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Major mistake corrected for the backward message formula. But someone should recompute the final results in the example, as I don't have time to do it. |
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Now that we have defined the forward probabilities, we continue to compute the backward probabilities. Again the matrices appear in the order as in the backward formula above.
'''=> Someone should recompute the final results because the formula for the backward message was previously wrong and was corrected above and in the following equations, but
<math>
\mathbf{b_{5:5}} = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}\begin{pmatrix}0.9 & 0.0 \\ 0.0 & 0.2 \end{pmatrix}\begin{pmatrix}1.0000 & 1.0000 \end{pmatrix}=\begin{pmatrix}0.5984 & 0.0543\end{pmatrix}=\begin{pmatrix}0.9168 & 0.0831 \end{pmatrix}
</math>
<math>
\mathbf{b_{4:5}} = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}\begin{pmatrix}0.9 & 0.0 \\ 0.0 & 0.2 \end{pmatrix}\begin{pmatrix}0.9168 & 0.0831 \end{pmatrix}=\begin{pmatrix}0.7308 & 0.2691\end{pmatrix}=\begin{pmatrix}0.8593 & 0.1407 \end{pmatrix}
</math>
<math>
\mathbf{b_{3:5}} = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}\begin{pmatrix}0.1 & 0.0 \\ 0.0 & 0.8 \end{pmatrix}\begin{pmatrix}0.8593 & 0.1407 \end{pmatrix}=\begin{pmatrix}0.0178 & 0.0845\end{pmatrix}=\begin{pmatrix}0.1739 & 0.8260 \end{pmatrix}
</math>
<math>
\mathbf{b_{2:5}} = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}\begin{pmatrix}0.9 & 0.0 \\ 0.0 & 0.2 \end{pmatrix}\begin{pmatrix}0.1739 & 0.8260 \end{pmatrix}=\begin{pmatrix}0.1405 & 0.0189\end{pmatrix}=\begin{pmatrix}0.8814 & 0.1185 \end{pmatrix}
</math>
<math>
\mathbf{b_{1:5}} = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}\begin{pmatrix}0.9 & 0.0 \\ 0.0 & 0.2 \end{pmatrix}\begin{pmatrix}0.8814 & 0.1185 \end{pmatrix}=\begin{pmatrix}0.4600 & 0.0462\end{pmatrix}=\begin{pmatrix}0.9087 & 0.0912 \end{pmatrix}
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