Probability of an observation message

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?

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Since X is discrete and your Y is a Gaussian mixture, the most simple thing to assume is that $$y_t | x_t = a \sim N(\mu_a, \sigma_a^2).$$
Now, given a sequence of observations and hidden states $(x_t, y_t)_{t=1}^T$, you can estimate $P(y_t|x_t = a)$ using the maximum likelihood principle. That is, take all times t such that $x_t = a$, ${\cal T}_a = \{t : x_t = a\}$, and estimate $(\mu_a, \sigma_a)$ from corresponding $\{y_t\}_{{\cal T}_a}$. Perform this for all possible values that X can take on.