MACHINE LEARNING TECHNIQUES - LASA

168
To do this, one proceeds iteratively and tries to find the optimal state at each time step. Such an
iterative procedure is advantageous in that, if one is provided with part of the observations as in
the above weather prediction example, one can use the observations { } 1 ,..., t
first t time steps to guide the inference.
o o made over the
The optimal state at each time step is obtained by combining inferences made with the forward
and backward procedures and is given by
γ ( j)
=
t
α
N
∑
i=
1
The most likely sequence is then obtained by computing:
j
( t) β ( t)
α
i
j
( t) β ( t)
( ) arg max( γ () i)
qt
1≤≤
i N
t
i
, see also Equation(7.4).
= (7.13)
The state sequence maximizing the probability of a path which accounts for the first t
observations and ends in state j is given by:
δ ( j) = max p( q... q , q = j, o... o)
(7.14)
t 1 1 1
1...
t −
q q
t t
t−1
Computing the above quantity requires taking into account the emission probabilities and the
transition probabilites. Again one proceeds iteratively through induction. This forms the core of
the Viterbi algorithm and is summarized in the table below:
Hence, when inferring the weather over the next five days, given information on the weather for
the last ten days, one would first compute the first 10 states sequence q1,.....
q using (7.14) and
10
then one would use (7.13) to infer the next five states q 11
,..... q 15
. Given q 11
,..... q 15
, one would
then draw from the associated emission probabilities to predict the particular weather (i.e. the
particular observation one should make) for the next 15 time slots.
:
© A.G.Billard 2004 – Last Update March 2011