j.-a. goulet 214
models. For the example presented in ﬁgure 13.1, if we have a prior
knowledge for the joint distribution of model parameters
f
(
✓
), we
can propagate this prior knowledge through the model in order
to quantify the unce r tai nty associated with predi c ti on s. T hi s un -
certainty propagati on can be, for instance, performed using either
Monte Carlo sampling (see
§
6.5) or ﬁrst-order linearization (see
§
3.4.2). When the uncertainty in the prior knowledge
f
(
✓
) is weakly
informative, it may le ad t o a large variability in model predi ct i ons .
In that situation, it becomes interesting to employ empirical obser-
vations to reduce the unce r tai nty related to the prior knowledge of
model parameters. The key with model calibration is that model
parameters are typically not directly observable. For instance, in
ﬁgure 13.1 it is often not possible to directly measure the bound-
ary condition properties or to measure the cable internal tension.
These properties have t o be i n fe rr ed f r om the obse rved structural
responses
D
y
. Another key aspect is that we typically build physics-
based models
g
(
✓, x
) because they can predict quantities that
cannot be observed. For example, we may want to learn about
model parameters
✓
using observations of the st at i c re sponse of a
structure deﬁned by the covariates
x
and then employ the model
to predict the unobserved responses
g
(
✓, x
⇤
), deﬁned by other co-
variates
x
⇤
. The vector
x
⇤
may describe unobs er ved lo cat i on s or
quantities such as the d yn ami c behavior instead of the stat i c one
employed for parameter estimation. O n e last possible problem setup
is associated with the selection of model classes for describing a
phenomenon.
There are three main challenges associated with model calibra-
tion: observation errors, prediction errors, and model complexity.
The ﬁrst challenge aris es because the observations available
D
y
are in most cases contaminated by observation errors. The second
is due to the discrepancy between the prediction of hard-coded
physics-based models
g
(
✓, x
) and the real system responses; that
is, even w he n par amet e r values are known, the model remains an
approximation of the re ali ty. The third challenge is related to the
diﬃculties associated with choosing a model structure having the
right c om ple x i ty for the task at hand. In
§
13.1, we explore the
impact of these challenges on the least-squares model calibration
approach, which is stil l ex t en si vely used in practice. Then, the
subsequent sections explore how to address some of these chal-
lenges using a hierarchical Bayesian approach combining concepts
presented in chapters 6 and 8.