IEEE Computational Intelligence Magazine - May 2022 - 46

BNNs constitute a promising paradigm allowing
the application of deep learning in areas where a
system is not allowed to fail to generalize without
emitting a warning.
algorithm allows training a much more complex model than
the distributions usually considered for variational inference.
VII. Performance Metrics of Bayesian Neural Networks
One big challenge with BNNs is how to evaluate their performance.
They do not directly output a point estimate prediction
yt
yx ,,
alternatively 60 , p@ or 611p ,s
=
s T
y! y
p
/ s $ tt @dd y
/ I6pp ^ h
tt @dd y
-+
-+
,
,
s T
y! y
For multiclass classifiers, the calibration curve can be independently
checked for each class against all the other classes. In this
case, the problem is reduced to a binary classifier.
Regression problems are slightly more complex since the
but a conditional probability distribution pD;^h from
which an optimal estimate yt can later be extracted. This means
that both the predictive performance, i.e., the ability of the
model to give correct answers, and the calibration, i.e., that the
network is neither overconfident nor underconfident about its
prediction, have to be assessed.
The predictive performance, sometimes called sharpness
in statistics, of a network can be assessed by treating the estimator
yt
type of data the network is meant to treat. Many different metrics,
e.g., mean square error (MSE), n
as the prediction. This procedure often depends on the
, distances and crossentropy,
are used in practice. Covering these metrics is out of
the scope of this tutorial. Instead, we refer the reader to [112]
for more details.
The standard method to assess the model calibration is a
calibration curve, also called a reliability diagram [32], [113]. It
is defined as a function :,
p 01 01 "
s 66
,
@@ that represents the
observed probability ,ps or empirical frequency, as a function of
the predicted probability ;pt
see Figure 12. If
p p,s
t
p p1s
t
, then the
model is overconfident. Otherwise, it is underconfident. A wellcalibrated
model should have
. Using this approach
requires to first choose a set of events E with different predicted
probabilities and then to measure the empirical frequency of
each event using a test set T.
For a binary classifier, the set of test events can be chosen as
the set of all sets of datapoints with predicted probabilities of
acceptance in interval
6 -+ @ for a chosen d , or
pp ,dd
network does not output a confidence level, as in a classifier,
but a distribution of possible outputs. The solution is to use an
intermediate statistic with a known probability distribution.
Assuming independence between the yt
for a sufficiently large
set of different randomly selected inputs x, one can assume that
the normalized sum of squared residuals (NSSR) follows a
Chi-square law:
NSSR yy yyy
=- R - s + |^^t
t
shh h
R
-1
t
2
Dim^y
.
(50)
This allows attributing to each data point in the test set T a
predicted probability that is the probability of observing a variance-normalized
distance between the prediction and the true
value equal to or lower than the measured NSSR. Formally, the
predicted probability is computed as:
t =
y
pX h
iiy NSSRh 6^ ,,ih T
2
where X2
Dim^ ^
yx !
(51)
Dim^h is the Chi-square cumulative distribution, with
Dim y^h degrees of freedom. The observed probability can be
computed as:
s
pi
=-3
=
|| 1
T j
I
1 / ^h[, )
tt
||
T
0 pp .
ji
(52)
We present in the Supplementary Material a practical computation
of such calibration curve for the sparse measure practical
example (Practical example II).
Giving the whole calibration curve for a given stochastic
model allows observing where the model is likely to be overconfident
or underconfident. It also allows, to a certain extent, to
recalibrate the model [113]. However, providing a summary measure
to ease comparison or interpretation might also be necessary.
The area under the curve (AUC) is a standard metric of the form:
AUC = # s tpd .p
1
An AUC of 0.5 indicates that the model is, on average, well
calibrated.
The distance from the actual calibration curve to the ideal
Predicted Probability
(a)
Predicted Probability
(b)
FIGURE 12 Examples of calibration curves for underconfident (a) and
overconfident (b) models.
46 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2022
calibration curve is also a good indicator for the calibration
of a model:
dppp pdp2ss#
1
=0
(,
)( ).tt t
(54)
(53)
t
@ for small datasets.
The empirical frequency is given by:
.
y I6pp ^ h
t
(49)
Observed Probability
Observed Probability
IEEE Computational Intelligence Magazine - May 2022

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - May 2022
