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We can describe the backward computation <math>\mathbf{b_{k+1:t}}</math> that starts at the end of the sequence in a similar manner. Let the end of the sequence be described by the index k, starting at 0. Therefore running from k down to t=0 and calculating each backward probability can be described by the following formula:
<math>\mathbf{b_{k+1:t}} = \alpha(\mathbf{T}\mathbf{O_{k+1}}\mathbf{b_{k+
Note that we use the non-transposed matrix of <math>\mathbf{T}</math> and that the order of the terms has changed. Also note that as a final product we have not a usual matrix multiplication, but a point product. This operation multiplies each value in one matrix with the corresponding value of the other. Finally note that the description in [[#RusselNorvig03|Russel & Norvig 2003 pp. 550]] excludes the point product, thought the procedure is required earlier.
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