The Viterbi Algorithm can be used to calculate the most likely path, based on observations in a Hidden Markov Model.
Using the same notations as Wikipedia, "each element
T1[i, j] of
T1 stores the probability of the most likely path so far
X = (x , x , ... , x[j]) with
x[j] = s[i] that generates
Y = (y , y , ... , y[j]).
Now suppose, that the initial distribution $\pi$ is such that for one state the probability is
1 (let's call it state
K), and for all other states it is
0. Also, let's say that in the emission matrix
B all elements are strongly between 0 and 1 (i.e.
0 < B[i, j] < 1 for all
In this way
0 < T[K,1] < 1 and for all other states
T[i,1] = 1.
For me, the part
T[K,1] < 1 feels intuitively wrong. How can it be, that the first state is
K, with probability
1, still the most likely path contains it with probability less than
I understand that the reason must be something like: "There are more different paths. One where the first state is
K, and the first observation is
y1. One, where the first state is
K, and the observation is
However, it is also true, that "no matter what the observation is, the first state is
K with probability
Question: What is the explanation for this (seeming) contradiction? (The best would be if it is "intuitive".)