Hidden Markov Model with Probabilistic Observations

Question

I have an HMM with $N$ states and $T$ possible obsevations where $A \in \mathbb{R}^{N \times N}$ is transition probability matrix and $B \in \mathbb{R}^{N \times T}$ is emission probability matrix. I follow the introductory text https://web.stanford.edu/~jurafsky/slp3/A.pdf and understand the concepts and formulas.

However, in my case observations are also probabilistic. It means instead of observing only one state, I have a probability vector of observed states at each discrete time. I can modify the likelihood and decoding algorithms by summing over the observations with probabilities as weights. So, if the probability of being in state $j$ after the first $t$ observations for a particular observation $o_i$ at time $t$ is $\alpha_t(j)$, then for a vector of observation probabilities $o_t$ I can write

$$ \bar{\alpha}_t(j) := \sum_{i=1}^T o_{ti} \alpha_t(j) $$

Hopefully that makes sense, because I'm quite new to the topic. However, I'm not sure how to do the training with this setting. Is there a name for this kind of problem? Can you suggest me a reference if there is? Alternatively, is there an easy way to modify the training formulation to be used in this problem?

Answer 1

I'm pretty sure you can just update the emission probability matrix to take the probablistic observations into account.

There are a lot of ways we could add probablistic "measurement error" to the observations. Let's say that $\gamma$ is the probability that the observation accurately reflects the emission, and $1 - \gamma$ the probability that the observation is a random sample from the $T$ possible observations, and $T=3$ in this example.

$B_i = [b_i^1, b_i^2, b_i^3]$ is the vector of emission probabilities from state $i$ before measurement error, and so are also the observations probabilities in cases where no measurement error occurs), and $E = [\frac{1}{3}, \frac{1}{3}, \frac{1}{3}]$ are the observation probabilities on cases where measurement error does occur. Since the probability of measurement error occurring is $1 - \gamma$, the marginal observation probabilities are therefore

$$ \hat B_i = \gamma B_i + (1-\gamma)E \\ = [ \gamma b_i^1 + (1-\gamma)\frac{1}{3}, \ \gamma b_i^2 + (1-\gamma)\frac{1}{3}, \ \gamma b_i^3 + (1-\gamma)\frac{1}{3} ] \\ = [ \gamma b_i^1 + \frac{(1-\gamma)}{3}, \ \gamma b_i^2 + \frac{(1-\gamma)}{3}, \ \gamma b_i^3 + \frac{(1-\gamma)}{3} ] $$