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Can someone explain why

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Is it because for the left argument, we would find the most likely sequence of states given observations and for the right argument, we would eventually find the most likely sequence of states amongst all sequences of states and observations. And because of this, we find the most likely sequence of states for both arguments so they are equal?

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    $\begingroup$ Please add some details about the meaning of the various symbols. $\endgroup$ – Dilip Sarwate Oct 8 '20 at 2:00
  • $\begingroup$ Viterbi Algorithm has been maturely used in digital communications for long, especially used for soft decoding of error-correction code, e.g. convolution code, trellis code, turbo code etc. You may easily find detailed explanation and examples in a typical textbook on digital communications or error-correction coding (or channel coding). $\endgroup$ – user295357 Oct 8 '20 at 15:51
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We can express that conditional probability as:

$$P(q|O,\lambda)=\frac{P(q,O|\lambda)}{P(O|\lambda)}$$

Since $P(O|\lambda)$ does not depend on $q$, the $\arg\max_q$ of this fraction will be the same as the $\arg\max_q$ of the numerator alone.

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