Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

STATISTICS


Thus, a Bayesian statistician considers the ML estimatesaˆMLof the parameters

to be the values that maximise the posteriorP(a|x,H) under the assumption of


a uniform prior. More importantly, however, a Bayesian wouldnotcalculate the


standard error or confidence interval on this estimate using the (classical) method


employed in subsection 31.3.4. Instead, a far more straightforward approach is


adopted. Let us assume, for the moment, that one is estimating just a single


parametera. Using (31.83), we may determine the valuesa−anda+such that


Pr(a<a−|x,H)=

∫a−

−∞

L(x;a)da=α,

Pr(a>a+|x,H)=

∫∞

a+

L(x;a)da=β.

where it is assumed that the likelihood has been normalised in such a way that

L(x;a)da= 1. Combining these equations gives


Pr(a−≤a<a+|x,H)=

∫a+

a−

L(x;a)da=1−α−β, (31.84)

and [a−,a+]istheBayesian confidence intervalon the value ofaat the confidence


level 1−α−β. As in the case of classical confidence intervals, one often quotes


the central confidence interval, for whichα=β. Another common choice (where


possible) is to use the two valuesa−anda+satisfying (31.84), for whichL(x;a−)=


L(x;a+).


It should be understood that a frequentist would consider the Bayesian confi-

dence interval as anapproximationto the (classical) confidence interval discussed


in subsection 31.3.4. Conversely, a Bayesian would consider the confidence inter-


val defined in (31.84) to be the more meaningful. In fact, the difference between


the Bayesian and classical confidence intervals is rather subtle. The classical con-


fidence interval is defined in such a way that if one took a large number of


samples each of sizeNand constructed the confidence interval in each case then


the proportion of cases in which the true value ofawould be contained within the


interval is 1−α−β. For the Bayesian confidence interval, one does not rely on the


frequentist concept of a large number of repeated samples. Instead, its meaning is


that, given the single samplex(and our hypothesisHfor the functional form of


the population), the probability thatalies within the interval [a−,a+]is1−α−β.


By adopting the Bayesian viewpoint, the likelihood functionL(x;a) may also

be used to obtain an approximationσˆaˆto the standard error in the ML estimator;


the approximation is given by


σˆaˆ=

(

∂^2 lnL
∂a^2





a=aˆ

)− 1 / 2

. (31.85)


Clearly, ifL(x;a) were a Gaussian centred ona=aˆthenσˆˆawould be its standard


deviation. Indeed, in this case, the resulting ‘one-sigma’ limits would constitute a

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