Viterbi Algorithm: initial state with ONE probability
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05-11-2019 - |
Pregunta
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[1] , x[2] , ... , x[j])
with x[j] = s[i]
that generates Y = (y[1] , y[2] , ... , 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 i
and j
.
In this way 0 < T[K,1] < 1
and for all other states T[i,1] = 1
.
Problem:
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 1
?
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 y2
. Etc.".
However, it is also true, that "no matter what the observation is, the first state is K
with probability 1
".
Question: What is the explanation for this (seeming) contradiction? (The best would be if it is "intuitive".)
No hay solución correcta