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In electrical engineering, computer science, statistical computing and bioinformatics, the Viterbi Path Counting algorithm is used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the Viterbi algorithm.

Explanation
The Viterbi Path Counting algorithm is suited to training on multiple training sequences. The algorithm is a particular case of a generalized expectation-maximization (GEM) algorithm. It can compute maximum likelihood estimates and posterior mode estimates for the parameters (transition and emission probabilities) of an HMM, when given only emissions as training data.

For a given cell $$ S_i $$ in the transition matrix, all paths to that cell are summed. There is a link (transition from that cell to a cell $$ S_j $$). The joint probability of $$ S_i $$, the link, and $$ S_j$$ can be calculated and normalized by the probability of the entire string. Call this $$ \chi $$.

Now, calculate the probability of all paths with all links emanating from $$ S_i $$. Normalize this by the probability of the entire string. Call this $$ \sigma $$.

Now divide $$ \chi $$ by $$ \sigma $$. This is dividing the expected transition from $$ S_i $$ to $$ S_j $$ by the expected transitions from $$ S_i $$. As the corpus grows, and particular transitions are reinforced, they will increase in value, reaching a local maximum. No way to ascertain a global maximum is known.