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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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marked as duplicate by Willie Wong Jun 13 '12 at 7:28

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up vote 1 down vote accepted

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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The initial state probability, mhh.. can I compute it by counting the occurence of $x_t = 1$ and $x_t = 0$? I have a list of the (actually latent) variables $X$ – Mahoni Jun 12 '12 at 22:57
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. – mkolar Jun 12 '12 at 23:29

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