Page 57 - Spring 2015

examples
An example in acoustics/seismic research is the investiga- tion of earthquakes in California based on a limited num- ber of globally deployed seismic sensors. Must we wait for the ensemble of repeatable, independent devastating earth- quakes at the same location to occur in order to infer the location of their epicenter?
Figure 1. Room-acoustic energy decay process. Experimental data values collected in an enclosure are expressed as the sequence D = [d0, d1,..., dK-1] and are plotted on a logarithmic scale. The para- metric model function is plausibly described by an exponential decay function H(Θ) having three parameters Θ = [θ0, θ1, θ2].
To introduce Bayesian analysis in acoustic studies, let us ap-
ply it to a data analysis task common not only in acoustic
investigations but also in many scientific and engineering
fields. This example begins with a room-acoustic measure-
ment that records a dataset expressed as a sequence, D=
[d0,d1,...,dK-1]. The data comprise a finite number of obser-
vation points as shown in Figure 1. These data represent
a sound energy decay process in an enclosed space. Based
on visual inspection of Figure 1, an experienced architec-
tural acoustician will formulate the hypothesis that “the data
This model contains a set of parameters Θ=[θ0,θ1,θ2], in which θ0 is the noise floor, θ1 is the initial amplitude and θ2 the decay constant. The aim is to estimate this set of pa- rameters Θ so as to match the modeled curve (solid line) to the data points (black dots) as closely as possible. The data analysis task is to estimate the relevant parameters Θ encap- sulated in the model H(Θ), particularly θ2, from the experi- mental observation D. This is known as an inverse problem.
bayes’ Theorem
Bayes’ theorem was published posthumously in 1763 in
Philosophical Transactions of the Royal Society (Bayes, 1763)
through the effort of Richard Price, an amateur mathemati-
cian and a close friend of Reverend Thomas Bayes (1702-
1761), two years after Bayes died. While sorting through
probably represent an exponential decay.” This hypothesis
by p(Θ) will be updated by objective new information, p(D|Θ), about the probable cause of those items of data. The new in- formation expressed in D comes from experiments.
In the present room-acoustic example, the quantity p(Θ) is
a probability density function (PDF) that represents one’s
initial knowledge (degree of belief) about the parameter val-
ues before taking into account the experimental data D. It
is therefore known as the prior probability for Θ. The term
p(D|Θ) reads “the probability of the data given the param- (1)
Figure EQUATIONS
can be translated into an analytical function of time (t) as an
“amplitude multiplied by (an exponential decay) + noise” or the so-called model
H(Θ)=θ +θe−θ t 01
2
(1)
eter Θ” and represents the degree of belief that the measured data D would have been generated for a given value of Θ. It represents, after observing the data, the likelihood of obtain-
Bayes’ unpublished mathematical papers, Price recognized
Figure EQUATIONS
the importance of an essay by Bayes giving a solution to
an inverse probability problem “in moving mathematically
from observations of the natural world inversely back to its
ultimate cause” (McGrayne, 2011). The general mathemati-
−θ t 2
cal form oHf t(hΘe )th=eoθrem+iθs deue to Laplace (1812), who was 01
(
p(Θ|D)p(D)= p(D|Θ)p(Θ). (2) (
In the Equation 2, the solidus (vertical bar) inside, e.g.,
p(Θ|D), is a conditional symbol and it reads “probability of
Θ conditional on D" or, in short, “probability of Θ given
D." Bayes’ theorem represents the principle of inverse prob-
also the first to apply Bayes’ theorem to social sciences, as- tronomy, and earth science.
Nowadays, Bayes’ theorem is straightforwardly derived from widely accepted axioms of probability theory, particularly through the product rule.
posterior × evidence = prior × likelihood as recognized by R. Price, is that an initial belief expressed
ability (Jeffreys, 1965). The significance of Bayes’ theorem,
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