I want to do inference in a Hidden Markov Modell (Gaussian Mixture), given observed continoues variables $Y$ and latent discrete variables $X$. For this I need to compute the probability of an observation message $\mu_{Y \rightarrow X}(x_t) =: \varrho_t)x_t = P(y_t|x_t)$.
But how can I obtain the probability $P(y_t|x_t)$ for some given time $t$? I only have the values of $Y$, the (actually hidden) corresponding values of $X$, and a transition probability table $P(x'|x)$ but no probabilities what so ever for $y_t$.
How can I approach this?