Transition Probabilities for each observation

[omerorsun]

I am trying to decompose the forward probabilities in each time period into their constituent terms: a_{t-1}(i)q_{ij}P(o_{t}|j)

in

[a_{t}(j)=\sum_{i=1}^{N}a_{t-1}(i)q_{ij}P(o_{t}|j)]

The main challenge is the calculation of P(o_{t}|j) in multivariate case. Sometimes the joint emission probability for each state is zero for a given observation. As a result, forward probabilities should all be NAs due to zero division. However, forward_backward gives me non-zero or non-NA estimate for these scenarios. What is the specific way seqHMM deals with this scenario so that I get an non-zero or non-NA estimate?

Thank you in advance for any direction you can give.

[helske]

I'm not sure I follow, can you show me an example? What do you mean by joint emission probability? All "individuals" have their own hidden state sequence and a_{t}(j) is computed separately for each (possibly multichannel) sequence, so there can't be a case where all P(o_{t}|j) with different j are zero (as they have to sum to one).

[omerorsun]

Let's say I have three manifest variables, X, Y, Z and observation with the values of 1, 0, 0 for these manifest variables. If some emission probabilities for these X, Y, and Z are P(1|State 1)=0, P(0|State 2)=0, P(0|State 3)=0, then the joint emission probability for State 1, State 2, State 3 will be 0 as follows:

P(1|State 1)P(0|State 1)P(0|State 1)=0

P(1|State 2)P(0|State 2)P(0|State 2)=0

P(1|State 3)P(0|State 3)P(0|State 3)=0

As a result, the forward probabilities for each observation will not sum to 1 as well.

[helske]

Thanks now I understand. Yes, that is possible if the emission matrix is poorly defined with respect to data or due to numerical issues during optimization, but I would think forward_backward gives error or warning in that case as well (just tested with hmm_biofam and it did)? If not, that sounds like a bug. Can you make minimal example where forward_backward behaves differently?

[omerorsun]

Thank you for your response. I will send you a minimal example from my dataset shortly.

My main motivation is to extract a_{t-1}(i)q_{ij}P(o_{t}|j) before summation. I will use them as observation-specific transition probabilities from one state to another. Is there a way to extract them separately before summation without the recalculation?

[helske]

Unfortunately, at the moment you can't extract those terms separately. It could be pretty straightforward to make a new function for that based on this: https://github.com/helske/seqHMM/blob/master/src/internalForward.cpp if speed of "manual" computation in R is an issue though. Unfortunately, I do not have time right now but pull requests are of course very welcome. Although I'm not sure how relevant those values are in general?

[omerorsun]

Thank you so much! I will experiment with the package. I have forked the package. Kind regards!

[omerorsun]

I am looking at https://github.com/helske/seqHMM/blob/master/src/internalForward.cpp
Line 13 and line 20 use %= operator rather than *= for the joint emission probabilities (P(o_{t}|j)). Is there a specific reason why this is the case given the equation for forward probabilities is a_{t-1}(i)q_{ij}P(o_{t}|j)

Thank you so much in advance!

[helske]

The % operator is an element-wise multiplication, so there I am just multiplying the vector of a_{t-1}(i)q_{ij}, j=1,m with each P(o_{t}|j).

[omerorsun]

Thank you so much for your quick response!