I am studying forward-backward algorithm following the wikipedia page. I have little background in statistics and have managed to understand (hopefully) most part of the algorithm. However the scaling factor introduced in the Forward probabilities section is confusing me.

It says from the second last equation that the "product of scaling factor from each timestamp is the total probability for observing the given events irrespective of the final states". However from my understanding that, in order to scale the state vector, the scaling factor should be a vector that has the same dimension as the state vector. And the value of each entry of the scaling factor should be the same and equal to the sum of all entries from the state vector. Thus the product of all scaling factor will result in a vector whose entries are all the same, and the total probability calculated this way is just meaningless. I must have misunderstood something during the process but I could not figure out myself. Please correct me and any help is appreciated.

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