I've come across this statement whilst reading about HMMs:

Given $\alpha_{n}(j) = P(Y_{0}^{n}, X_{n} = j)$ and $\beta_{n}(j) = P(Y_{n+1}^{N} | X_{n} = j)$, then

$$P(Y_{0}^{N}) = \sum_{j} \alpha_{n}(j)\beta_{n}(j)$$

This doesn't come with any explanation though.

My question is, why is this the case?

I know that $P(Y)=∑_{i=1}=α_{T}(i)$. So, I'm unsure how the product of both $\alpha$ and $\beta$ give the same result.



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