4
J. F. Reis et al.
object maps time to probability of functioning and provides an estimation of the
device reliability in time.
Moreover, Bayesian techniques can be employed to infer quantities that cannot
be directly measured. For instance, a broker working at the Wall Street stock market
may be interested in pricing the assets of a company before deciding to buy/sell its
shares, but there are no exact algorithms that allow to estimate the price of shares of a
certain company. In order to achieve this task, one has to collect as much information
as possible, and, most often, one has to come up with assumptions, as companies
do not release information that are key to their business (such as industrial goals,
market strategies, etc.).
Therefore, the price of shares cannot be measured, but it can only be inferred by
gathering all the available information, including personal assumptions. Bayesian
techniques, such as Bayesian networks [1], Bayesian inference methods and so on,
may then be exploited in asset pricing problems, helping investors in the endeavour
of increasing their capital.
Bayesian techniques also find a very interesting application in medicine. Indeed,
very often doctors need to visualise the interior of a patient’s body system in order
to identify the disease and prescribe the most appropriate therapy. Of course doctors
also aim at stressing the patient as little as possible, so they need to exploit the less
intrusive techniques.
Computed tomography (CT), ultrasound, electrocardiogram (ECG) and Magnetic Resonance Imaging (MRI) are just a few examples of how information about
the internal state of the human body is investigated. All these techniques rely on
measurements, electric signals, magnetic-field variations, etc., which are not the
QoI doctors are looking for. Through Bayesian inference techniques, it is possible
to reconstruct images, even three-dimensional models, of an organ and thus provides
information that helps identify the disease.
The examples reported in this section represent of course a small insight of a way
broader world. Uncertainty quantification techniques may find their application in
industrial processes, astronomy, physics, game theory, control theory, sociology and
many others. In the rest of this chapter, we will try to refer to practical examples
whenever it is possible.
1.2 Illustrative Problem
Throughout the chapter, we will make an extensive use of the heat diffusion problem
to illustrate the advantages and the drawback of UQ techniques. This is intended to
guide the reader in the journey of understanding the potential of UQ techniques.
The diffusion of heat through a continuum is of the utmost interest in many
practical applications. The phenomenon has been deeply investigated since long
time ago. A general study of the heat equation with applications can be found in
[2, 3].
J. F. Reis et al.
object maps time to probability of functioning and provides an estimation of the
device reliability in time.
Moreover, Bayesian techniques can be employed to infer quantities that cannot
be directly measured. For instance, a broker working at the Wall Street stock market
may be interested in pricing the assets of a company before deciding to buy/sell its
shares, but there are no exact algorithms that allow to estimate the price of shares of a
certain company. In order to achieve this task, one has to collect as much information
as possible, and, most often, one has to come up with assumptions, as companies
do not release information that are key to their business (such as industrial goals,
market strategies, etc.).
Therefore, the price of shares cannot be measured, but it can only be inferred by
gathering all the available information, including personal assumptions. Bayesian
techniques, such as Bayesian networks [1], Bayesian inference methods and so on,
may then be exploited in asset pricing problems, helping investors in the endeavour
of increasing their capital.
Bayesian techniques also find a very interesting application in medicine. Indeed,
very often doctors need to visualise the interior of a patient’s body system in order
to identify the disease and prescribe the most appropriate therapy. Of course doctors
also aim at stressing the patient as little as possible, so they need to exploit the less
intrusive techniques.
Computed tomography (CT), ultrasound, electrocardiogram (ECG) and Magnetic Resonance Imaging (MRI) are just a few examples of how information about
the internal state of the human body is investigated. All these techniques rely on
measurements, electric signals, magnetic-field variations, etc., which are not the
QoI doctors are looking for. Through Bayesian inference techniques, it is possible
to reconstruct images, even three-dimensional models, of an organ and thus provides
information that helps identify the disease.
The examples reported in this section represent of course a small insight of a way
broader world. Uncertainty quantification techniques may find their application in
industrial processes, astronomy, physics, game theory, control theory, sociology and
many others. In the rest of this chapter, we will try to refer to practical examples
whenever it is possible.
1.2 Illustrative Problem
Throughout the chapter, we will make an extensive use of the heat diffusion problem
to illustrate the advantages and the drawback of UQ techniques. This is intended to
guide the reader in the journey of understanding the potential of UQ techniques.
The diffusion of heat through a continuum is of the utmost interest in many
practical applications. The phenomenon has been deeply investigated since long
time ago. A general study of the heat equation with applications can be found in
[2, 3].