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( x )=ln P 1
1
q df X|− 1
dx
( x )
2 p 2 f X| 1 ( x )
P 1
d 2 H S
dx 2
p df X| 1
dx
( x )
.
2 P 1
(4.32)
This expression is quite intractable as it involves general class-conditional
PDFs. Important simplifications can be made by considering the case of mu-
tually symmetric class distributions [216], that is, where qf X|− 1 ( x ) has the
same shape of pf X| 1 (
x ).Wereadilyseethat P 1 ( x )= P 1 ( x ) for this type
of two-class problems. In particular, the Gaussian two-class problem with
equal standard deviations and priors is an example of such type of distribu-
tions. For mutually symmetric class distributions one has
q df X|− 1
dx
p df X| 1
dx
( x )=
( x );
(4.33)
hence,
2 p df X| 1
dx
.
p 2 f X| 1 ( x )
P 1
d 2 H S
dx 2
P 1
( x )=
( x )ln
2 P 1 +
(4.34)
1
Let
Q ( x )= df
dx
P
+ f 2
P
ln
,
(4.35)
1
2 P
P 1 ( x ).The
function Q ( x ) plays a key role in the analysis of the error entropy critical
points as carefully discussed in [216]. In fact, for increasingly distant classes
(increasing d with x →−∞
pf X| 1 ( x ) and P
where for notation simplicity we took f
) Q ( x ) can be shown to be negative and thus
error entropy has a minimum at x . On the other hand, if the classes get
closer (decreasing d with x
0) Q ( x ) may change sign and thus x turns
out to be a maximizer of error entropy. Such is the case when the mutually
symmetric class distributions are by themselves symmetric like in Gaussian-
classes problems.
Summarizing, for a large class of problems optimization of SEE performs
optimaly whether in the sense of minimization (MEE) for suciently sepa-
rated classes or maximization for close classes. Of course, it would also be
relevant to know when to choose between optimization strategy, that is, to
know what is the minimum-to-maximum turn-about distance between the
classes. Unfortunately, the answer has to be searched case by case for each
pair of mutually symmetric distributions, taking into account that for a scaled
version of X , Y = ΔX ,the ratio x between the Q ( x )=0solution and
the scale Δ , is a constant. In fact one has
 
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