Page 56 - Spring 2015

Ning Xiang
Postal:
Graduate Program in Architectural Acoustics Rensselaer Polytechnic Institute 110 8th Street Troy, New York 12180 USA
Email:
xiangn@rpi.edu
Cameron Fackler
Postal:
3M Personal Safety Division E•A•RCAL Laboratory 7911 Zionsville Road Indianapolis, Indiana 46268-1650 USA
Email:
cameron.fackler@mmm.com
objective bayesian Analysis in Acoustics
Many scientists analyze the experimental data using well-understood models or hypotheses as a way of thinking like a Bayesian.
Introduction
Experimental investigations are vital for scientists and engineers to better under- stand underlying theories in acoustics. Scientific experiments are often subject to uncertainties and randomness. Probability theory is the appropriate mathematical language for quantification of uncertainties and randomness, and involves rules of calculation for the manipulation of probability values. This article introduces probability theory in application to recent research in acoustics, taking the Bayes- ian view of probability. Bayesian probability theory, centered upon Bayes’ theorem, includes all valid rules of statistics for relating and manipulating probabilities; in- terpreted as logic, it is a quantitative theory of inference (Jaynes, 2003).
Probability, Randomness
When considering the interpretation of probability, debate is now in its third cen- tury between the two main schools of statistical inference, the frequentist school and the Bayesian school. The frequentists consider probability to be a proportion in a large ensemble of repeatable observations. This interpretation has been domi- nant in statistics until recent decades. However, when expressing, for instance, the probability of Mars being lifeless (Jefferys and Berger 1992), the probability cannot be interpreted as frequencies of repeatable observations; there is only one Mars.
According to the Bayesian interpretation, probabilities represent quantitative measures of the state of knowledge or the degree of belief of an individual in the truth of a proposition (Caticha, 2012). In other words, the Bayesian view posits that a probability represents quantitatively how strongly we should believe that something is true. Probability is defined as a real valued quantity ranging between 0 and 1, and according to the Bayesian view, it represents the degree of appropri- ate belief, from total disbelief (0) to total certainty (1). Bayesian probability theory is not confined to applications involving large ensembles of repeatable events or random variables. Bayesian probability makes it possible to reason in a consistent and rational manner about single events. This interpretation is not in conflict with the conventional interpretation that probabilities are associated with randomness. Conceivably, an unknown influence may affect the process under investigation in unpredictable ways. When considering such an influence as “random,” the ran- domness is expressed through the value of a quantity being unpredictable, un- certain, or unknown. With the Bayesian interpretation, probability can be used in wider applications, including but not restricted to situations in which the fre- quentist interpretation is applicable (Caticha, 2012). The methods of probability theory constitute consistent reasoning in scenarios where insufficient information for certainty is available. Thus, probability is the tool for dealing with uncertainty, lack of scientific data, and ignorance.
54 | Acoustics Today | Spring 2015 , volume 11, issue 2 ©2015 Acoustical Society of America. All rights reserved.