Biology Reference
In-Depth Information
Exercise 9.11.
Continue the application of the forward algorithm from Example
9.8
to carry out the termination step and compute
P
(
x
)
. Verify that this termination step
produces the same value for
P
(
x
)
as the value computed by the forward algorithm in
Example
9.7
.
The posterior probabilities computed here can be used to complement the results
from the Viterbi decoding or as a possible alternative decoding method referred to as
posterior decoding
. They could prove useful when there are multiple sequences with
probabilities close to the maximal probability sequence(s) generated by the process of
Viterbi decoding, in which case it may not be justified to only consider the sequence(s)
of maximal probability. The posterior probabilities give the likelihood (based on the
entire observed sequence
x
that the symbol
x
t
in position
t
has been
emitted by the hidden state
k
. Posterior decoding can be quite useful for decoding a
hidden process with two states (or two groups of states), which is exactly the case we
are concerned with in this chapter. To see this, in the case of the Dishonest Casino
example, given the sequence
x
, we plot the probabilities
P
=
x
1
x
2
···
x
l
)
(π
t
=
U
|
x
)
for each
t
l
. The “hills” in the resulting graph would indicate segments in the
sequence
x
that are likely to be emitted from the
U
state. The remaining segments are
more likely to have been emitted by the
F
state. Analytically, the decoded sequence
will be
=
0
,
1
,...,
∼
π
t
=
argmax
k
P
(π
t
=
k
|
x
).
(9.7)
Figure
9.8
exemplifies this approach for a simulated sequence of 500 symbols. The
gray areas highlight the time steps at which the unfair die was used for the simulations.
FIGURE 9.8
A plot of the posterior probabilities of being in a state generated by the unfair die for a
sequence of 500 simulated runs. The gray highlight identifies the runs obtained from the
simulation by using the unfair die. The transition probabilities of the HMM are
a
UU
=
0
96. The default values for the emission
probabilities (chosen to match those from Example 9.4) were used.
.
95
;a
UF
=
0
.
05
;a
FU
=
0
.
04,and
a
UU
=
0
.
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