Possible Duplicate:
Probability of an observation message
In a given Gaussian mixture model with observed continues variables $Y$ and latent discrete variables $X$ I want to apply the forward-backward algorithm in order to compute the marginal posteriors $P(x_t|y_{1:T})$.
Since this is computed as $$\frac{\alpha_t(x_t) \beta_t(x_t)}{P(Y)}$$
I was wondering how do I obtain the value of $P(Y)$? The only probabilities I have given is a transition probability $P(x'|x)$.
If you observe the sequence of latent variables $\{x_t\}_{t=1}^T$, then you can estimate $P(y_t|x_t)$. Computing $P(Y)$ after that is straight forward (assuming that you know the initial state probability $P(x_0)$).