IEEE Geoscience and Remote Sensing Magazine - June 2023 - 74

where ()) !zx R denotes the kth class predictions of the
input sample .x) However, the classification predictions of
k
the softmax function in neural networks are usually poorly
calibrated due to overconfidence of the neural networks
[130], while the predictions cannot characterize the domain
shifts by discarding the original features of the neural
network [131]. As a result, the accuracy of uncertainty quantification
is influenced.
To overcome these challenges, several uncertainty
quantification approaches introduced prior networks to
parameterize the distribution over a simplex. For example,
Dirichlet prior networks are adopted widely to quantify
the uncertainty from a Dirichlet distribution with tractable
analytic properties [112], [113], [114]. The Dirichlet distribution
is a prior distribution over categories that represents
the density of the predicted probabilities. The Dirichlet
distribution-based methods directly analyze the logit magnitude
of the neural networks, quantifying the data uncertainty
with awareness of domain distributions in Dirichlet
distribution representations. For training of the Dirichlet
prior networks, model parameters are optimized by minimizing
the Kullback-Leibler divergence between the model
and Dirichlet distribution, focusing on the in- and out-ofdistribution
data, respectively [129].
Except for the previous network-based approaches, ensemble
methods can also approximate uncertainty by averaging
x
x
Deterministic NN
Deterministic NN
Deterministic NN
Deterministic NN
ξ∗
y1∗
y∗ = Mean [ξ∗]
σ∗ = S.D. [ξ∗]
(a)
x
BNN
BNN
BNN
BNN
x
y∗ = Mean [y1, y2, ..., yn]∗∗ ∗
(b)
y2∗∗
yn
σ∗ = S.D. [y1, y2, ..., yn]∗∗ ∗
y1∗
y2∗
y∗ = Mean [y1, y2, ..., yn]∗∗ ∗
(c)
x ,t
the first three methods deliver the prediction y)
or
y )n
)
yn∗
σ∗ = S.D. [y1, y2, ..., yn]∗∗ ∗
(d)
FIGURE 10. A visualization of uncertainty quantification methods. (a) Previous network-based methods. (b) Ensemble methods. (c) Monte
Carlo methods. (d) External methods. For an input sample
ty v) from the average of a series of model outputs (i.e., p)
the external methods directly output the results of prediction and uncertainty quantification. BNN: Bayesian neural network.
74
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE JUNE 2023
and the quantified uncertainand
their standard deviation (S.D.) results, respectively. On the contrary,
y∗
σ∗

IEEE Geoscience and Remote Sensing Magazine - June 2023

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