Hidden Markov Model with Probabilistic Observations 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?

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