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*Assume that we have the Hidden Markov Model (HMM). If each of the states can take on k different values and a total of m different observations are possible (across all states), how many parameters are required to fully define this HMM? Justify your answer.*

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__Question:__

### Assume that we have the Hidden Markov Model (HMM). If each of the states can take on k different values and a total of m different observations are possible (across all states), how many parameters are required to fully define this HMM? Justify your answer.

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__Answer:__

There are a total
of three probability distributions that define the HMM, the initial probability
distribution, the transition probability distribution, and the emission
probability distribution.

- There are a total
of
*k*states, so*k*parameters are required to define the initial**probability distribution**.

- For the
**transition distribution**, we can transition from any one of*k*states to any of the*k*states (including staying in the same state), so*k*parameters are required.^{2}

- We need a total of
*km*parameters for the**emission probability distribution**, since each of the*k*states can emit each of the*m*observations.

Thus, the total
number of parameters required are

*. Note that the number of parameters does not depend on the length of the HMMs.***k + k**^{2}+ km
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