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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 $P(x'|x)$.

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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 $P(x_0)$).

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    If you have multiple sequences you could count how many times does 0 or 1 occur at the beginning. Otherwise just assume that the initial state probability for 0 and 1 is equal.2012-06-12