# Transition Probabilities in HMM

In the Wiki page on the Viterbi Algorithm (https://en.wikipedia.org/wiki/Viterbi_algorithm) there is an example of an HMM describing patients being in states "fever" or "healthy".

What I wish to understand is how I can calculate the transition probabilities, if I have the following info: Let's assume that the doctor has previously calculated the statistics for the time a patient is healthy, and has a fever.
He concluded that the annual number of days for being "healthy" and with "fever" are distributed normally, with $(\mu_h=350,\sigma_h=5)$ and $(\mu_f=15,\sigma_f=2)$ respectively.

Intuitively the transition probability from "healthy" to "healthy" is much higher than the transition probability from "fever" to "fever".
But how would I calculate the exact probabilities? (Assuming observations are done on a daily basis)

• If somebody is either healthy or has a fever, then the standard deviations should be the same – Henry Dec 12 '17 at 8:38

The Viterbi algorithm takes as input the transition probabilities; it doesn't estimate them. To use it to get an estimate for them, you could give an initial value for $A$, run the algorithm, look at the proportion of transitions, rerun the algorithm, etc. Repeating that until convergence gives you an iterative algorithm.
$$A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$$
By thinking about the proportion of transitions in the same way as in the iterative approach. You would need to find a way to sample $\{X_n\}_{n=1}^{365}$ such that $|X_n = \textrm{"fever"}| \sim \textrm{N}(15,2)$ and $|X_n = \textrm{"healthy"}| \sim \textrm{N}(350,5)$.
However, this is impossible for many reasons. First, the number of days is limited, so $|X_n = \textrm{"fever"}|=365 - |X_n = \textrm{"healthy"}|$. Second, the days are discrete, so using a normal will give you values that don't work with probability one. Third, it doesn't provide any dependency between how the "fever" days are distributed. Are the uniformly scattered, or clustered? I think you are trying to find the transition probabilities without providing any sort of temporal structure.