assigned as the class label. This is all done under the assumption that priors over HMM are uniform.
A problem can arise when one of the class is rare or unusual and its prior probability may be very low in comparison to other classes and therefore the uniform priors may not be a good assumption. Therefore, I am interested in posterior probability and not just the likelihood to capture the combined effect. My observations are continuous (features extracted from sensor) and not discrete values. My questions are:
Can inference be done using a bayesian network type approach that include multiplication of prior with likelihood?
In my case the prior will be the count of activities available per HMM. Can that be estimated using a dirichlet prior to avoid zero-count problem for rare class (assuming I approximate an HMM for a rare class). Does that make sense?
The multivariate observation data is approximated using single gaussian (and not mixtures), in that case likelihood will be gaussian?, can it be mixed with dirichlet prior to compute posterior probability? or the likelihood is still multinomial as it represents K different outcomes from K different HMMs?
Sorry if I have mixed with some of the basic concepts, I am new and I seek guidance to move further.