Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
(65) To apply the idea in (63), we will always be multiplying a 1 × n matrix times a square n × n matrix, to
get the new 1 × n probability distribution for the events of the n state Markov process.
(66) For example, suppose we have a coffee machine that (upon inserting money and pressing a button) will
do one of 3 things:
(q1) produce a cup of coffee,
(q2) return the money with no coffee,
(q3) keep the money and do nothing.
Furthermore, after an occurrence of (q2), following occurrences of (q2) or (q3) aremuchmorelikelythan
they were before. We could capture something like this situation with the following initial distribution
for q1,q2 and q3 respectively,
I = [0.7 0.2 0.1]
and if the transition matrix is:
⎡
0.7
⎢
T= ⎣0.1
0.2
0.7
⎤
0.1
⎥
0.2⎦
0 0 1
a. What is the probability of state sequence q1q2q1?
P(q1q2q1) = P(q1)P(q2|q1)p(q1|q2) = 0.7 · 0.2 · 0.1 = 0.014
b. What is the probability of the states ΩX at a particular time t?
At time 0 (maybe, right after servicing) the probabilities of the events in ΩX are given by I.
At time 1, the probabilities of the events in ΩX are given by
IT= 0.7 · 0.7 + 0.2 · 0.1 + 0.1 · 0
0.7 · 0.2 + 0.2 · 0.7 + 0.1 · 0
0.7 · 0.1 + 0.2 · 0.2 + 0.1 · 1
= 0.49 + 0.02
0.14 + 0.14
0.07 + 0.04 + .1
= 0.51 0.28 .21
At time 2, the probabilities of the events in ΩX are given by IT 2 .
At time t, the probabilities of the events in ΩX are given by IT t .
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